In order to simplify the discussion, we restrict ourself to the exponent $p = 2$ and to the exponential weight function $m = e^{a\langle x \rangle^s}$, $s \in (0, \gamma]$, $a > 0$. We thus work in the Banach lattice $X := L_m^2$. We observe that

$$ \frac{\nabla m}{m} = s a x \langle x \rangle^{s-2} \sim s a |x|^{s-1}, $$

$$ \frac{\Delta m}{m} = s a d \langle x \rangle^{s-2} + s(s-2) a |x|^2 \langle x \rangle^{s-4} + (s a)^2 |x|^2 \langle x \rangle^{2s-4} \sim (s a)^2 |x|^{2s-2}, $$

$$ \mathrm{div} b = d \langle x \rangle^{\gamma-2} + (\gamma-2) |x|^2 \langle x \rangle^{\gamma-4} \sim (d + \gamma - 2) |x|^{\gamma-2}, $$

$$ b \cdot \frac{\nabla m}{m} = s a x \langle x \rangle^{s-2} \cdot x \langle x \rangle^{\gamma-2} \sim s a |x|^{s+\gamma-2}, $$

so that the contribution of $(c - \mathrm{div}b/2)$ is always negligible at infinity, and we get

$$ (7.63) \qquad \varphi_i \sim (s a)^2 |x|^{2s-2} - s a |x|^{s+\gamma-2}. $$

We denote

$$ a' := s a > 0 \quad \text{if} \quad s \in (0, \gamma), $$

$$ a' := a\gamma - 2(a\gamma)^2 > 0 \quad \text{if} \quad s = \gamma \text{ and } a \in (0, 1/(\sqrt{2}\gamma)). $$

We then face to three cases :

(i) $\gamma > 1$: taking $s \in ((2 - \gamma)_+, \gamma)$, we have $\varphi_i \sim -a'|x|^{s+\gamma-2} \to -\infty$ with $s + \gamma - 2 > 0$;

(ii) $\gamma = 1$: taking $s = \gamma$, $a < 1/(\sqrt{2}\gamma)$, we have $\varphi_i \to -a'$;

(iii) $\gamma \in (0, 1)$: taking $s = \gamma$, $a < 1/(\sqrt{2}\gamma)$, we have $\varphi_i \sim -a'|x|^{2\gamma-2} \to 0$ with $2\gamma - 2 < 0$.

**Condition (H1).** In any of the above cases, we have from (7.59)

$$ ((\lambda - \mathcal{L})f, f) = \int |\nabla f|^2 m^2 + \int (\lambda - \varphi_1) f^2 m^2, $$

for $\lambda \in \mathbb{R}$, with $\inf(\lambda - \varphi_1) > 0$ for $\lambda \ge \kappa_1$ and $\kappa_1 > 0$ large enough. We deduce that $\lambda - \mathcal{L}$ is coercive for $\lambda \ge \kappa_1$. With the same arguments as in section 7.1, we conclude that $\mathcal{L}$ is the generator in $L_m^2$ of a positive semigroup $S_{\mathcal{L}}$, so that (H1) holds.

**Condition (H2).** When $\gamma > 1$, the same arguments as in Section 7.2 imply that condition (H2) holds for some $\kappa_0 \in \mathbb{R}$. When $\gamma \in (0, 1]$, we have $\mathcal{L}^*1 = c - \mathrm{div}b \ge 0$ from (7.58) and (H2) holds with $\kappa_0 = 0$.

**Conditions (H4) and (H5).** The strong maximum principle holds here because for instance we may apply the same barrier function argument as presented in Section 7.2. The reverse Kato's inequality condition is proved by using local arguments, so that it holds for the same reasons as in the previous section.

**Condition (H3).** We define the multiplication operator $\mathcal{A}$ and the elliptic operator $\mathcal{B}$ by

$$ \mathcal{A} := M\chi_R, \quad \mathcal{B} := \mathcal{L} - \mathcal{A}, $$

for $M, R > 0$ and $\chi_R(x) := \chi(x/R)$ with $\chi \in \mathcal{D}(\mathbb{R}^d)$, $\mathbf{1}_{B_1} \le \chi \le \mathbf{1}_{B_2}$. We fix $\kappa_B < \kappa_0$ in case (i), $\kappa_B := -a'/4$ in case (ii) and $\kappa_B := 0$ in case (iii), and we set $a'' := a'/2$. Choosing $M, R > 0$ large enough, from Lemma 7.7 and the discussion which follows, we deduce that

$$ (7.64) \qquad ((\mathcal{B} - \alpha)f, f) \le - \int |\nabla f|^2 m^2 - a'' \int f^2 (\mathbf{1}_{B_1} + \mathbf{1}_{B_1^c} |x|^{s+\gamma-2}) m^2, $$

for any $\alpha \ge \kappa_B$ and any nice function $f$. We classically deduce that $\alpha - \mathcal{B}$ is coercive and thus invertible. We discuss the three different cases.

- In the first case $\gamma > 1$, so that $s + \gamma - 2 > 0$, we see that the operator $\mathcal{R}_B(\alpha)$ is compact from Rellich-Kondrachov theorem, so that also $\mathcal{W}(\alpha) := \mathcal{R}_B(\alpha)\mathcal{A}$ for any $\alpha \ge \kappa_B$. We may thus apply Lemma 2.8-(2) and we deduce that (H3) holds for both the primal and the dual problems.

- In the case $\gamma = 1$, so that $2\gamma - 2 \le 0$, the operator $\mathcal{R}_B(\alpha)$ is not compact anymore. However, for any sequence $(f_n)$ which is bounded in $L_m^2$, we define the sequence $(g_n)$ by $g_n := \mathcal{A}f_n$, and $(g_n)$ is bounded in $L_m^2$ with $\tilde{m} := e^{\tilde{a}\langle x \rangle^\gamma}$, $\tilde{a} \in (a, 1/\sqrt{2\gamma})$. Using the dissipativity estimate (7.64) in
based on the Harnack inequality as presented in the previous section. Let us then consider $f \in D(\mathcal{L}^k) \cap X_+ \setminus \{0\}$ such that $(\lambda - \mathcal{L})f \ge 0$ with $k$ large enough ($k > d/2$ must be suitable) and $\lambda \ge \lambda_1$ large enough but fixed ($\lambda \ge \kappa_1$ is suitable). Using a very classically bootstrap argument based on iterated application of the Calderon-Zygmond elliptic regularity theorem and the Morrey estimate, we have $f \in C(\mathbb{R}^d)$. By assumption, there thus exist $x_0 \in \mathbb{R}^d$, and two constants $\tau, r > 0$ such that $f \ge \tau$ on $B(x_0, r)$ and we take choose $x_0 = 0$ in order to simplify the notations. We next fix $R > r$ and we observe that the function

$$g(x) := \tau^*(g_0(|x|) - g_0(R)), \quad g_0(s) := \exp(\sigma r^2/2 - \sigma s^2/2),$$

satisfies

$$ (\tau^*)^{-1}(\lambda - \mathcal{L})g = (\lambda - c)(g_0 - g_0(R)) + (d\sigma - \sigma b \cdot x - \sigma^2 r^2)g_0 \\ \le [2(|\lambda| + \|c\|_{L^\infty(B_R)}) + \sigma(d + \|b \cdot x\|_{L^\infty(B_R)} - \sigma^2 r^2)]g_0 \le 0 $$

on $\mathcal{O} := B(0, R) \setminus B(0, r)$ for $\sigma > 0$ large enough. We next fix $\tau^*$ such that $g = \tau$ on $\partial B(x_0, r)$. We also observe that from (7.52), $\lambda - \mathcal{L}$ is coercive on $\mathcal{O}$, in the sense that

$$ \forall h \in H_0^1(\mathcal{O}) \quad ((\lambda - \mathcal{L})h, h)_{L^2(\mathcal{O})} \ge \|h\|_{L^2(\mathcal{O})}. $$

In particular, $\lambda - \mathcal{L}$ satisfies the weak maximum principle as explained in the proof of (7.9). Arguing as in the proof of Lemma 4.11, we deduce that $f \ge g > 0$ on $\mathcal{O}$, what we also see directly by observing that $h := (g - f)_+ \in H_0^1(\mathcal{O})$, $(\lambda - \mathcal{L})h \le 0$ and using that the weak maximum principle implies $h \le 0$, thus $h \equiv 0$ and finally $f \ge g$. Because $R > r$ can be chosen arbitrarily large, we conclude with $f > 0$ on $\mathbb{R}^d$.

**Condition (H5).** The reverse Kato's inequality condition is proved by using local arguments, so that it holds for the same reasons as in the previous section. Similarly, because the argument are local, the conclusion of Lemma 7.1 holds here.

As a consequence, using Theorem 2.21, Theorem 4.13, Theorem 5.16 and Theorem 5.23, we may summarize our analysis in the following result.

**Theorem 7.6.** Consider the elliptic operator (7.50) in the whole space and assume that the coefficients satisfy (7.51). Then the conclusions (S1), (S2) and (S32) hold in $L^2$ as well as (E2) in $L_{\phi_1}^1$.

While the conclusion (S1)-(S2) is essentially due to Lions [251, 8th course], we believe that the ergodicity property (E2) is new. We do not present an exponential constructive estimate, which we believe is possible to prove, but would require significantly more development.

**7.3. Diffusion in $\mathbb{R}^d$ with weak potential confinement.** We consider in this section the same elliptic operator (7.50) with now a weak confinement condition assuming that $c$ converges to a constant. With no loss of generality, we may assume $c \to 0$. More precisely, we consider the elliptic operator

$$(7.54) \qquad \mathcal{L}f := \Delta f + b \cdot \nabla f + rcf,$$

with $c \in C_0(\mathbb{R}^d)$, $b \in C_0(\mathbb{R}^d)$ and $r \in \mathbb{R}_+$ a parameter. When not necessary in the discussion we will take $r = 1$. The associated first eigenvalue problem in such a situation has been studied in [251, 8th and 9th courses] to which we refer for more details. We define

$$ \lambda_1 = \lambda_1(r) := \inf\{\kappa \in \mathbb{R}; (\lambda - \mathcal{L})^{-1} \text{ well defined and positive for any } \lambda \ge \kappa\}. $$

Proceeding exactly as in the proof of (H1) in the preceding section, we see that the operator $\lambda - \mathcal{L}$ is invertible for any $\lambda > \|(rc + |b|^2/4)_+\|_{L^\infty}$, and then its inverse is positive. Because the proof of (H2) in the preceding section also applies here, we deduce that the infimum $\lambda_1$ of the set $\mathcal{I}$ of real resolvent values is well defined with $\lambda_1 \in (\kappa_0, \kappa_1)$, for some constructive constants $\kappa_i \in \mathbb{R}$.
We split now the discussion into two cases.

**Case 1.** We start considering the case $b = 0$. In that case, $\mathcal{L}$ is self-adjoint so that $\lambda_1$ is also characterized by

$$ \lambda_1 = \sup_{\|f\|_{L^2}=1} \mathcal{E}(f), $$
$L_m^2$, we see that $\mathcal{B} - \alpha$ is dissipative in $L_m^2$ for any $\alpha \ge \kappa_B$, and more precisely the sequence $(h_n)$ defined by $h_n := \mathcal{R}_B(\alpha)g_n$ satisfies

$$ \int |\nabla h_n|^2 m^2 + \tilde{a}'' \int h_n^2 (\mathbf{1}_{B_1} + \mathbf{1}_{B_1^c} |x|^{2\gamma-2}) \tilde{m}^2 \le \int g_n^2 \tilde{m}^2. $$

Using that $|x|^{2\gamma-2}\tilde{m}^2/m^2 \to \infty$ as $|x| \to \infty$, that implies that $(h_n)$ is relatively compact in $L_m^2$. More precisely, the above estimates show that $\mathcal{W}(\alpha) := \mathcal{R}_B(\alpha)\mathcal{A} : L_m^2 \to H_m^1 \cap L_{m^2}^2$ with $m^2 := m^{1/2}\tilde{m}^{1/2}$ and in particular we have established that $\mathcal{W}(\alpha) := \mathcal{R}_B(\alpha)\mathcal{A}$ is a compact operator in $L_m^2$ uniformly on $\alpha \ge \kappa_B$ because of the Rellich-Kondrachov theorem and the fact that $m = o(m^2)$. Since $\mathcal{R}_B(\alpha)$ is bounded in $\mathcal{B}(L_m^2)$ uniformly for any $\alpha > \kappa_B$, the operator $\mathcal{L}$ satisfies the splitting structure (HS1) and, applying Lemma 2.8-(2), we deduce that (H3) holds for both the primal and the dual problems.

At this stage, when $\gamma \ge 1$, we obtain a solution $(\lambda_1, f_1, \phi_1)$ to the first eigentriplet problem (7.3) by using Theorem 2.21.

**Condition (HS3).** In the case $\gamma \in (0, 1)$, the same as in the case $\gamma = 1$ holds except that $\mathcal{R}_B(\alpha)$ is not uniformly bound in $\mathcal{B}(L_m^2)$ for $\alpha \ge \kappa_B$ because we are in the critical case $\kappa_B = \kappa_0$. We do not know how to adapt the stationary approach in that situation and we thus aim to use a dynamical approach through the use of Theorem 3.4 with the above splitting $\mathcal{L} = \mathcal{A} + \mathcal{B}$ and $N := [d/4] + 1$. We set $X = X_1 := L_m^2$ and $X_0 := L^1$. The proof of condition (HS3) is an immediate consequence of the following estimate.

**Proposition 7.8.** We define $\Theta_\zeta(t) := e^{-\zeta t^{\gamma/(2-\gamma)}}$. For $N := [d/4] + 1$, there hold

(i) $S_B \in L_t^\infty(\mathcal{B}(X_1))$;

(ii) $S_B \mathcal{A} \Theta_\zeta^{-1} \in L_t^\infty(\mathcal{B}(X_i))$ for $i = 0, 1$ and any $\zeta \in (0, \zeta^*)$;

(iii) $(S_B \mathcal{A})^{(\ast N)} \Theta_\zeta^{-1} \in L_t^\infty(\mathcal{B}(X_0, X_1))$ for any $\zeta \in (0, \zeta^*/2)$.

The proof of Proposition 7.8 is similar to the proofs of [231, Lem. 2.1], [231, Lem. 2.2], [231, Lem. 2.3] and [231, Lem. 2.4]. For the sake of completeness we however present the main lines of the proof. We start with a technical result that we will use during the proof of Proposition 7.8.

**Lemma 7.9.** Consider two Banach spaces $X_0, X_1$ and a function $u : \mathbb{R}_+ \to \mathcal{B}(X_0) + \mathcal{B}(X_1)$ which satisfies

(a) $u \Theta^{-1} \in L^\infty(0, \infty; \mathcal{B}(X_0) \cap \mathcal{B}(X_1))$;

(b) $u \varphi \in L^\infty(0, \infty; \mathcal{B}(X_0, X_1))$;

for any exponentially decaying function $\Theta = \Theta_\zeta = e^{-\zeta t^\zeta}$, $\forall \zeta \in (0, \zeta^*)$, and for the power function $\varphi := t^{-\alpha}$, with $\zeta^* > 0$, $\zeta \in (0, 1]$ and $\alpha \ge 0$ fixed. Then

(c) there exists $N$ such that $u^{(\ast N)} \tilde{\Theta} \in L^\infty(0, \infty; \mathcal{B}(X_0, X_1))$,

for any $\tilde{\Theta} = \Theta_\zeta$, $\zeta \in (0, \zeta^*/2)$.

**Proof of Lemma 7.9.** A similar argument is developed in [190, Lem. 2.17], [274, Lem. 2.4], [276, Prop. 2.5] and [231, Lem. 2.4].

**Step 1.** Consider two functions $v$ and $w$ which satisfy the estimate (a). For $\mathcal{X} = X_0$ or $\mathcal{X} = X_1$, we compute

$$ \begin{aligned} \|v * w(t)\|_{\mathcal{X} \to \mathcal{X}} &\le \int_0^t \|v(t-s)w(s)\|_{\mathcal{X} \to \mathcal{X}} ds \\ &\le \int_0^t C_{\mathcal{X}}^v \Theta(t-s) C_{\mathcal{X}}^w \Theta(s) ds \le C_{\mathcal{X}}^v C_{\mathcal{X}}^w t \Theta(t), \end{aligned} $$

with obvious notation and where we have used that $\Theta(t-s)\Theta(s) \le \Theta(t)$ for any $0 < s < t$. Since for any $\zeta' \in (0, \zeta)$, there exists a constant $C$ such that $t\Theta_\zeta(t) \le C\Theta_{\zeta'}(t)$ for any $t \ge 0$, we see that the function $v * w$ satisfies the same estimate (a) for any $\Theta = \Theta_\zeta$, $\zeta \in (0, \zeta^*)$.
with

$$ \mathcal{E}(f) := (\mathcal{L}f, f) = r \int c f^2 - \int |\nabla f|^2. $$

We make the following elementary observations:

• We claim that $\lambda_1 \ge 0$. Taking $f_n(x) := n^{-d/2}u(x/n)$ for some function $u \in H^1(\mathbb{R}^d)$, $\|u\|_{L^2} = 1$, we compute

$$ \begin{aligned} -\mathcal{E}(f_n) &= \int |\nabla f_n|^2 - \int_{B_R} r c f_n^2 - \int_{B_R^c} r c f_n^2 \\ &\le \frac{1}{n^2} \int |\nabla u|^2 + \|rc\|_{L^\infty(B_R)} \int_{B_{R/n}} u^2 + \|rc\|_{L^\infty(B_R^c)}, \end{aligned} $$

for any $R > 0$, so that

$$ -\lambda_1 \le \limsup(-\mathcal{E}(f_n)) \le 0. $$

• We claim that $\lambda_1 = 0$ when $c \le 0$. In that case, we have $\mathcal{E}(f) \le 0$ for any $f \in H^1(\mathbb{R}^d)$, and we deduce the reverse inequality $\lambda_1 \le 0$. In particular, as a function $\lambda_1 = \lambda_1(r)$ of $r \ge 0$, we have $\lambda_1(0) = 0$. We also claim that $\lambda_1(r) \to \infty$ as $r \to \infty$ when $c_+ \ne 0$. We may indeed fix $f \in H^1(\mathbb{R}^d)$, $\|f\|_{L^2} = 1$, $\sup f \subset \sup c_+$, and we compute

$$ \mathcal{E}(f) = r \int_{\mathbb{R}^d} c_+ f^2 - \int |\nabla f|^2 \to \infty, \quad \text{as } r \to \infty. $$

• We finally observe that $\lambda_1 : \mathbb{R}_+ \to \mathbb{R}_+$ is convex since it is defined as the supremum of linear functions $r \to \mathcal{E}(f)$ for any fixed $f \in H^1(\mathbb{R}^d)$. As a consequence, we have the following alternative: - $\lambda_1 \equiv 0$;

- $\exists r_0 \in [0, \infty)$ such that $\lambda_1(r) = 0$ for $r \le r_0$ and $\lambda_1(r) > 0$ for $r > r_0$.

Concerning the value of $r_0$, it may happen that $r_0 > 0$, and that is the case when $c \in L^{d/2}$ because of the Sobolev inequality, or that $r_0 = 0$, and that is the case for instance when $c \ge 0$, $c(x) = |x|^{-m}$ for $x \in B_R^c$, $m \in (0, 2)$, $R > 0$. To prove that last claim, we may take the same sequence $(f_n)$ as above, and we compute

$$ \begin{aligned} \mathcal{E}(f_n) &\ge \int_{B_R^c} r |x|^{-m} f_n^2 - \int |\nabla f_n|^2 dx \\ &= \frac{r}{n^m} \int_{B_{R/n}^c} |x|^{-m} u^2 - \frac{1}{n^2} \int |\nabla u|^2 dx > 0, \end{aligned} $$

for $n$ large enough (whatever is the value of $r > 0$).

**About condition (H3).** It is established in [251] that when $\lambda_1 = 0$, the condition (H3) is not satisfied and there does not exist a first eigenfunction $f_1 \in L^2(\mathbb{R}^d)$ to the operator $\mathcal{L}$ defined by (7.54). We refer to [251, 8th course] for a proof of that result. On the other hand, we claim that the condition (H3) is satisfied when $\lambda_1 > 0$. Consider indeed three sequences $(\lambda_n)$ of $\mathbb{R}$, $(f_n)$ of $H^1(\mathbb{R}^d)$ and $(\varepsilon_n)$ of $L^2(\mathbb{R}^d)$ such that $(\lambda_n - \mathcal{L})f_n = \varepsilon_n$, $\varepsilon_n$, $f_n \ge 0$, $\|f_n\|_{L^2} = 1$, for any $n \ge 1$, $\lambda_n \to \lambda_1$ and $\varepsilon_n \to 0$ in $L^2$ as $n \to \infty$. We then have

$$ \lambda_n - \mathcal{E}(f_n) = \langle (\lambda_n - \mathcal{L})f_n, f_n \rangle = \langle \varepsilon_n, f_n \rangle \to 0, $$

as $n \to \infty$. By definition of $\mathcal{E}$ and boundedness of $c$, we see that $(f_n)$ is bounded in $H^1$. As a consequence, up to the extraction of a subsequence, we have $f_n \to f_1 \ge 0$ in $L_{\text{loc}}^2$ and thus next $(\lambda_1 - \mathcal{L})f_1 = 0$ in the variational sense and

$$ \int c f_n^2 \to \int c f_1^2, \quad \|\nabla f_1\|_{L^2} \le \liminf \|\nabla f_n\|_{L^2}, $$

where we have used the dominated convergence theorem of Lebesgue and the fact that $c \to 0$ at infinity in order to get the first convergence. We finally deduce

$$ \mathcal{E}(f_1) \ge \limsup \mathcal{E}(f_n) = \lambda_1 > 0, $$

so that $f_1 \ne 0$, and the condition (H3) is verified.

As a conclusion, for a self-adjoint operator, condition (H3) is automatically fulfilled by its adjoint and the conditions (H4) and (H5) have been proved in a general situation, including the present
Step 2. Consider two functions $v$ and $w$ which satisfy the estimates (a) and (b) with $\alpha \ge 1$. We compute

$$
\begin{align*}
\|v * w(t)\|_{X_0 \to X_1} &\le \int_0^{t/2} \|v(t-s)w(s)\|_{X_0 \to X_1} ds + \int_{t/2}^t \|v(t-s)w(s)\|_{X_0 \to X_1} ds \\
&\le \int_0^{t/2} C_{01}^v (t-s)^{-\alpha} C_0^w \Theta(s) ds + \int_{t/2}^t C_1^v \Theta(t-s) C_{01}^w s^{-\alpha} ds \\
&= [C_1^v C_{01}^w + C_{01}^v C_0^w] \Theta(0) t^{-\alpha+1} \int_0^{1/2} (1-\tau)^{-\alpha} d\tau,
\end{align*}
$$

with obvious notation and we have used that $\Theta$ is a decaying function. As a consequence, the function $v * w$ satisfies estimate (b) with an exponent $\alpha - 1$ instead of $\alpha$.

Step 3. Consider two functions $v$ and $w$ which satisfy the estimates (a) and (b) with $\alpha \in [0, 1)$. We compute

$$
\begin{align*}
\|v * w(t)\|_{X_0 \to X_1} &\le \int_0^{t/2} \|v(t-s)w(s)\|_{X_0 \to X_1} ds + \int_{t/2}^t \|v(t-s)w(s)\|_{X_0 \to X_1} ds \\
&\le \int_0^{t/2} C_1^v \Theta(t-s) C_{01}^w s^{-\alpha} ds + \int_{t/2}^t C_{01}^v (t-s)^{-\alpha} C_0^w \Theta(s) ds \\
&\le C_1^v C_{01}^w \Theta(t/2) \int_0^{t/2} s^{-\alpha} ds + C_{01}^v C_0^w \Theta(t/2) \int_{t/2}^t (t-s)^{-\alpha} ds \\
&= [C_1^v C_{01}^w + C_{01}^v C_0^w] \Theta(t/2) \frac{t^{1-\alpha}}{1-\alpha},
\end{align*}
$$

with the same obvious notation and we have used again that $\Theta$ is a decaying function.

Step 4. Iterating $n := [\alpha]$ times steps 1 and 2, we get that $u^{(*n)}$ still satisfies estimate (a) and satisfies the estimate (b) for the exponent $\alpha - [\alpha] \in (0, 1)$. We then conclude that (c) holds with $N := n + 1$ and any $\zeta \in (0, \zeta^*/2)$ by using the third step. $\square$

Proof of Proposition 7.8. We classically establish that $\mathcal{B}$ generates a positive semigroup $S_{\mathcal{B}}$ in both spaces $X_i$ and we thus concentrate on the announced estimates. On the one hand, proceeding as for the proof of (7.64), we have

$$
(7.65) \qquad \int (\mathcal{B}f)(\operatorname{sign} f)m \le -a'' \int |f|(\mathbf{1}_{B_1} + \mathbf{1}_{B_1^c} |x|^{s+\gamma-2})m,
$$

for any nice function $f$ and any weight function $m = m_a$, with $m_a(x) := e^{a\langle x \rangle^\gamma}$, $a \in (a_1, a_2)$, $0 < a_1 < a_2 < 1/(\sqrt{2}\gamma)$, where we define $a'' := a\gamma/2 - (a\gamma)^2$. That exactly means that $\mathcal{B}$ is weakly dissipative in $L_m^1$ as defined in (3.19). From the discussion in Section 3.3, we deduce that $S_{\mathcal{B}}$ is a semigroup of contractions and satisfies the associated decay estimate (3.23), (3.24), and more precisely

$$
(7.66) \qquad \|S_{\mathcal{B}}(t)f\|_{L_{m_a}^1} \le \|f\|_{L_{m_a}^1}, \quad \|S_{\mathcal{B}}(t)f\|_{L_{m_a}^1} \le \Theta_{\zeta}(t) \|f\|_{L_{m_a'}^1},
$$

for any $a, a' \in (a_1, a_2)$, $a < a'$, $\zeta \in (0, \zeta_*)$, $\zeta_* := (a' - a)^{(2-2\gamma)/(2-\gamma)}(a'\gamma(1 - a'\gamma))^{\gamma/(2-\gamma)}$. We refer to [231, Lem. 2.1] for details. Using that $\mathcal{A}: L^1 \to L_m^1$ is bounded, that establishes (ii) in $X_0$. Similarly, starting from (7.61) and proceeding as in the proof of (7.64), we get

$$
(7.67) \qquad (\mathcal{B}f, f)_{L_m^2} \le - \int |\nabla(fm)|^2 - a'' \int f^2 (\mathbf{1}_{B_1} + \mathbf{1}_{B_1^c} |x|^{s+\gamma-2}) m^2,
$$

for any nice function $f$ and any weight function $m = m_a$ as above. Throwing away the first term at the RHS and arguing as we did in $L_m^1$, we obtain that $S_{\mathcal{B}}$ satisfies

$$
(7.68) \qquad \|S_{\mathcal{B}}(t)f\|_{L_{m_a}^2} \le \|f\|_{L_{m_a}^2}, \quad \|S_{\mathcal{B}}(t)f\|_{L_{m_a}^2} \le \Theta_{\zeta}(t) \|f\|_{L_{m_a'}^2},
$$

for any $a, a' \in (a_1, a_2)$. Using that $\mathcal{A}: L_{m_a}^2 \to L_{m_a'}^2$ is bounded, that establishes (i) and (ii) in $X_1$.
framework. The same conclusions of existence, uniqueness and asymptotic stability of the first
eigentriplet solution $(\lambda_1, f_1, \phi_1)$ as in section 7.2 hold true when $\lambda_1 > 0$.

**Case 2.** We consider the **general case** $b \in C_0(\mathbb{R}^d)$.

• We claim that $\lambda_1 \ge 0$. Adapting the second constructive argument in the proof of (H2) in Section 7.1, we consider $\chi \in C_c^1(\mathbb{R}_+) \cap W^{2,\infty}(\mathbb{R}_+)$ such that $\mathbf{1}_{[0,1/2]} \le \chi \le \mathbf{1}_{[0,1]}$, $\chi' \le 0$ on $[0,1]$, $\chi(s) := (1-s)^2/2$ on $[\eta, 1]$ with $\eta \in (1/2, 1)$ large enough in such a way that

$$
(7.55) \qquad \chi''(s) + (d-1)\chi'(s)/s \ge 1/2, \quad \forall s \in (\eta, 1),
$$

and define $f_0(x) := \chi(|x-x_0|/n)$ for $|x_0|$ large enough to be chosen later. We have supp $f_0 \subset B_n(x_0)$ for any $n \ge 1$ and we compute

$$
\mathcal{L}f_0(x) = \frac{1}{n^2} \left\{ \chi''(r/n) + \frac{d-1}{r/n} \chi'(r/n) \right\} + \frac{1}{n} b(x) \cdot \hat{y} \chi'(r/n) + c(x) \chi(r/n)
$$

where $y = x - x_0$ and $r = |y|$. On $B_{n\eta}(x_0)$, we have

$$
\mathcal{L}f_0(x) \geq - \frac{\|\chi''\|_{\infty}}{n^2} - \frac{d-1}{n^2} \left\| \frac{\chi'(r)}{r} \right\|_{\infty} - \frac{\|\chi'\|_{\infty}}{n} \sup_{B_n(x_0)} |b| - \|\chi\|_{\infty} \sup_{B_n(x_0)} |c|.
$$

On $B_n(x_0) \setminus B_{n\eta}(x_0)$, thanks to (7.55), we have

$$
\mathcal{L}f_0(x) \geq \frac{1}{2n^2} - \frac{\|\chi'\|_{\infty}}{n} \sup_{B_n(x_0)} |b| - \|\chi\|_{\infty} \sup_{B_n(x_0)} |c|.
$$

Let now fix $\varepsilon > 0$ and choose first $n$ large enough so that

$$
- \frac{\|\chi''\|_{\infty}}{n^2} - \frac{d-1}{n^2} \left\| \frac{\chi'(r)}{r} \right\|_{\infty} \geq - \frac{\varepsilon}{2} \inf_{(0,\eta)} \chi.
$$

Then, using that $b, c \in C_0(\mathbb{R}^d)$, we can take $|x_0|$ large enough so that

$$
- \frac{\|\chi'\|_{\infty}}{n} \sup_{B_n(x_0)} |b| - \|\chi\|_{\infty} \sup_{B_n(x_0)} |c| \geq - \frac{\varepsilon}{2} \inf_{(0,\eta)} \chi
$$

and

$$
\frac{\|\chi'\|_{\infty}}{n} \sup_{B_n(x_0)} |b| + \sup_{B_n(x_0)} |c| \le \frac{1}{2n^2}.
$$

Gathering the above inequalities, we obtain

$$
\mathcal{L}f_0 \geq -\varepsilon f_0,
$$

and the condition (H2) is verified with $\kappa_0 = -\varepsilon$. Because $\varepsilon > 0$ can be choose arbitrarily small,
we conclude with $\lambda_1 \ge 0$.

• We claim that $\lambda_1 = 0$ when $\sigma_2 \le 0$. Indeed, we have already seen that

$$
\langle \mathcal{L}f, f \rangle = - \int_{\mathbb{R}^d} |\nabla f|^2 + \int_{\mathbb{R}^d} \sigma_2 f^2,
$$

from which we deduce that

$$
\frac{d}{dt} \|S_t f\|^2 = 2 \langle \mathcal{L}f, f \rangle \le 0.
$$

This ensures that (H1) is verified with $\kappa_1 = 0$ and so $\lambda_1 \le 0$.

• We claim that $\lambda_1 > 0$ when $c_+ \ne 0$ and $r > 0$ is large enough. For simplifying notations and up to translation and dilatation, we may reduce to the case $c \ge c_0 \mathbf{1}_{B(0,1)}$ with $c_0 > 0$. Adapting the second constructive argument in the proof of (H2) in Section 7.1, we consider $\chi \in C_c^1(\mathbb{R}_+) \cap W^{2,\infty}(\mathbb{R}_+)$, $\mathbf{1}_{[0,1/2]} \le \chi \le \mathbf{1}_{[0,1]}$, $\text{supp}\chi = [0,1]$, $\chi''(1) = 1$, $\chi' \le 0$ on $[0,1]$ and we set $f_0(x) := \chi(|x|)$. We compute

$$
\mathcal{L}f_0 = \chi''(|x|) + \chi'(|x|)\left(\frac{d-1}{|x|} + b \cdot \hat{x}\right) + rc(x)\chi(|x|).
$$

On the one hand, we fix $\eta \in (1/2, 1)$, $1 - \eta$ small enough, in such a way that

$$
\|\chi'\|_{L^{\infty}(\eta,1)} (2(d-1) + \|b\|_{L^{\infty}}) \le 1/4, \quad 1/2 \le \|\chi''\|_{L^{\infty}(\eta,1)},
$$

and thus

$$
\mathcal{L}f_0 \ge \frac{1}{4} \ge \frac{1}{4} f_0 \quad \text{on} \quad B(0, \eta)^c.
$$
On the other hand, throwing away the second term at the RHS in (7.67), for any trajectory $f_t = S_B(t)f_0$, $f_0$ in the domain of $\mathcal{B}$ in $L_m^2$, we have

$$ \frac{1}{2} \frac{d}{dt} \int_{\mathbb{R}^d} f_t^2 m^2 dx \leq - \int_{\mathbb{R}^d} |\nabla(f_t m)|^2 dx. $$

Using Nash's inequality which for some constant $C_N \in (0, \infty)$ stipulates that

$$ \int_{\mathbb{R}^d} g^2 dx \leq C_N \left( \int_{\mathbb{R}^d} |\nabla g|^2 dx \right)^{\frac{d}{d+2}} \left( \int_{\mathbb{R}^d} |g| dx \right)^{\frac{4}{d+2}}, \quad \forall g, $$

with $g := f_t m$ and the first estimate in (7.68), we deduce

$$ (7.69) \qquad F'(t) \leq -2 C'_N F(t)^{-4/d} G(t)^{1+\frac{2}{d}} \leq -2 C'_N F(t)^{-4/d} G(0)^{1+\frac{2}{d}}, $$

with $C'_N := C_N^{-1-2/d}$ and where for brevity of notations we have set

$$ F(t) := \|f_t\|_{L^2(m)}^2, \qquad G(t) := \|f_t\|_{L^1(m)}. $$

Integrate the differential inequality (7.69), we find

$$ \|S_B(t)f_0\|_{L_m^2}^2 \lesssim t^{-d/4} \|f_0\|_{L_m^1}, \quad \forall t > 0, $$

and using that $\mathcal{A}: L^1 \to L_m^1$, we next obtain

$$ (7.70) \qquad S_B(t) \mathcal{A} t^{d/4} \in L^\infty(0, \infty; \mathcal{B}(X_0, X_1)). $$

Setting with $u(t) := S_B(t)\mathcal{A}$, we see that $u$ satisfies (a) in Lemma 7.9 thanks to (ii) in $X_0$ and $X_1$. Furthermore, $u$ satisfies (b) in Lemma 7.9 thanks to (7.70). Using Lemma 7.9, we conclude that condition (iii) holds. $\square$

We come back to the proof of (HS3). Gathering (i) and (ii) in $X_1$ in Proposition 7.8, we get that $(S_B\mathcal{A})^{(*\ell)} * S_B \in L_t^\infty(\mathcal{B}(X_1))$ for any $\ell \in \{0, \dots, N-1\}$, $N := [d/4] + 1$. Using that $\Theta \in L^1(0, \infty)$ and (iii) in Proposition 7.8, we deduce that $(S_B\mathcal{A})^{(*N)} \in L^1(0, \infty; \mathcal{B}(X_0, X_1))$.

We may now handle the existence part of the first eigenvalue problem. On the one hand, recalling (H2), we have $\mathcal{L}^*\phi_0 \geq 0$ with $\phi_0 = 1$ so that the condition (i) in Theorem 3.4 holds. On the other hand, the condition (ii) in Theorem 3.4 is an immediate consequence of (HS3) as emphasized in Remark 3.5-(1). As a conclusion, the hypotheses of Theorem 3.4 are thus met, and we deduce that there exists $(\lambda_1, f_1) \in \mathbb{R}_+ \times L_{m+}^2$ solution to the first eigenvalue problem. Because the strong maximum principle (H4) holds, we have $f_1 > 0$ on $\mathbb{R}^d$.

In order to prove the existence of a first positive eigenfunction for the dual problem, we argue in the following way. We start observing that we have the alternative: $\lambda_1 = 0$ or $\lambda_1 > 0$.

- In the first case, we may argue as in Remark 4.19. We indeed have in the same time $\mathcal{L}^*\phi_0 \geq 0$ and $\langle \mathcal{L}^*\phi_0, f_1 \rangle = \langle \phi_0, \mathcal{L}f_1 \rangle = 0$, so that $\mathcal{L}^*\phi_0 = 0$ because $f_1 > 0$. The function $\phi_1 := \phi_0$ is thus a solution to the first dual eigenvalue problem.

- In the second case $\lambda_1 > 0$, we may argue as in the case $\gamma = 1$ above. On the one hand, the operator $\mathcal{R}_B(\alpha)$ is uniformly bounded in $L_m^2$ for any $\alpha \geq \kappa_B := \lambda_1/2 > 0$ and on the other hand the operator $\mathcal{W}(\alpha) := \mathcal{R}_B(\alpha)\mathcal{A}: L_m^2 \to H_m^1 \cap L_{m+}^2$ is uniformly bounded for any $\alpha \geq \kappa_B$ with $m = o(m^\gamma)$, so that $H_m^1 \cap L_{m+}^2 \subset L_m^2$ is compact. We may thus apply Theorem 2.21 and we conclude to the existence of a solution $(\lambda'_1, f'_1, \phi'_1)$ to the eigentriplet problem.

The conditions (H4) and (H5) being true in a general situation as well as the conclusions of Lemma 7.1, as an intermediate conclusion, we have established under the general condition $\gamma > 0$ in (7.56) that yet the same conclusions as in section 7.2 hold true.

**Quantitative stability.** We now establish a quantitative stability estimate using the Dobbins-Harris approach presented in Section 6 and yet used in the case of a bounded domain in Section 7.1. We first consider the more difficult case $\gamma \in (0, 1)$, so that $\lambda_1 \geq \kappa_0 = 0$, and then explain the modifications to be made in order to deal with the case $\gamma \geq 1$. As explained just above, $\lambda_1 = 0$ corresponds to the conservative case $(\lambda_1, \phi_1) = (0, 1)$ which has been considered in [231]. We thus focus on the case $\lambda_1 > 0$ for which an adapted version of Theorem 7.3 already imply the exponential asymptotic stability (E31) in $L_m^2$ with non constructive rate. We do not develop further this argument but rather establish a constructive sub-exponential asymptotic stability.
On the other hand, we fix $r > 0$, large enough, in such a way that

$$ \|\chi''\|_{L^\infty} + \|\chi'\|_{L^\infty}(2(d-1) + \|b\|_{L^\infty}) \le \kappa(r) := \frac{1}{2}rc_0 \inf_{[0,\eta)} \chi, $$

and thus

$$ \mathcal{L}f_0 \ge \kappa(r) \ge \kappa(r)f_0 \quad \text{on } B(0, \eta). $$

As a conclusion, we have established that condition (ii) in the statement Lemma 2.4 holds with $\kappa_0 := \min(1/4, \kappa(r))$, and that ends the constructive proof of condition (H2) by using Lemma 2.4. That implies in particular the claim since then $\lambda_1 \ge \kappa_0 > 0$.

• We finally claim that (H3) holds when $\lambda_1 > 0$. To see that, we consider again three sequences $(\lambda_n)$ of $\mathbb{R}$, $(f_n)$ of $H^1(\mathbb{R}^d)$ and $(\varepsilon_n)$ of $L^2(\mathbb{R}^d)$ such that $(\lambda_n - \mathcal{L})f_n = \varepsilon_n$, $\varepsilon_n, f_n \ge 0$, $\|f_n\|_{L^2} = 1$, for any $n \ge 1$, $\lambda_n \searrow \lambda_1$ and $\varepsilon_n \to 0$ in $L^2$ as $n \to \infty$. As a consequence, we have

$$ \lambda_n + \int |\nabla f_n|^2 - \int f_n b \cdot \nabla f_n - \int c f_n^2 = ((\lambda_n - \mathcal{L})f_n, f_n) = \langle \varepsilon_n, f_n \rangle \to 0, $$

as $n \to \infty$. Using the boundedness of $c$, $b$ and $\lambda_n$, we see that $(f_n)$ is bounded in $H^1$. As a consequence, up to the extraction of a subsequence, we have $f_n \to f_1 \ge 0$ in $L^2_{\text{loc}}$. We assume by contradiction that $f_1 \equiv 0$. We deduce that

$$ \int c f_n^2 \to 0, \quad \int f_n b \cdot \nabla f_n \to 0, $$

where we have used the dominated convergence theorem of Lebesgue and the fact that $b, c \to 0$ at infinity. We thus obtain

$$ 0 < \lambda_1 \le \lambda_n + \int |\nabla f_n|^2 = \int f_n b \cdot \nabla f_n + \int c f_n^2 + \langle \varepsilon_n, f_n \rangle \to 0, $$

and our contradiction. So that $f_1 \ne 0$, and the condition (H3) is verified.

For the dual problem, from the above analysis, we know that there exist two sequences $(\phi_n)$ of $H^1(\mathbb{R}^d)$, $(\varepsilon_n)$ of $L^2(\mathbb{R}^d)$ such that $(\lambda_n - \mathcal{L}^*)\phi_n = \varepsilon_n$, $\varepsilon_n, \phi_n \ge 0$ and $\|\phi_n\|_{L^2} = 1$, for any $n \ge 1$, and $\varepsilon_n \to 0$ in $L^2$ as $n \to \infty$. But we face the same situation as previously, since again

$$ \lambda_n + \int |\nabla \phi_n|^2 - \int \phi_n b \cdot \nabla \phi_n - \int c \phi_n^2 = ((\lambda_n - \mathcal{L}^*)\phi_n, \phi_n) = (\varepsilon_n, \phi_n) \to 0, $$

and thus the same conclusion, namely $\phi_n \to \phi_1$, with $\phi_1 \in H^1(\mathbb{R}^d)$, $\phi_1 \ge 0$, $\phi_1 \ne 0$.

The conditions (H4) and (H5) have been proved in a general situation, including the present framework. For the same reason as in section 7.2, the same conclusions hold true when $r > 0$ is large enough (and thus $\lambda_1 > 0$) and more precisely we have established the following result.

**Theorem 7.7.** Consider the elliptic operator (7.50) in the whole space and assume that the coefficients satisfy $b, c \in C_0(\mathbb{R}^d)$, $c_+ \ne 0$. For $r > 0$ large enough, the conclusions (S1), (S2) and (S32) hold in $L^2$ (with $\lambda_1 > 0$) as well as (E2) in $L^1_{\phi_1}$.

While the conclusion (S1)-(S2) is essentially due to Lions [251, 9th course], we believe that the ergodicity property (E2) is new.

**7.4. Diffusion in $\mathbb{R}^d$ with drift confinement.** We now consider the elliptic operator

$$ \mathcal{L}f := \Delta f + b \cdot \nabla f + cf, $$

with a drift confinement as it is the case for the Fokker-Planck operator. More precisely, and for the sake of simplicity, we assume here

$$ (7.56) \qquad b = \nabla U, \quad U(x) = \frac{1}{\gamma} \langle x \rangle^{\gamma}, \quad \gamma > 0. $$

When $\gamma = 2$ and $c = x$, that operator corresponds to the classical harmonic Fokker-Planck operator which is known to be related to the standard Poincaré inequality and to the standard log-Sobolev inequality, see [29, 26, 342] or more recently [27, 230] and the references therein. When $c = \text{div}b$, the operator $\mathcal{L}$ is on divergence form and $\mathcal{L}^*1 = 0$, so that $(0,1) \in \mathbb{R} \times L^\infty(\mathbb{R}^d)$ is a solution to the dual first eigenvalue problem. Existence of stationary solution $f_1$ (which is also the first
**Step1 - Lyapunov condition.** We take $m = e^{|x|^\gamma}$ with $0 < a < \gamma^{-1}$. From Lemma 7.7 or a direct computation, we have

$$ 
\begin{aligned}
\mathcal{L}^* m &= \Delta m + (c - \operatorname{div} b)m - b \cdot \nabla m \\
&\leq (C\langle x \rangle^{\gamma-2} + |c| - 2a^*\langle x \rangle^{2\gamma-2})m \\
&\leq C_0 \mathbf{1}_{B_{\varrho_0}} - a^*\langle x \rangle^{2\gamma-2}m,
\end{aligned}
$$

for three positive constants $C = C(d)$, $C_0 = C_0(c, C, a, \gamma)$, $\varrho_0 = \varrho_0(c, C, a, \gamma)$ and with $a^* := (a\gamma - (a\gamma)^2)/2 > 0$. We now set $m_1 := m$ and $m_0 := a^*\langle x \rangle^{2\gamma-2}m$. We fix $T > 0$ and for $0 \le f_0 \in L_m^1$, we denote $f_t := \tilde{S}_L(t)f_0$. Recording that $\lambda_1 \ge 0$ and using the above pointwise estimate, we deduce

$$ (7.71) \qquad \int f_T m_1 + \int_0^T \int f_t m_0 dt \le \int f_0 m_1 + C_1 \int_0^T \int_{B_{\varrho_0}} f_t dt. $$

Because the same kind of pointwise estimate holds for $\mathcal{L}^*m_0$, we have

$$ \int f_T m_0 \le \int f_t m_0 + C_1 \int_t^T \int_{B_{\varrho_0}} f_s ds $$

and integrating in time, we get

$$ T \int f_T m_0 \le \int_0^T \int f_t m_0 dt + C_0 \int_0^T s \int_{B_{\varrho_0}} f_s ds. $$

Coming back to the first estimate, we deduce

$$ (7.72) \qquad \int f_T m_1 + T \int f_T m_0 \le \int f_0 m_1 + (C_1 + C_0 T) \int_0^T \int_{B_{\varrho_0}} f_t dt. $$

**Step 2 - Pointwise estimates on $\phi_1$.** We define $\mathcal{B} := \mathcal{L} - C_0\chi_{\varrho_0}$ which is the generator of a positive semigroup of contraction in $L_m^1$ because of the above discussion. For $\lambda > 0$, $0 \le g \in L_m^1$ and $0 \le f \in L_m^1$ the solution to $(\lambda - \mathcal{B})f = g$, we compute

$$ \int gm = \int f(\lambda - \mathcal{B}^*)m \ge \int f(\lambda m + m_0) \ge \int fm_0, $$

from what we deduce

$$ \|\mathcal{R}_B(\lambda)f\|_{L_{m_0}^1} \le \|f\|_{L_m^1}, \quad \forall f \in L_m^1. $$

Now, we consider two weight functions $m_1$ and $m_3$ with $m_i := e^{a_i|x|^\gamma}$, $0 < a_1 < a_3 < \gamma^{-1}$, we denote $m_0 := a_1^*\langle x \rangle^{2\gamma-2}m_1$ and we compute

$$ \begin{aligned} \|\mathcal{A}\mathcal{R}_B(\lambda)f\|_{L_{m_3}^1} &\le C_0\|\mathcal{R}_B(\lambda)f\|_{L_{m_3}^1(B_{2\varrho_0})} \\ &\le C_1\|\mathcal{R}_B(\lambda)f\|_{L_{m_0}^1} \le C_1\|f\|_{L_{m_1}^1}. \end{aligned} $$

By duality, we obtain

$$ (7.73) \qquad \|\mathcal{R}_{B^*}(\lambda)\phi\|_{L_{m_1}^{\infty}} \le \|\phi\|_{L_{m_0}^{\infty}} \quad \text{and} \quad \|\mathcal{R}_{B^*}\mathcal{A}^*(\lambda)\phi\|_{L_{m_1}^{\infty}} \le C_1 \|\phi\|_{L_{m_3}^{\infty}}, $$

for any $\lambda > 0$ and $\phi \in L_{m_0}^\infty$. We also deduce from Proposition 7.8 the regularization estimate $(\mathcal{A}^*\mathcal{R}_{B^*})^N : L_{m_1}^2 \to L^\infty$. Let us now consider $0 \le \phi_1 \in L_{m_1}^2$ the first eigenvector for the dual problem built in the preceding paragraph. From the eigenvalue equation

$$ \mathcal{B}^*\phi_1 + \mathcal{A}^*\phi_1 = \mathcal{L}^*\phi_1 = \lambda_1\phi_1, $$

we deduce that $\phi_1 = (\mathcal{R}_{B^*}\mathcal{A}^*)\phi_1$, and iterating

$$ \phi_1 = (\mathcal{R}_{B^*}\mathcal{A}^*)^{N+1}\phi_1 = \mathcal{R}_{B^*}(\mathcal{A}^*\mathcal{R}_{B^*})^N\mathcal{A}^*\phi_1. $$

From the above regularization estimate and the first estimate in (7.73), we thus deduce that $\phi_1 \in L_{m_1}^\infty$. Moreover, normalizing $\phi_1$ and using the second estimate in (7.73), we may obtain

$$ (7.74) \qquad \|\phi_1\|_{L_{m_3-1}^{\infty}} = 1 \quad \text{and} \quad \|\phi\|_{L_{m_1-1}^{\infty}} \le C_1. $$
eigenfunction) and its stability have been widely studied. We refer for instance to [343, 331, 170, 28] as well as to [230, 273, 189] which techniques will be adapted here.

In the present situation, we impose that the contribution of $c$ has lower influence at the infinity that the drift term $b$ and we assume

$$ (7.57) \qquad c \in L_{\text{loc}}^{\infty}(\mathbb{R}^d), \quad \exists C_0, R_0 > 0, \forall x \in B_{R_0}^c, |c(x)| = o(|x|^{2(\gamma-1)}). $$

We further assume that

$$ (7.58) \qquad c \ge \operatorname{div} b \quad \text{when} \quad \gamma \in (0, 1]. $$

The action of the drift term will be revealed through the choice of a convenient “confining space”. More precisely, for a weight function $m: \mathbb{R}^d \to [1, \infty)$, we will work in a weighted Lebesgue space. Our analysis is based on the following elementary computation which can be readily adapted from [230, Lem. 2.1], [273, Lem. 3.8] and [189, Lem. 3.8].

**Lemma 7.8.** For any $p \in [1, \infty)$, any weight function $m$ and any smooth, rapidly decaying function $f$, we have

$$ (7.59) \qquad \int (\mathcal{L}f) f |f|^{p-2} m^p = -(p-1) \int |\nabla f|^2 |f|^{p-2} m^p + \int |f|^p m^p \varphi_1, $$

with

$$ (7.60) \qquad \varphi_1 := (p-1) \frac{|\nabla m|^2}{m^2} + \frac{\Delta m}{m} + \left(c - \frac{1}{p} \operatorname{div} b\right) - b \cdot \frac{\nabla m}{m}. $$

as well as

$$ (7.61) \qquad \int (\mathcal{L}f) f |f|^{p-2} m^p = -(p-1) \int |\nabla (fm)|^2 |fm|^{p-2} + \int |f|^p m^p \varphi_2, $$

with

$$ (7.62) \qquad \varphi_2 := 2\left(1 - \frac{1}{p}\right) \frac{|\nabla m|^2}{m^2} + \left(\frac{2}{p} - 1\right) \frac{\Delta m}{m} + \left(c - \frac{1}{p} \operatorname{div} b\right) - b \cdot \frac{\nabla m}{m}. $$

In order to simplify the discussion, we restrict ourself to the exponent $p = 2$ and to the exponential weight function $m = e^{a\langle x \rangle^s}$, $s \in (0, \gamma]$, $a > 0$. We thus work in the Banach lattice $X := L_m^2$. We observe that

$$ \frac{\nabla m}{m} = s a x \langle x \rangle^{s-2} \sim s a |x|^{s-1}, $$

$$ \frac{\Delta m}{m} = s a d \langle x \rangle^{s-2} + s(s-2)a|x|^2 \langle x \rangle^{s-4} + (sa)^2 |x|^2 \langle x \rangle^{2s-4} \sim (sa)^2 |x|^{2s-2}, $$

$$ \operatorname{div} b = d \langle x \rangle^{\gamma-2} + (\gamma - 2) |x|^2 \langle x \rangle^{\gamma-4} \sim (d + \gamma - 2) |x|^{\gamma-2}, $$

$$ b \cdot \frac{\nabla m}{m} = s a x \langle x \rangle^{s-2} \cdot x \langle x \rangle^{\gamma-2} \sim s a |x|^{s+\gamma-2}, $$

so that the contribution of $(c - \operatorname{div} b/2)$ is always negligible at infinity, and we get

$$ (7.63) \qquad \varphi_i \sim (sa)^2 |x|^{2s-2} - s a |x|^{s+\gamma-2}. $$

We denote

$$ a' := sa > 0 \quad \text{if} \quad s \in (0, \gamma), $$

$$ a' := a\gamma - 2(a\gamma)^2 > 0 \quad \text{if} \quad s = \gamma \text{ and } a \in (0, 1/(\sqrt{2}\gamma)). $$

We then face to three cases :

(i) $\gamma > 1$: taking $s \in ((2 - \gamma)_+, \gamma)$, we have $\varphi_i \sim -a'|x|^{s+\gamma-2} \to -\infty$ with $s + \gamma - 2 > 0$;

(ii) $\gamma = 1$: taking $s = \gamma$, $a < 1/(\sqrt{2}\gamma)$, we have $\varphi_i \to -a'$;

(iii) $\gamma \in (0, 1)$: taking $s = \gamma$, $a < 1/(\sqrt{2}\gamma)$, we have $\varphi_i \sim -a'|x|^{2\gamma-2} \to 0$ with $2\gamma - 2 < 0$.

**Condition (H1).** In any of the above cases, we have from (7.59)

$$ ((\lambda - \mathcal{L})f, f) = \int |\nabla f|^2 m^2 + \int (\lambda - \varphi_1) f^2 m^2, $$
We deduce

$$ 
\begin{aligned}
1 &= \max\left(\sup_{B_{\varrho_2}} \frac{|\phi_1|}{m_3}, \sup_{B_{\varrho_2}^c} \frac{|\phi_1|}{m_3}\right) \\
  &\le \max\left(\sup_{B_{\varrho_2}} \frac{|\phi_1|}{m_3}, C_{0,1} \sup_{B_{\varrho_2}^c} \frac{m_1}{m_3}\right),
\end{aligned}
$$

so that $\sup_{B_{\varrho_2}} \frac{|\phi_1|}{m_3} = 1$ by choosing $\varrho_2 := \max(\varrho_0, \varrho_1)$ with $C_1 e^{(a_1-a_3)\varrho_1^\gamma} = 1$. As a consequence, there exists $x_0 \in B_R$ such that $\phi_1(x_0) \ge 1$. On the other hand, using standard regularity result for elliptic equation in the ball $B_{2R}$, we obtain that $\phi_1 \in C^{0,1}(B_R) \cap W^{2,p}(B_R)$ for any $p \in [1, \infty)$ with constructive bound. Making use next of the Harnack inequality as at the end of Section 7.1 or using barrier functions as in the proof of [231, Lem. 6.2], we classically deduce that

$$ (7.75) \qquad \phi_1 \ge z_{\varrho} \mathbf{1}_{B_{\varrho}}, \quad \forall \varrho > 0, $$

for a constructive constant $z_{\varrho} > 0$ (where we emphasize here and below the $\phi_1$ always denote the normalized by (7.74) dual eigenvector).

**Step 3 - Doblin-Harris estimate.** We fix $T > 0$ (for instance $T := 1$) and $A > 0$ arbitrary. For $0 \le f_0 \in L_m^1$ such that $\|f_0\|_{L_m^1} \le A\|f_0\|_{L_{\phi_1}^1}$, we denote $f_t := \tilde{S}_L(t)f_0$. On the one hand, we have

$$ \int f_t \phi_1 = \int f_0 \phi_1 \quad \text{and} \quad \int f_t m \le C_T \int f_0 m, $$

for any $t \in [0, T]$, the second estimate being an immediate consequence of (7.71). On the other hand, we define $\varepsilon(r) := \sup_{|x| \ge r} (m(x)/\phi_1(x))$ and we compute

$$ 
\begin{aligned}
\int_{B_{\rho}} f_t \phi_1 &= \int f_t \phi_1 - \int_{B_{\rho}^c} f_t \phi_1 \ge \int f_t \phi_1 - \varepsilon(\rho) \int f_t m \\
&\ge \int f_0 \phi_1 - C_T \varepsilon(\rho) \int f_0 m \ge \left(1 - \frac{C_T}{A} \varepsilon(\rho)\right) \int f_0 \phi_1 \ge \frac{1}{2} \int f_0 \phi_1,
\end{aligned}
$$

for any $t \in (0, T)$, by choosing $\rho := \rho(T, A) > 0$ large enough. In particular, there exists $x_0(t) \in B_\rho$ such that

$$ f(t, x_0(t)) \ge \vartheta := \frac{1}{2} \frac{1}{\|\phi_1\|_{L^1(B_\rho)}} \int f_0 \phi_1. $$

Next, arguing exactly as in Section 7.1 or as in the proof of [231, Lem. 6.2], we deduce

$$ (7.76) \qquad \tilde{S}_T f_0 \ge \eta_{T,A} \mathbf{1}_{B_1}[f_0]_{L_{\phi_1}^1}, $$

for some constructive constant $\eta_{T,A} > 0$.

## 8. TRANSPORT EQUATIONS

The main aim of this part is to analysis the long time asymptotic of the solutions to the transport equation

$$ (8.1) \qquad \partial_t f + \operatorname{div}_y (af) = \mathcal{H}[f] - Kf \quad \text{in } (0, \infty) \times \mathcal{O}, $$

on the function $f = f(t, y)$, $t \ge 0$, $y \in \mathcal{O}$, with $\mathcal{O} \subset \mathbb{R}^D$, $D \ge 1$, a smooth open connected set. We assume that $a = a(y)$, $a : \mathcal{O} \to \mathbb{R}^D$, $K = K(y)$, $K : \mathcal{O} \to \mathbb{R}_+$ and that the collision operator $\mathcal{K}$ is linear and defined by

$$ (8.2) \qquad (\mathcal{H}g)(y) := \int_{\mathcal{O}} k g_* dy_*, $$

for some kernel $k : \mathcal{O} \times \mathcal{O} \to \mathbb{R}_+$ and for any (conveniently) bounded function $g : \mathcal{O} \to \mathbb{R}$. Here and below, we use the common shorthands

$$ g_* := g(y_*), \quad k := k(y, y_*), \quad k_* := k(y_*, y). $$

When $\mathcal{O} \ne \mathbb{R}^D$, the equation is complemented with a boundary condition which imposes the value of the trace $\gamma_-f$ of $f$ on the incoming subsets of the boundary and takes the form

$$ (8.3) \qquad (\gamma_-f)(t, y) = \mathcal{R}[f(t, \cdot), \gamma_+f(t, \cdot)](y) \quad \text{on } (0, \infty) \times \Sigma_- $$
for $\lambda \in \mathbb{R}$, with $\inf(\lambda - \varphi_1) > 0$ for $\lambda \ge \kappa_1$ and $\kappa_1 > 0$ large enough. We deduce that $\lambda - \mathcal{L}$ is coercive for $\lambda \ge \kappa_1$. With the same arguments as in section 7.1, we conclude that $\mathcal{L}$ is the generator in $L_m^2$ of a positive semigroup $S_{\mathcal{L}}$, so that (H1) holds.

**Condition (H2).** When $\gamma > 1$, the same arguments as in Section 7.2 imply that condition (H2) holds for some $\kappa_0 \in \mathbb{R}$. When $\gamma \in (0, 1]$, we have $\mathcal{L}^*1 = c - \text{div}b \ge 0$ from (7.58) and (H2) holds with $\kappa_0 = 0$.

**Conditions (H4) and (H5).** The strong maximum principle holds here because for instance we may apply the same barrier function argument as presented in Section 7.2. The reverse Kato's inequality condition is proved by using local arguments, so that it holds for the same reasons as in the previous section.

**Condition (H3).** We define the multiplication operator $\mathcal{A}$ and the elliptic operator $\mathcal{B}$ by

$$ \mathcal{A} := M\chi_R, \quad \mathcal{B} := \mathcal{L} - \mathcal{A}, $$

for $M, R > 0$ and $\chi_R(x) := \chi(x/R)$ with $\chi \in \mathcal{D}(\mathbb{R}^d)$, $\mathbf{1}_{B_1} \le \chi \le \mathbf{1}_{B_2}$. We fix $\kappa_B < \kappa_0$ in case (i), $\kappa_B := -a'/4$ in case (ii) and $\kappa_B := 0$ in case (iii), and we set $a'' := a'/2$. Choosing $M, R > 0$ large enough, from Lemma 7.8 and the discussion which follows, we deduce that

$$ (7.64) \qquad ((\mathcal{B} - \alpha)f, f) \le -\int |\nabla f|^2 m^2 - a'' \int f^2 (\mathbf{1}_{B_1} + \mathbf{1}_{B_1^c} |x|^{s+\gamma-2}) m^2, $$

for any $\alpha \ge \kappa_B$ and any nice function $f$. We classically deduce that $\alpha - \mathcal{B}$ is coercive and thus invertible. We discuss the three different cases.

- In the first case $\gamma > 1$, so that $s + \gamma - 2 > 0$, we see that the operator $\mathcal{R}_B(\alpha)$ is compact from Rellich-Kondrachov theorem, so that also $W(\alpha) := \mathcal{R}_B(\alpha)\mathcal{A}$ for any $\alpha \ge \kappa_B$. We may thus apply Lemma 2.8-(2) and we deduce that (H3) holds for both the primal and the dual problems.

- In the case $\gamma = 1$, so that $2\gamma - 2 \le 0$, the operator $\mathcal{R}_B(\alpha)$ is not compact anymore. However, for any sequence $(f_n)$ which is bounded in $L_m^2$, we define the sequence $(g_n)$ by $g_n := \mathcal{A}f_n$, and $(g_n)$ is bounded in $L_m^2$ with $\tilde{m} := e^{\tilde{a}(x)^\gamma}$, $\tilde{a} \in (a, 1/\sqrt{2\gamma})$. Using the dissipativity estimate (7.64) in $L_m^2$, we see that $\mathcal{B} - \alpha$ is dissipative in $L_m^2$ for any $\alpha \ge \kappa_B$, and more precisely the sequence $(h_n)$ defined by $h_n := \mathcal{R}_B(\alpha)g_n$ satisfies

$$ \int |\nabla h_n|^2 m^2 + \tilde{a}'' \int h_n^2 (\mathbf{1}_{B_1} + \mathbf{1}_{B_1^c} |x|^{2\gamma-2}) \tilde{m}^2 \le \int g_n^2 \tilde{m}^2. $$

Using that $|x|^{2\gamma-2}\tilde{m}^2/m^2 \to \infty$ as $|x| \to \infty$, that implies that $(h_n)$ is relatively compact in $L_m^2$. More precisely, the above estimates show that $W(\alpha) := \mathcal{R}_B(\alpha)\mathcal{A} : L_m^2 \to H_m^1 \cap L_{m^\sharp}^2$ with $m^\sharp := m^{1/2}\tilde{m}^{1/2}$ and in particular we have established that $W(\alpha) := \mathcal{R}_B(\alpha)\mathcal{A}$ is a compact operator in $L_m^2$ uniformly on $\alpha \ge \kappa_B$ because of the Rellich-Kondrachov theorem and the fact that $m = o(m^\sharp)$. Since $\mathcal{R}_B(\alpha)$ is bounded in $\mathcal{B}(L_m^2)$ uniformly for any $\alpha > \kappa_B$, the operator $\mathcal{L}$ satisfies the splitting structure (HS1) and, applying Lemma 2.8-(2), we deduce that (H3) holds for both the primal and the dual problems.

At this stage, when $\gamma \ge 1$, we obtain a solution $(\lambda_1, f_1, \phi_1)$ to the first eigentriplet problem (7.3) by using Theorem 2.21.

**Condition (HS3).** In the case $\gamma \in (0, 1)$, the same as in the case $\gamma = 1$ holds except that $\mathcal{R}_B(\alpha)$ is not uniformly bound in $\mathcal{B}(L_m^2)$ for $\alpha \ge \kappa_B$ because we are in the critical case $\kappa_B = \kappa_0$. We do not know how to adapt the stationary approach in that situation and we thus aim to use a dynamical approach through the use of Theorem 3.4 with the above splitting $\mathcal{L} = \mathcal{A} + \mathcal{B}$ and $N := [d/4] + 1$. We set $X = X_1 := L_m^2$ and $X_0 := L^1$. The proof of condition (HS3) is an immediate consequence of the following estimate.

**Proposition 7.9.** We define $\Theta_\zeta(t) := e^{-\zeta t^{\gamma/(2-\gamma)}}$. For $N := [d/4] + 1$, there hold

(i) $S_B \in L_t^\infty(\mathcal{B}(X_1))$;

(ii) $S_B \mathcal{A} \Theta_\zeta^{-1} \in L_t^\infty(\mathcal{B}(X_i))$ for $i = 0, 1$ and any $\zeta \in (0, \zeta^*)$;

(iii) $(S_B \mathcal{A})^{(N)} \Theta_\zeta^{-1} \in L_t^\infty(\mathcal{B}(X_0, X_1))$ for any $\zeta \in (0, \zeta^*/2)$.

The proof of Proposition 7.9 is similar to the proofs of [230, Lem. 2.1], [230, Lem. 2.2], [230, Lem. 2.3] and [230, Lem. 2.4]. For the sake of completeness we however present the main lines of the proof. We start with a technical result that we will use during the proof of Proposition 7.9.
Let us explain the meaning of the different terms involved in (8.3). We denote by $\Sigma := \partial\mathcal{O}$ the boundary set, by $d\sigma_y$ the Lebesgue measure on $\Sigma$, by $n : \Sigma \to \mathbb{S}^{D-1}$ the normal outward vector field, we write $n = n_y = n(y)$, and by $\Sigma_-$ the incoming, $\Sigma_+$ the outgoing and $\Sigma_0$ the singular subsets of the boundary defined by

$$ \Sigma_{\pm} := \{y \in \Sigma; \pm a(y) \cdot n_y > 0\}, \quad \Sigma_0 := \{y \in \Sigma; a(y) \cdot n_y = 0\}. $$

We denote $\gamma f = f|_{(0,\infty)\times\Sigma}$ the trace of $f$ and $\gamma_{\pm}f := \mathbf{1}_{(0,\infty)\times\Sigma_{\pm}} \gamma f$ the trace restrictions on the incoming and outgoing sets. We then assume that the boundary operator $\mathcal{R}$ splits into two pieces $\mathcal{R}(g, h) = \mathcal{R}_{\mathcal{O}}(g) + \mathcal{R}_{\Sigma}(h)$, where

$$ (8.4) \qquad (\mathcal{R}_{\mathcal{O}}g)(y) = \int_{\mathcal{O}} g(y_*)r_{\mathcal{O}}(y, dy_*), \quad (\mathcal{R}_{\Sigma}h)(y) = \int_{\Sigma_+} h(y_*)r_{\Sigma}(y, dy_*), $$

for a domain transition kernel $r_{\mathcal{O}} : \Sigma_- \times \mathcal{B}_{\mathcal{O}} \to [0, \infty]$, a boundary transition kernel $r_{\Sigma} : \Sigma_- \times \mathcal{B}_{\Sigma_+} \to [0, \infty]$ and for any (conveniently) bounded functions $g : \mathcal{O} \to \mathbb{R}$ and $h : \Sigma_+ \to \mathbb{R}$, where $\mathcal{B}_E$ stands for the set of Borel subsets of $E$.

In the next sections we will first consider the trace problem for a general force field $a$ and next the well-posedness for the transport equation with given inflow at the boundary and with reflection condition at the boundary. We will also revisit the characteristic method for general force field $a$. We will next consider the Krein-Rutman problem still for a general force field $a$, but making strong simplification assumptions on the kernel operators $\mathcal{K}$ and $\mathcal{R}$. We will next explain how the classical age structured equation falls into the present framework. We will come back to more specific physical situations concerning the growth-fragmentation equation and the kinetic relaxation equation with more general and physically relevant hypothesis on the kernel in parts 9 and 10.

## 8.1. The trace problem.

In this section, we are concerned with the trace problem associated to a (mainly stationary) transport equation for a general vector field $a : \mathcal{O} \to \mathbb{R}^D$ for which we only assume

$$ (8.5) \qquad a \in W_{\text{loc}}^{1,1}(\bar{\mathcal{O}}), $$

where we recall that $\mathcal{O} \subset \mathbb{R}^D$, $D \ge 1$, is a smooth open connected set. The regularity needed on the domain is formulated in the following way: we assume that there exists $n : \mathcal{O} \to \mathbb{R}^D$, $y \mapsto n(y)$ a vector field belonging to $W^{1,\infty}(\mathcal{O})$ and which coincides with the previously defined unit outgoing normal vector field on $\Sigma$ and satisfies $\|n\|_{L^\infty} = 1$. In that situation, it is well-known that the above vector field is the restriction of a vector field $a \in W_{\text{loc}}^{1,1}(\mathbb{R}^D)$ (where we abuse notations denoting the restriction and the extension in the same way). We also consider the associated differential equation

$$ (8.6) \qquad \frac{dY}{dt} = a(Y), \quad Y(0) = y, $$

and then define the characteristic flow $Y_t = Y(t, y)$, for any $y \in \mathcal{O}$, which is the solution to (8.6) on a maximal time interval $(t_-(y), t_+(y))$ where $t_-(y) < 0 < t_+(y)$ are defined by $t_- := -t_b$ and $t_+ := t_f$, the backward exit time is defined by

$$ (8.7) \qquad t_b(y) := \sup\{\tau > 0; Y_{-t}(y) \in \mathcal{O}, \forall t \in [0, \tau]\} \in (0, +\infty] $$

and the forward exit time is defined by

$$ (8.8) \qquad t_f(y) := \sup\{\tau > 0; Y_t(y) \in \mathcal{O}, \forall t \in [0, \tau]\} \in (0, +\infty]. $$

The real number $t_{\ell t}(y) := t_b(y) + t_f(y) \in (0, \infty]$ corresponds to the “life time” of the characteristic flow in $\mathcal{O}$ going by $y$. The construction of the flow ($Y_t$) is classical when $a$ is a Lipschitz function and we refer to [139, Thm. II.3] for a more general situation which corresponds to the assumptions we will make in the present work (see also Lemma 8.14 below).

For a solution $g : \mathcal{O} \to \mathbb{R}$ to the transport equation

$$ (8.9) \qquad a \cdot \nabla_y g = G \quad \text{in } \mathcal{O}, $$

for a given source term $G : \mathcal{O} \to \mathbb{R}$, we wish to define the trace $\gamma g$ of $g$ on the boundary set $\Sigma$. Similarly, for a solution $g : (0, T) \times \mathcal{O} \to \mathbb{R}$, $T \in (0, +\infty]$, to the transport equation

$$ (8.10) \qquad \partial_t g + a \cdot \nabla_y g = G \quad \text{in } (0, T) \times \mathcal{O}, $$
**Lemma 7.10.** Consider two Banach spaces $X_0, X_1$ and a function $u : \mathbb{R}_+ \to \mathcal{B}(X_0) + \mathcal{B}(X_1)$ which satisfies

(a) $u\Theta^{-1} \in L^\infty(0, \infty; \mathcal{B}(X_0) \cap \mathcal{B}(X_1));$

(b) $u\wp \in L^\infty(0, \infty; \mathcal{B}(X_0, X_1));$

for any exponentially decaying function $\Theta = \Theta_\zeta = e^{-\zeta t^\alpha}, \forall \zeta \in (0, \zeta^*),$ and for the power function $\wp := t^{-\alpha},$ with $\zeta^* > 0, \zeta \in (0, 1]$ and $\alpha \ge 0$ fixed. Then

(c) there exists $N$ such that $u^{(N)}\tilde{\Theta} \in L^\infty(0, \infty; \mathcal{B}(X_0, X_1)),$
for any $\tilde{\Theta} = \Theta_\zeta, \zeta \in (0, \zeta^*/2).$

*Proof of Lemma 7.10.* A similar argument is developed in [189, Lem. 2.17], [273, Lem. 2.4], [275, Prop. 2.5] and [230, Lem. 2.4].

**Step 1.** Consider two functions $v$ and $w$ which satisfy the estimate (a). For $\mathcal{X} = X_0$ or $\mathcal{X} = X_1$, we compute

$$ \begin{aligned} \|v * w(t)\|_{X \to \mathcal{X}} &\le \int_0^t \|v(t-s)w(s)\|_{X \to \mathcal{X}} ds \\ &\le \int_0^t C_{\mathcal{X}}^v \Theta(t-s) C_{\mathcal{X}}^w \Theta(s) ds \le C_{\mathcal{X}}^v C_{\mathcal{X}}^w t \Theta(t), \end{aligned} $$

with obvious notation and where we have used that $\Theta(t-s)\Theta(s) \le \Theta(t)$ for any $0 < s < t$. Since for any $\zeta' \in (0, \zeta)$, there exists a constant $C$ such that $t\Theta_\zeta(t) \le C\Theta_{\zeta'}(t)$ for any $t \ge 0$, we see that the function $v * w$ satisfies the same estimate (a) for any $\Theta = \Theta_\zeta, \zeta \in (0, \zeta^*)$.

**Step 2.** Consider two functions $v$ and $w$ which satisfy the estimates (a) and (b) with $\alpha \ge 1$. We compute

$$ \begin{aligned} \|v * w(t)\|_{X_0 \to X_1} &\le \int_0^{t/2} \|v(t-s)w(s)\|_{X_0 \to X_1} ds + \int_{t/2}^t \|v(t-s)w(s)\|_{X_0 \to X_1} ds \\ &\le \int_0^{t/2} C_{01}^v (t-s)^{-\alpha} C_0^w \Theta(s) ds + \int_{t/2}^t C_1^v \Theta(t-s) C_{01}^w s^{-\alpha} ds \\ &= [C_1^v C_{01}^w + C_{01}^v C_0^w] \Theta(0) t^{-\alpha+1} \int_0^{1/2} (1-\tau)^{-\alpha} d\tau, \end{aligned} $$

with obvious notation and we have used that $\Theta$ is a decaying function. As a consequence, the function $v * w$ satisfies estimate (b) with an exponent $\alpha - 1$ instead of $\alpha$.

**Step 3.** Consider two functions $v$ and $w$ which satisfy the estimates (a) and (b) with $\alpha \in [0, 1)$. We compute

$$ \begin{aligned} \|v * w(t)\|_{X_0 \to X_1} &\le \int_0^{t/2} \|v(t-s)w(s)\|_{X_0 \to X_1} ds + \int_{t/2}^t \|v(t-s)w(s)\|_{X_0 \to X_1} ds \\ &\le \int_0^{t/2} C_1^v \Theta(t-s) C_{01}^w s^{-\alpha} ds + \int_{t/2}^t C_{01}^v (t-s)^{-\alpha} C_0^w \Theta(s) ds \\ &\le C_1^v C_{01}^w \Theta(t/2) \int_0^{t/2} s^{-\alpha} ds + C_1^v C_0^w \Theta(t/2) \int_{t/2}^t (t-s)^{-\alpha} ds \\ &= [C_1^v C_{01}^w + C_{01}^v C_0^w] \Theta(t/2) \frac{t^{1-\alpha}}{1-\alpha}, \end{aligned} $$

with the same obvious notation and we have used again that $\Theta$ is a decaying function.

**Step 4.** Iterating $n := [\alpha]$ times steps 1 and 2, we get that $u^{(n)}$ still satisfies estimate (a) and satisfies the estimate (b) for the exponent $\alpha - [\alpha] \in (0, 1)$. We then conclude that (c) holds with $N := n + 1$ and any $\zeta \in (0, \zeta^*/2)$ by using the third step. $\square$

*Proof of Proposition 7.9.* We classically establish that $\mathcal{B}$ generates a positive semigroup $S_\mathcal{B}$ in both spaces $X_i$ and we thus concentrate on the announced estimates. On the one hand, proceeding as for the proof of (7.64), we have

$$ (7.65) \qquad \int (\mathcal{B}f)(\operatorname{sign} f)m \le -a'' \int |f|(\mathbf{1}_{B_1} + \mathbf{1}_{B_1^c} |x|^{s+\gamma-2})m, $$
for a given source term $G : (0, T) \times \mathcal{O} \to \mathbb{R}$, we wish to define the trace $\gamma g$ of $g$ on the boundary set $(0, T) \times \Sigma$. It is worth emphasizing that the trace will be in fact only defined out of the singular set $\Sigma_0$ and thus only on the boundary set $\Sigma \setminus \Sigma_0$.

We start by recalling several possible definitions of the trace of a function $g$ satisfying (8.9) when

$$ (8.11) \qquad a \in W_{\text{loc}}^{1,s}(\bar{\mathcal{O}}), \quad g \in L_{\text{loc}}^{p}(\bar{\mathcal{O}}), \quad G \in L_{\text{loc}}^{q}(\bar{\mathcal{O}}), \quad s, p, q \in [1, \infty]. $$

Here and below, we denote by $L(E)$ the Lebesgue space of measurable functions $g : E \to \bar{\mathbb{R}} := [-\infty, +\infty]$ with typically $E = \mathcal{O}$ or $E \subset \Sigma$, and by $L^0(E) = L^0(E, \mu) \subset L(E)$ the subset of almost everywhere finite measurable functions on a measurable space $(E, \mathcal{A}, \mu)$.

**Definition 8.1.** We say that a function $g$ on $\mathcal{O}$ satisfying (8.9) and (8.11) admits a trace if one of the following assertions holds true:

* **Extension of the restriction on the boundary.** There exists $\gamma g \in L_{\text{loc}}^r(\Sigma \setminus \Sigma_0)$, $r \in [1, \infty]$, such that

$$ g_{n|\Sigma \setminus \Sigma_0} \to \gamma g \quad \text{in} \quad L_{\text{loc}}^r(\Sigma \setminus \Sigma_0) $$
for any sequence $(g_n)$ satisfying

$$ (8.12) \qquad g_n \in C_c^1(\bar{\mathcal{O}}), \quad g_n \to g \quad \text{in} \quad L_{\text{loc}}^p(\bar{\mathcal{O}}), \quad a(y) \cdot \nabla_y g_n \to G \quad \text{in} \quad L_{\text{loc}}^q(\bar{\mathcal{O}}). $$

* **Characteristics.** There exists a measurable function $\gamma g$ on $\Sigma \setminus \Sigma_0$ such that for a.e. $y \in \mathcal{O}$ satisfying $t_-(y) > -\infty$, there holds

$$ (8.13) \qquad g(y) = \gamma g(Y(t_-(y), y)) + \int_{t_-(y)}^0 G(Y(t, y)) dt, $$
and for a.e. $y \in \mathcal{O}$ satisfying $t_+(y) < \infty$, there holds

$$ (8.14) \qquad g(y) = \gamma g(Y(t_+(y), y)) - \int_0^{t_+(y)} G(Y(t, y)) dt. $$

* **Green formula.** There exists $\gamma g \in L_{\text{loc}}^r(\Sigma \setminus \Sigma_0)$, $r \in [1, \infty]$, such that

$$ (8.15) \qquad \int_{\mathcal{O}} (G \varphi + g \operatorname{div}(a\varphi)) dy = \int_{\Sigma} \gamma g \varphi a(y) \cdot n(y) d\sigma_y, $$
for any $\varphi \in C_c^1(\bar{\mathcal{O}} \setminus \Sigma_0)$.

* **Renormalized Green formula.** There exists a measurable function $\gamma g$ on $\Sigma \setminus \Sigma_0$ such that

$$ (8.16) \qquad \int_{\mathcal{O}} (\beta'(g) G \varphi + \beta(g) \operatorname{div}(a\varphi)) dy = \int_{\Sigma} \beta(\gamma g) \varphi a \cdot n \, d\sigma_y, $$
for any $\varphi \in C_c^1(\bar{\mathcal{O}})$ and any $\beta \in C^1(\mathbb{R})$ such that $\beta' \in L^\infty(\mathbb{R})$.

**Remark 8.2.** (1) In order that the first definition makes sense, we implicitly assume that there exists at least one sequence $(g_n)$ which satisfies (8.12). That last fact corresponds to the density of $C_c^1(\bar{\mathcal{O}})$ in the Sobolev space $\{g \in L^p(\mathcal{O}); a(y) \cdot \nabla_y g \in L^q(\mathcal{O})\}$, which is true as we will see in Lemma 8.5 below under the regularity assumptions made on a and $\mathcal{O}$. It is worth emphasizing that the last convergence in (8.12) may require additional integrability assumption, typically $a \in W^{1,s}(\mathcal{O})$ with $1/r \ge 1/p + 1/s$. Such a definition has been introduced in [37] for a $C^1$ vector field $a$. It is also the point of view adopted by Cessenat in [99, 100] in the case of the neutronic operator, see also [126, chap. XXI] or Agoshkov [2, 3, 4].

(2) In order that the second definition makes sense, we implicitly assume that the set of points $y \in \mathcal{O}$ such that the characteristic $Y_t(y)$ hits the boundary on $\Sigma_0$ has zero measure in $\mathcal{O}$. It is indeed the case thanks to the Sard theorem under enough regularity assumption on a and $\mathcal{O}$, see [37, Prop. 2.3]. It is worth emphasizing that what we really need in order to write (8.13) and (8.14) is that $t \mapsto G(Y(t, y)) \in L^1(t_-(y), t_+(y))$ for a.e. $y \in \mathcal{O}$. We also mention that this characteristics description leads to a layer cake formula linking the integral of a function on the domain to the integral of its trace on the boundary. Such a definition has been widely used in kinetic theory for constructing DiPerna-Lions renormalized solution, see [138, 199, 18] and the references therein. For the classical kinetic operator this trace approach is developed by Arkeryd, Cercignani and co-authors in [86, 96, 16, 17] while for more general (but still regular) vector fields, the approach has been developed in [37, 348, 43, 184] and more recently by Arlotti et al. in [20, 21, 22, 23, 24].
for any nice function $f$ and any weight function $m = m_a$, with $m_a(x) := e^{a\langle x \rangle^\gamma}$, $a \in (a_1, a_2)$, $0 < a_1 < a_2 < 1/(\sqrt{2}\gamma)$, where we define $a'' := a\gamma/2 - (a\gamma)^2$. That exactly means that $\mathcal{B}$ is weakly dissipative in $L_m^1$ as defined in (3.18). From the discussion in Section 3.3, we deduce that $S_B$ is a semigroup of contractions and satisfies the associated decay estimate (3.22), (3.23), and more precisely

$$ (7.66) \qquad \|S_B(t)f\|_{L_{m_a}^1} \le \|f\|_{L_{m_a}^1}, \quad \|S_B(t)f\|_{L_{m_a}^1} \le \Theta_\zeta(t)\|f\|_{L_{m_a'}^1}, $$

for any $a, a' \in (a_1, a_2)$, $a < a'$, $\zeta \in (0, \zeta_*)$, $\zeta_* := (a' - a)^{(2-2\gamma)/(2-\gamma)}(a'\gamma(1 - a'\gamma))^{\gamma/(2-\gamma)}$. We refer to [230, Lem. 2.1] for details. Using that $\mathcal{A}: L^1 \to L_m^1$ is bounded, that establishes (ii) in $X_0$. Similarly, starting from (7.61) and proceeding as in the proof of (7.64), we get

$$ (7.67) \qquad (\mathcal{B}f, f)_{L_m^2} \le -\int |\nabla(fm)|^2 - a'' \int f^2(\mathbf{1}_{B_1} + \mathbf{1}_{B_1^c} |x|^{s+\gamma-2})m^2, $$

for any nice function $f$ and any weight function $m = m_a$ as above. Throwing away the first term at the RHS and arguing as we did in $L_m^1$, we obtain that $S_B$ satisfies

$$ (7.68) \qquad \|S_B(t)f\|_{L_{m_a}^2} \le \|f\|_{L_{m_a}^2}, \quad \|S_B(t)f\|_{L_{m_a}^2} \le \Theta_\zeta(t)\|f\|_{L_{m_a'}^2}, $$

for any $a, a' \in (a_1, a_2)$. Using that $\mathcal{A}: L_{m_a}^2 \to L_{m_a'}^2$ is bounded, that establishes (i) and (ii) in $X_1$. On the other hand, throwing away the second term at the RHS in (7.67), for any trajectory $f_t = S_B(t)f_0$, $f_0$ in the domain of $\mathcal{B}$ in $L_m^2$, we have

$$ \frac{1}{2} \frac{d}{dt} \int_{\mathbb{R}^d} f_t^2 m^2 dx \le - \int_{\mathbb{R}^d} |\nabla(f_t m)|^2 dx. $$

Using Nash's inequality which for some constant $C_N \in (0, \infty)$ stipulates that

$$ \int_{\mathbb{R}^d} g^2 dx \le C_N \left( \int_{\mathbb{R}^d} |\nabla g|^2 dx \right)^{\frac{d}{d+2}} \left( \int_{\mathbb{R}^d} |g| dx \right)^{\frac{4}{d+2}}, \quad \forall g, $$

with $g := f_t m$ and the first estimate in (7.68), we deduce

$$ (7.69) \qquad F'(t) \le -2 C'_N F(t)^{-4/d} G(t)^{1+2/d} \le -2 C'_N F(t)^{-4/d} G(0)^{1+2/d}, $$

with $C'_N := C_N^{-1-2/d}$ and where for brevity of notations we have set

$$ F(t) := \|f_t\|_{L^2(m)}^2, \quad G(t) := \|f_t\|_{L^1(m)}. $$

Integrate the differential inequality (7.69), we find

$$ \|S_B(t)f_0\|_{L_m^2}^2 \lesssim t^{-d/4} \|f_0\|_{L_m^1}, \quad \forall t > 0, $$

and using that $\mathcal{A}: L^1 \to L_m^1$, we next obtain

$$ (7.70) \qquad S_B(t) \mathcal{A} t^{d/4} \in L^\infty(0, \infty; \mathcal{B}(X_0, X_1)). $$

Setting with $u(t) := S_B(t)\mathcal{A}$, we see that $u$ satisfies (a) in Lemma 7.10 thanks to (ii) in $X_0$ and $X_1$. Furthermore, $u$ satisfies (b) in Lemma 7.10 thanks to (7.70). Using Lemma 7.10, we conclude that condition (iii) holds. $\square$

We come back to the proof of (HS3). Gathering (i) and (ii) in $X_1$ in Proposition 7.9, we get that $(S_B\mathcal{A})^{(*\ell)} * S_B \in L_m^\infty(\mathcal{B}(X_1))$ for any $\ell \in \{0, \dots, N-1\}$, $N := [d/4] + 1$. Using that $\Theta \in L^1(0, \infty)$ and (iii) in Proposition 7.9, we deduce that $(S_B\mathcal{A})^{(*N)} \in L^1(0, \infty; \mathcal{B}(X_0, X_1))$.

We may now handle the existence part of the first eigenvalue problem. On the one hand, recalling (H2), we have $L^*\phi_0 \ge 0$ with $\phi_0 = 1$ so that the condition (i) in Theorem 3.4 holds. On the other hand, the condition (ii) in Theorem 3.4 is an immediate consequence of (HS3) as emphasized in Remark 3.5-(1). As a conclusion, the hypotheses of Theorem 3.4 are thus met, and we deduce that there exists $(\lambda_1, f_1) \in \mathbb{R}_+ \times L_{m_+}^2$ solution to the first eigenvalue problem. Because the strong maximum principle (H4) holds, we have $f_1 > 0$ on $\mathbb{R}^d$.

In order to prove the existence of a first positive eigenfunction for the dual problem, we argue in the following way. We start observing that we have the alternative: $\lambda_1 = 0$ or $\lambda_1 > 0$.

- In the first case, we may argue as in Remark 4.18. We indeed have in the same time $L^*\phi_0 \ge 0$ and $\langle L^*\phi_0, f_1 \rangle = \langle \phi_0, L f_1 \rangle = 0$, so that $L^*\phi_0 = 0$ because $f_1 > 0$. The function $\phi_1 := \phi_0$ is thus a solution to the first dual eigenvalue problem.
(3) In order that the third definition makes sense, we need that $a \cdot n \gamma g \in L_{\text{loc}}^1(\Sigma)$ and $p \ge s'$. In some situation, this third definition is in some sense the weakest: it makes sense also when $\gamma g \in M_{\text{loc}}^1(\Sigma \setminus \Sigma_0)$ for instance and can be relevant under the weak assumption $a$, $\text{diva} \in L_{\text{loc}}^{p'}(\bar{O})$ as it is the case in the early works on weak solution to the Vlasov-Poisson equation in [322, 194, 1, 361]. It is also easier to handle than the two first definitions because of the way it connects the function $g$ and its trace.

(4) We will adopt the last definition which extends up to the boundary the renormalization technique introduced in [139]. It is more general and adapted to the weak regularity assumption made on the vector field $a$ than the two first definitions and we recover the third definition by just letting $\beta(s) \to s$ when the conditions of integrability make the limit well defined. Such a kind of definition has been introduced in [270, 272] for kinetic equations and in [70, 10] for transport equations.

We start with a trace result in a $L^\infty$ framework. We denote by $C_{\text{pw}}^1(\mathbb{R})$ the space of continuous functions $\beta: \mathbb{R} \to \mathbb{R}$ with piecewise continuous derivative.

**Theorem 8.3.** Assume that $g \in L^\infty(\mathcal{O})$, $a \in W_{\text{loc}}^{1,1}(\bar{O})$ and $G \in L_{\text{loc}}^1(\bar{O})$ satisfy the transport equation (8.9) in the distributional sense. Then, there exists a unique function

$$ \gamma g \in L^\infty(\Sigma \setminus \Sigma_0; d\sigma_y), \quad \|\gamma g\|_{L^\infty} \le \|g\|_{L^\infty}, $$

which satisfies the renormalized Green formula

$$ (8.17) \qquad \int_{\mathcal{O}} (\beta'(g) G \varphi + \beta(g) \text{div}(a\varphi)) dy = \int_{\Sigma} \beta(\gamma g) \varphi a \cdot n d\sigma_y, $$

for any $\varphi \in C_c^1(\bar{O})$ and any $\beta \in C_{\text{pw}}^1(\mathbb{R})$. As a consequence, renormalization and trace operations commute:

$$ (8.18) \qquad \gamma \beta(g) = \beta(\gamma g), \quad \forall \beta \in C_{\text{pw}}^1(\mathbb{R}). $$

**Remark 8.4.** (1) Because of the very general assumption (8.5) made on the vector field $a: \mathcal{O} \to \mathbb{R}^D$ which is exactly the one made in the DiPerna-Lions theory for transport equation in the whole space developed in [139], the above trace result slightly improves the similar trace result established by Boyer in [70, Thm. 3.1], where an additional assumption $a \cdot n \in L^\zeta(\partial\mathcal{O})$, $\zeta > 1$, is made.

(2) An alternative approach has been developed by Ambrosio and co-authors by assuming weaker bound on Da but stronger bound on $a$. More precisely, denoting by $M_\infty$ the set of vector fields $a \in L^\infty(\mathcal{O})$ such that $\text{diva} \in M^1(\mathcal{O})$, it is established in [10, Prop. 3.2] that there exists a linear and bounded mapping $\text{Tr}: M_\infty(\mathcal{O}) \to L^\infty(\partial\mathcal{O})$ such that $\text{Tra} = n \cdot a|_{\partial\mathcal{O}}$ when $a \in C^1(\bar{O})$. The proof relies on Ambrosio's extension to a BV framework in [8] of the famous Di Perna-Lions improvement [139, Lem. II.1] of Freidrichs' type Lemma on the estimate of the commutator between directional derivative and convolution (see Lemma 8.5 below). Moreover, it is also established in [10] (see in particular [10, Thm. 4.2]) that

$$ \text{Tr}(a\beta(g)) = \beta\left(\frac{\text{Tr}(ag)}{\text{Tra}}\right)\text{Tr}(a), \quad \forall \beta \in C^1(\mathbb{R}), $$

for any $a \in \text{BV}(\mathcal{O}) \cap L^\infty(\mathcal{O})$ and $g \in L^\infty(\mathcal{O})$ such that $ag \in M_\infty$. The above formula is then nothing but (8.18) when $a \in W^{1,1}(\mathcal{O}) \cap L^\infty(\mathcal{O})$.

Before coming to the proof of Theorem 8.3, we state one technical but fundamental result. We define the mollifier $(\rho_\varepsilon)_{\varepsilon>0}$ by

$$ (8.19) \qquad \rho_\varepsilon(z) = \frac{1}{\varepsilon^d} \rho(z/\varepsilon), \quad 0 \le \rho \in \mathcal{D}(\mathbb{R}^d), \quad \text{supp} \rho \subset B_1, \quad \int_{\mathbb{R}^N} \rho(z) dz = 1, $$

and for any $u \in L_{\text{loc}}^1(\bar{O})$, $v_\varepsilon \in C_c(\mathbb{R}^D)$, $\text{supp} v_\varepsilon \subset B_\varepsilon$, we introduce the convolution-translation function $u *_\varepsilon v_\varepsilon$ defined by

$$ (8.20) \qquad (u *_\varepsilon v_\varepsilon)(y) := \int_{\mathcal{O}} u(z) v_\varepsilon(y - 2\varepsilon n(y) - z) dz. $$

**Lemma 8.5.** For $g \in L_{\text{loc}}^p(\bar{O})$, $p \in [1, \infty]$, $a \in W_{\text{loc}}^{1,p'}(\bar{O})$ and $G \in L_{\text{loc}}^1(\bar{O})$ satisfying (8.9) in the distributional sense, the sequence $(g_\varepsilon)$ defined by $g_\varepsilon := g *_\varepsilon \rho_\varepsilon$ satisfies

$$ g_\varepsilon \in W_{\text{loc}}^{1,\infty}(\bar{O}), \quad G_\varepsilon := a \cdot \nabla g_\varepsilon \to a \cdot \nabla g \quad \text{in } L_{\text{loc}}^1(\bar{O}), $$
- In the second case $\lambda_1 > 0$, we may argue as in the case $\gamma = 1$ above. On the one hand, the operator $\mathcal{R}_B(\alpha)$ is uniformly bounded in $L_m^2$ for any $\alpha \ge \kappa_B := \lambda_1/2 > 0$ and on the other hand the operator $\mathcal{W}(\alpha) := \mathcal{R}_B(\alpha)A : L_m^2 \to H_m^1 \cap L_{m^*}^2$ is uniformly bounded for any $\alpha \ge \kappa_B$ with $m = o(m^*)$, so that $H_m^1 \cap L_{m^*}^2 \subset L_m^2$ is compact. We may thus apply Theorem 2.21 and we conclude to the existence of a solution $(\lambda'_1, f'_1, \phi'_1)$ to the eigentriplet problem.

The conditions (H4) and (H5) being true in a general situation as well as the conclusions of Lemma 7.1, as an intermediate conclusion, we have established under the general condition $\gamma > 0$ in (7.56) that yet the same conclusions as in section 7.2 hold true.

**Quantitative stability.** We now establish a quantitative stability estimate using the Doblinitz-Harris approach presented in Section 6 and yet used in the case of a bounded domain in Section 7.1. We first consider the more difficult case $\gamma \in (0,1)$, so that $\lambda_1 \ge \kappa_0 = 0$, and then explain the modifications to be made in order to deal with the case $\gamma \ge 1$. As explained just above, $\lambda_1 = 0$ corresponds to the conservative case $(\lambda_1, \phi_1) = (0,1)$ which has been considered in [230]. We thus focus on the case $\lambda_1 > 0$ for which an adapted version of Theorem 7.3 already imply the exponential asymptotic stability (E3$_1$) in $L_m^2$ with non constructive rate. We do not develop further this argument but rather establish a constructive sub-exponential asymptotic stability.

**Step1 - Lyapunov condition.** We take $m = e^{a|x|^\gamma}$ with $0 < a < \gamma^{-1}$. From Lemma 7.8 or a direct computation, we have

$$ \begin{aligned} \mathcal{L}^* m &= \Delta m + (c - \operatorname{div} b)m - b \cdot \nabla m \\ &\le (C\langle x \rangle^{\gamma-2} + |c| - 2a^*\langle x \rangle^{2\gamma-2})m \\ &\le C_0 \mathbf{1}_{B_{\varepsilon_0}} - a^*\langle x \rangle^{2\gamma-2}m, \end{aligned} $$

for three positive constants $C = C(d)$, $C_0 = C_0(c, C, a, \gamma)$, $\varrho_0 = \varrho_0(c, C, a, \gamma)$ and with $a^* := (a\gamma - (a\gamma)^2)/2 > 0$. We now set $m_1 := m$ and $m_0 := a^*\langle x \rangle^{2\gamma-2}m$. We fix $T > 0$ and for $0 \le f_0 \in L_m^1$, we denote $f_t := \tilde{S}_L(t)f_0$. Recording that $\lambda_1 \ge 0$ and using the above pointwise estimate, we deduce

$$ (7.71) \qquad \int f_T m_1 + \int_0^T \int f_t m_0 dt \le \int f_0 m_1 + C_1 \int_0^T \int_{B_{\varepsilon_0}} f_t dt. $$

Because the same kind of pointwise estimate holds for $\mathcal{L}^*m_0$, we have

$$ \int f_T m_0 \le \int f_t m_0 + C_1 \int_t^T \int_{B_{\varepsilon_0}} f_s ds $$

and integrating in time, we get

$$ T \int f_T m_0 \le \int_0^T \int f_t m_0 dt + C_0 \int_0^T s \int_{B_{\varepsilon_0}} f_s ds. $$

Coming back to the first estimate, we deduce

$$ (7.72) \qquad \int f_T m_1 + T \int f_T m_0 \le \int f_0 m_1 + (C_1 + C_0 T) \int_0^T \int_{B_{\varepsilon_0}} f_t dt. $$

**Step 2 - Pointwise estimates on $\phi_1$.** We define $\mathcal{B} := \mathcal{L} - C_0\chi_{\varepsilon_0}$ which is the generator of a positive semigroup of contraction in $L_m^1$ because of the above discussion. For $\lambda > 0$, $0 \le g \in L_m^1$ and $0 \le f \in L_m^1$ the solution to $(\lambda - \mathcal{B})f = g$, we compute

$$ \int gm = \int f(\lambda - \mathcal{B}^*)m \ge \int f(\lambda m + m_0) \ge \int fm_0, $$

from what we deduce

$$ \|\mathcal{R}_B(\lambda)f\|_{L_{m_0}^1} \le \|f\|_{L_m^1}, \quad \forall f \in L_m^1. $$

Now, we consider two weight functions $m_1$ and $m_3$ with $m_i := e^{a_i|x|^\gamma}$, $0 < a_1 < a_3 < \gamma^{-1}$, we denote $m_0 := a_1^*\langle x \rangle^{2\gamma-2}m_1$ and we compute

$$ \begin{aligned} \|\mathcal{A}\mathcal{R}_B(\lambda)f\|_{L_{m_3}^1} &\le C_0\|\mathcal{R}_B(\lambda)f\|_{L_{m_3}^1(B_{2\varepsilon_0})} \\ &\le C_1\|\mathcal{R}_B(\lambda)f\|_{L_{m_0}^1} \le C_1\|f\|_{L_{m_1}^1}. \end{aligned} $$
as $\varepsilon \to 0$, and

$$g_{\varepsilon} \to g \text{ in } L_{\text{loc}}^{p}(\bar{O}), \quad \text{if } p < \infty,$$

$$g_{\varepsilon} \to g \text{ in } L_{\text{loc}}^{1}(\bar{O}), \quad (g_{\varepsilon}) \text{ bounded in } L_{\text{loc}}^{\infty}(\bar{O}), \quad \text{if } p = \infty.$$

We skip the proof of Lemma 8.5 since it follows by just repeating the proofs of [139, Lem. II.1], [271, Lem. 1] or [70, Lem. 3.1].

**Proof of Theorem 8.3.** Let us fix $\chi \in \mathcal{D}(\bar{O})$ such that $0 \le \chi \le 1$ and denote $R > 0$ a real number such that $\text{supp} \chi \subset B_R$. We observe that $\chi\text{sign}(a \cdot n) \in L^1(\Sigma)$. From Gagliardo trace theorem [170, Teor. 1.II], there exists $\psi \in W^{1,1}(\mathcal{O})$ such that $\gamma\psi = \chi\text{sign}(a \cdot n)$ and $\text{supp} \psi \subset B_R$. Denoting $T_1: \mathbb{R} \to [-1, 1]$ the truncation function which is odd and is defined by $T_1(\sigma) = \sigma \wedge 1$ for any $\sigma \ge 0$, we see that $\gamma T_1(\psi) = T_1(\gamma\psi) = \gamma\psi$, and thus we may assume $\psi \in L^\infty(\mathcal{O})$ up to replacing $\psi$ by $T_1(\psi)$. As a consequence, there exists a sequence $(\psi_k)$ of $W^{1,\infty}(\mathcal{O})$ such that $\psi_k \to \psi$ in $W^{1,1}(\mathcal{O})$, with $(\psi_k)$ bounded in $L^\infty(\mathcal{O})$, $\text{supp}\psi_k \subset B_R$, and $\gamma\psi_k \to \chi\text{sign}(a \cdot n)$ in $L^1(\Sigma)$, with $(\gamma\psi_k)$ bounded in $L^\infty(\Sigma)$.

Let us then consider the sequences $(g_\varepsilon)$ and $(G_\varepsilon)$ defined in Lemma 8.5. The classical Green formula for Lipschitz functions writes

$$\begin{align*}
\int_{\Sigma} (g_{\varepsilon|\Sigma} - g_{\varepsilon'|\Sigma})^2 |a \cdot n| \chi \, d\sigma_y \\
&= \int_{\Sigma} (g_{\varepsilon|\Sigma} - g_{\varepsilon'|\Sigma})^2 a \cdot n \psi_k \, d\sigma_y + \int_{\Sigma} (g_{\varepsilon|\Sigma} - g_{\varepsilon'|\Sigma})^2 [|a \cdot n| \chi - a \cdot n \psi_k] \, d\sigma_y \\
&= \int_{\mathcal{O}} [2(g_{\varepsilon} - g_{\varepsilon'}) (G_{\varepsilon} - G_{\varepsilon'}) \psi_k \, dy + (g_{\varepsilon} - g_{\varepsilon'})^2 \text{div}(a\psi_k)] \, dy \\
&\quad + \int_{\Sigma} (g_{\varepsilon|\Sigma} - g_{\varepsilon'|\Sigma})^2 [|a \cdot n| \chi - a \cdot n \psi_k] \, d\sigma_y \\
&\le 4 \|\psi_k\|_{L^\infty} \|g\|_{L^\infty} \|G_{\varepsilon} - G_{\varepsilon'}\|_{L^1(B_R)} + \|\psi_k\|_{W^{1,\infty}} \int_{B_R} (|a| + |\text{div}a|)(g_{\varepsilon} - g_{\varepsilon'})^2 \, dy \\
&\quad + 2\|g\|_{L^\infty}^2 \|(a \cdot n)\gamma\psi_k - \chi|a \cdot n\|_{L^1(\Sigma)},
\end{align*}$$

for any $\varepsilon > 0$ and $k \ge 1$. We deduce that $(g_{\varepsilon|\Sigma})$ is a Cauchy sequence in $L^2(|a \cdot n| \chi d\sigma)$. From the fact that $(g_\varepsilon)$ is bounded in $L^\infty(\mathcal{O})$, we deduce that the sequence $(\gamma g_\varepsilon)$ is also bounded in $L^\infty(\Sigma)$. As a consequence, there exists a function $\gamma g \in L^\infty(\Sigma)$ such that $\gamma g_\varepsilon \to \gamma g$ in $L^2(|a \cdot n| \chi d\sigma)$. Next, we may write the Green formula

$$\int_{\mathcal{O}} [G_{\varepsilon} \varphi + g_{\varepsilon} \text{div}(a\varphi)] \, dy = \int_{\Sigma} \gamma g_{\varepsilon} \varphi a \cdot n \, d\sigma_y,$$

for any test function $\varphi \in C_c^1(\bar{O})$, and we may pass to the limit as $\varepsilon \to 0$. We deduce that the Green formula

$$(8.21) \qquad \int_{\mathcal{O}} (G \varphi + g \operatorname{div}(a\varphi)) \, dy = \int_{\Sigma} \gamma g \varphi a(y) \cdot n(y) \, d\sigma_y,$$

holds for any $\varphi \in C_c^1(\bar{O})$. That clearly uniquely defines the trace function $\gamma g$ on $\Sigma \setminus \Sigma_0$.

Now, on the one hand, from the DiPerna-Lions renormalizing theory [139, proof of Corollary II.1], we know that $\beta(g) \in L^\infty(\mathcal{O})$ satisfies the transport equation

$$(8.22) \qquad a(y) \cdot \nabla_y \beta(g) = \beta'(g)G \quad \text{in } \mathcal{D}'(\mathcal{O}),$$

for any renormalizing function $\beta \in \text{Lip}(\mathbb{R})$ and any test function $\varphi \in C_c^1(\mathcal{O})$. Using the already established trace result, we know that there exists $\gamma\beta(g) \in L^\infty(\Sigma \setminus \Sigma_0)$ such that

$$(8.23) \qquad \int_{\mathcal{O}} [\beta'(g)G\varphi + \beta(g)\operatorname{div}(a\varphi)] \, dy = \int_{\Sigma} \gamma\beta(g)\varphi a \cdot n \, d\sigma_y,$$

for any test function $\varphi \in C_c^1(\bar{O})$. On the other hand, from the classical Green formula for Lipschitz functions and because $\beta(g_\varepsilon)|_\Sigma = \beta(g_{\varepsilon|\Sigma})$, we have

$$\int_{\mathcal{O}} [\beta'(g_{\varepsilon})G_{\varepsilon} \varphi + \beta(g_{\varepsilon})\operatorname{div}(a\varphi)] \, dy = \int_{\Sigma} \beta(g_{\varepsilon|\Sigma}) \varphi a \cdot n \, d\sigma_y,$$
By duality, we obtain

$$ (7.73) \qquad \|\mathcal{R}_{B^*}(\lambda)\phi\|_{L_{m_1}^{\infty}} \le \|\phi\|_{L_{m_0}^{\infty}} \quad \text{and} \quad \|\mathcal{R}_{B^*} A^*(\lambda)\phi\|_{L_{m_1}^{\infty}} \le C_1 \|\phi\|_{L_{m_3}^{\infty}}, $$

for any $\lambda > 0$ and $\phi \in L_{m_0}^{\infty}$. We also deduce from Proposition 7.9 the regularization estimate $(A^*\mathcal{R}_{B^*})^N : L_{m_1}^2 \to L^{\infty}$. Let us now consider $0 \le \phi_1 \in L_{m_1}^2$ the first eigenvector for the dual problem built in the preceding paragraph. From the eigenvalue equation

$$ B^*\phi_1 + A^*\phi_1 = \mathcal{L}^*\phi_1 = \lambda_1\phi_1, $$

we deduce that $\phi_1 = (\mathcal{R}_{B^*} A^*)\phi_1$, and iterating

$$ \phi_1 = (\mathcal{R}_{B^*} A^*)^{N+1} \phi_1 = \mathcal{R}_{B^*} (A^* \mathcal{R}_{B^*})^N A^* \phi_1. $$

From the above regularization estimate and the first estimate in (7.73), we thus deduce that $\phi_1 \in L_{m_1}^{\infty}$. Moreover, normalizing $\phi_1$ and using the second estimate in (7.73), we may obtain

$$ (7.74) \qquad \|\phi_1\|_{L_{m_3-1}^{\infty}} = 1 \quad \text{and} \quad \|\phi\|_{L_{m_1-1}^{\infty}} \le C_1. $$

We deduce

$$ \begin{aligned} 1 &= \max\left(\sup_{B_{\varrho_2}} \frac{|\phi_1|}{m_3}, \sup_{B_{\varrho_2}^c} \frac{|\phi_1|}{m_3}\right) \\ &\le \max\left(\sup_{B_{\varrho_2}} \frac{|\phi_1|}{m_3}, C_{0,1} \sup_{B_{\varrho_2}^c} \frac{m_1}{m_3}\right), \end{aligned} $$

so that $\sup_{B_{\varrho_2}} \frac{|\phi_1|}{m_3} = 1$ by choosing $\varrho_2 := \max(\varrho_0, \varrho_1)$ with $C_1 e^{(a_1-a_3)\varrho_1^\gamma} = 1$. As a consequence, there exists $x_0 \in B_R$ such that $\phi_1(x_0) \ge 1$. On the other hand, using standard regularity result for elliptic equation in the ball $B_{2R}$, we obtain that $\phi_1 \in C^{0,1}(B_R) \cap W^{2,p}(B_R)$ for any $p \in [1, \infty)$ with constructive bound. Making use next of the Harnack inequality as at the end of Section 7.1 or using barrier functions as in the proof of [230, Lem. 6.2], we classically deduce that

$$ (7.75) \qquad \phi_1 \ge z_{\varrho} \mathbf{1}_{B_{\varrho}}, \quad \forall \varrho > 0, $$

for a constructive constant $z_{\varrho} > 0$ (where we emphasize here and below the $\phi_1$ always denote the normalized by (7.74) dual eigenvector).

**Step 3 - Doblin-Harris estimate.** We fix $T > 0$ (for instance $T := 1$) and $A > 0$ arbitrary. For $0 \le f_0 \in L_m^1$ such that $\|f_0\|_{L_m^1} \le A\|f_0\|_{L_{\phi_1}^1}$, we denote $f_t := \tilde{S}_t(t)f_0$. On the one hand, we have

$$ \int f_t \phi_1 = \int f_0 \phi_1 \quad \text{and} \quad \int f_t m \le C_T \int f_0 m, $$

for any $t \in [0, T]$, the second estimate being an immediate consequence of (7.71). On the other hand, we define $\varepsilon(r) := \sup_{|x| \ge r}(m(x)/\phi_1(x))$ and we compute

$$ \begin{aligned} \int_{B_{\rho}} f_t \phi_1 &= \int f_t \phi_1 - \int_{B_{\rho}^c} f_t \phi_1 \ge \int f_t \phi_1 - \varepsilon(\rho) \int f_t m \\ &\ge \int f_0 \phi_1 - C_T \varepsilon(\rho) \int f_0 m \ge \left(1 - \frac{C_T}{A} \varepsilon(\rho)\right) \int f_0 \phi_1 \ge \frac{1}{2} \int f_0 \phi_1, \end{aligned} $$

for any $t \in (0, T)$, by choosing $\rho := \rho(T, A) > 0$ large enough. In particular, there exists $x_0(t) \in B_{\rho}$ such that

$$ f(t, x_0(t)) \ge \vartheta := \frac{1}{2} \frac{1}{\|\phi_1\|_{L^1(B_{\rho})}} \int f_0 \phi_1. $$

Next, arguing exactly as in Section 7.1 or as in the proof of [230, Lem. 6.2], we deduce

$$ (7.76) \qquad \tilde{S}_T f_0 \ge \eta_{T,A} \mathbf{1}_{B_1}[f_0]_{L_{\phi_1}^1}, $$

for some constructive constant $\eta_{T,A} > 0$.

In conclusion, using Theorem 6.4 or Theorem 6.7, we have established the following result.
for any renormalizing function $\beta \in \text{Lip}(\mathbb{R})$ and any test function $\varphi \in C_c^1(\bar{\mathcal{O}})$. Using that

$$ \beta'(g_{\varepsilon})G_{\varepsilon} \rightarrow \beta'(g)G, \quad \beta(g_{\varepsilon}) \rightarrow \beta(g), \quad \beta(g_{\varepsilon|\Sigma}) \rightarrow \beta(\gamma g) $$

respectively in $L_{\text{loc}}^1(\bar{\mathcal{O}})$ and in $L_{\text{loc}}^1(\Sigma)$, and that the two last sequences are bounded in $L^{\infty}$, we may pass to the limit $\varepsilon \to 0$ in the last Green formula, and we thus get

$$ \int_{\mathcal{O}} [\beta'(g)G\varphi + \beta(g)\text{div}(a\varphi)] dy = \int_{\Sigma} \beta(\gamma g) \varphi a \cdot n d\sigma_y. $$

Together with (8.23) and by uniqueness of the trace function, we conclude to $\gamma\beta(g) = \beta(\gamma g)$. $\square$

Let us state several variants of the preceding trace result. For the transport evolution equation (8.10) a first possible trace result writes as follows.

**Theorem 8.6.** Assume that $g \in L^{\infty}((0, T) \times \mathcal{O})$, $a \in L^1(0, T; W_{\text{loc}}^{1,1}(\bar{\mathcal{O}}))$ and $G \in L_{\text{loc}}^1([0, T] \times \bar{\mathcal{O}})$ satisfy the evolution transport equation (8.10) in the distributional sense. Then,

$$ g \in C([0, T]; L_{\text{loc}}^1(\bar{\mathcal{O}})) $$

and there exists a unique function

$$ \gamma g \in L^{\infty}((0, T) \times \Sigma \setminus \Sigma_0; dt \otimes d\sigma_y), \quad \|\gamma g\|_{L^{\infty}} \le \|g\|_{L^{\infty}}, $$

which satisfies the renormalized Green formula

$$ (8.24) \qquad \int_{t_0}^{t_1} \int_{\mathcal{O}} (\beta'(g) G \varphi + \beta(g) [\partial_t \varphi + \text{div}(a\varphi)]) dydt \\ = \left[ \int_{\mathcal{O}} \beta(g(t, \cdot)) \varphi dy \right]_{t_0}^{t_1} + \int_{t_0}^{t_1} \int_{\Sigma} \beta(\gamma g) \varphi a \cdot n d\sigma_y dt, $$

for any $\varphi \in C_c^1([0, T] \times \bar{\mathcal{O}})$, any $t_0, t_1 \in [0, T]$ and any $\beta \in C_{\text{pw}}^1(\mathbb{R})$. In particular renormalization and trace operations commute: (8.18) holds.

We skip the proof of Theorem 8.6 which is very similar to the proof Theorem 8.3 using the slight modifications that one can find in [271, Thm. 2] or [70, Thm. 3.1]. Under the slightly more regularity assumption $a \in W_{\text{loc}}^{1,1}([0, T] \times \bar{\mathcal{O}})$, Theorem 8.6 is a direct corollary of Theorem 8.3 applied to the for field $(1, a(t, y))$ on the open set $(0, T) \times \mathcal{O}$.

For some additional function $b: \mathcal{O} \to \mathbb{R}$, another possible variant is the following trace result for the stationary transport equation

$$ (8.25) \qquad a \cdot \nabla_y g + bg = G \quad \text{in } \mathcal{O} $$

in the renormalized framework as introduced by DiPerna and Lions in [139]. Assuming $a \in W_{\text{loc}}^{1,1}(\bar{\mathcal{O}})$, $b, G \in L_{\text{loc}}^1(\bar{\mathcal{O}})$, we say that $g \in L_{\text{loc}}^1(\bar{\mathcal{O}})$ is a renormalized solution to the transport equation (8.25) if

$$ (8.26) \qquad a \cdot \nabla_y \beta(g) + b\beta'(g)g = \beta'(g)G, $$

in the distributional sense for any renormalizing function $\beta \in C_*^1(\mathbb{R})$ the set of $C^1(\mathbb{R})$ functions such that $\beta$ admits some finite limits in $\pm\infty$ and $s \mapsto \langle s \rangle \beta'(s)$ is bounded on $\mathbb{R}$, in particular $C_*^1(\mathbb{R}) \subset C_b^1(\mathbb{R})$. We also denote by $\beta \in C_{\text{pw},*}^1(\mathbb{R})$ the $C^1$ piecewise variant of $C_*^1(\mathbb{R})$. We will repeatedly use the family of functions $\beta_\delta \in C_*^1(\mathbb{R})$ defined by $\beta_\delta(s) := s/(1 + \delta s^2)^{1/2}$ for any $\delta \in (0, 1]$. We observe that $\beta'_\delta(s) = (1 + \delta s^2)^{-3/2}$, so that $s\beta'(s) \to 0$ as $s \to \pm\infty$.

Let us start formulating some basic facts on renormalized solutions to equation (8.25).

**Lemma 8.7.** Assume $a \in W_{\text{loc}}^{1,1}(\bar{\mathcal{O}})$, $b, G \in L_{\text{loc}}^1(\bar{\mathcal{O}})$.

(1) If $g \in L_{\text{loc}}^1(\bar{\mathcal{O}})$ and $\alpha(g)$ satisfies equation (8.26) for one renormalizing function $\alpha: \mathbb{R} \to (-1, 1)$ which is bijective and belongs to $C_{\text{pw},*}^1(\mathbb{R})$ then $\beta(g)$ satisfies equation (8.26) for any renormalizing function $\beta \in C_{\text{pw},*}^1(\mathbb{R})$.

(2) If $g_1, g_2 \in L_{\text{loc}}^1(\bar{\mathcal{O}})$ are two renormalized solutions to the transport equations

$$ a \cdot \nabla_y g_i + bg_i = G_i \in L_{\text{loc}}^1(\bar{\mathcal{O}}), $$

then $g := g_1 + g_2$ is a renormalized solution to the transport equation (8.25) with $G := G_1 + G_2$.
**Theorem 7.11.** Consider the elliptic operator (7.50) in the whole space and assume that the coefficients b and c satisfy (7.56), (7.57), (7.58). Then the associated semigroup satisfies the quantitative estimate $(\mathbf{E}3_1)$ when $\gamma \ge 1$ or $(\mathbf{E}3_2)$ when $\gamma \in (0, 1)$ in the Lebesgue spaces $L_m^2$, with the exponential weight function $m$ as defined above, in particular $(\mathbf{S}1)$, $(\mathbf{S}2)$ and $(\mathbf{S}3_3)$ hold true.

We are not aware of any result of this kind in a non conservative framework ($c \neq \text{div}b$). Our result generalizes to a non conservative framework the previous known results in a conservative framework, see for instance [27, 230] and the references therein.

## 8. TRANSPORT EQUATIONS

The main aim of this part is to investigate the long time asymptotic of the solutions to the transport equation

$$ (8.1) \qquad \partial_t f + \text{div}_y(af) = \mathcal{K}[f] - Kf \quad \text{in } (0, \infty) \times \mathcal{O}, $$

on the function $f = f(t, y)$, $t \ge 0$, $y \in \mathcal{O}$, with $\mathcal{O} \subset \mathbb{R}^D$, $D \ge 1$, a smooth open connected set. We assume that $a = a(y)$, $a: \mathcal{O} \to \mathbb{R}^D$, $K = K(y)$, $K: \mathcal{O} \to \mathbb{R}_+$ and that the collision operator $\mathcal{K}$ is linear and defined by

$$ (8.2) \qquad \mathcal{K}[g](y) := \int_{\mathcal{O}} k g_* dy_*, $$

for some kernel $k: \mathcal{O} \times \mathcal{O} \to \mathbb{R}_+$ and for any (conveniently) bounded function $g: \mathcal{O} \to \mathbb{R}$. Here and below, we use the common shorthands

$$ g_* := g(y_*), \quad k := k(y, y_*), \quad k_* := k(y_*, y). $$

When $\mathcal{O} \neq \mathbb{R}^D$, the equation is complemented with a boundary condition which imposes the value of the trace $\gamma_-f$ of $f$ on the incoming subsets of the boundary and takes the form

$$ (8.3) \qquad (\gamma_-f)(t, y) = \mathcal{R}[f(t, \cdot), \gamma_+f(t, \cdot)](y) \quad \text{on } (0, \infty) \times \Sigma_- $$

Let us explain the meaning of the different terms involved in (8.3). We denote by $\Sigma := \partial\mathcal{O}$ the boundary set, by $d\sigma_y$ the Lebesgue measure on $\Sigma$, by $n: \Sigma \to \mathbb{S}^{D-1}$ the normal outward vector field, we write $n = n_y = n(y)$, and by $\Sigma_-$ the incoming, $\Sigma_+$ the outgoing and $\Sigma_0$ the singular subsets of the boundary defined by

$$ \Sigma_{\pm} := \{y \in \Sigma; \pm a(y) \cdot n_y > 0\}, \quad \Sigma_0 := \{y \in \Sigma; a(y) \cdot n_y = 0\}. $$

We denote $\gamma f = f|_{(0,\infty)\times\Sigma}$ the trace of $f$ and $\gamma_{\pm}f := \mathbf{1}_{(0,\infty)\times\Sigma_{\pm}} \gamma f$ the trace restrictions on the incoming and outgoing sets. We then assume that the boundary operator $\mathcal{R}$ splits into two pieces $\mathcal{R}[g, h] = \mathcal{R}_{\mathcal{O}}[g] + \mathcal{R}_{\Sigma}[h]$, where

$$ (8.4) \qquad \mathcal{R}_{\mathcal{O}}[g](y) = \int_{\mathcal{O}} g(y_*)r_{\mathcal{O}}(y, dy_*), \quad \mathcal{R}_{\Sigma}[h](y) = \int_{\Sigma_+} h(y_*)r_{\Sigma}(y, dy_*), $$

for a domain transition kernel $r_{\mathcal{O}}: \Sigma_- \times \mathcal{B}_{\mathcal{O}} \to [0, \infty]$, a boundary transition kernel $r_{\Sigma}: \Sigma_- \times \mathcal{B}_{\Sigma_+} \to [0, \infty]$ and for any (conveniently) bounded functions $g: \mathcal{O} \to \mathbb{R}$ and $h: \Sigma_+ \to \mathbb{R}$, where $\mathcal{B}_E$ stands for the set of Borel subsets of $E$.

In the next sections we will first consider the trace problem for a general force field $a$ and next the well-posedness for the transport equation with given inflow at the boundary and with reflection condition at the boundary. We will also revisit the characteristic method for general force field $a$. We will next consider the Krein-Rutman problem still for a general force field $a$, but making strong simplification assumptions on the kernel operators $\mathcal{K}$ and $\mathcal{R}$. We will finally explain how the classical age structured equation falls into the present framework. We will come back to more specific physical situations concerning the growth-fragmentation equation and the kinetic relaxation equation with more general and physically relevant hypothesis on the kernel in parts 9 and 10.
(3) If $g$ is a renormalized solution to the transport equation (8.25) and $\Phi, c \in L^\infty(\mathcal{O})$ satisfy

$$a \cdot \nabla_y \Phi = c$$

in the distributional sense, then $h := ge^{-\Phi}$ satisfies

$$(8.27) \qquad a \cdot \nabla_y h + (b+c)h = Ge^{-\Phi}$$

in the renormalized sense.

**Proof of Lemma 8.7.** We briefly sketch the proof and for more details we refer to [139], in particular to [139, Lem. II.2]. It is worth mentioning that only the case $b \in L^\infty(\mathcal{O})$ is considered in [139], but it readily extends to our framework. Assertion (1) is just a consequence of the chain rule $\beta'(s) = (\beta \circ \alpha^{-1})'(\alpha(s))\alpha'(s)$ for smooth enough solutions and thus for any solution thanks to Lemma 8.5 (see the proof of [139, Cor. II.1]) and to a standard approximation procedure in order to deal with piecewise $C^1$ functions. In order to establish (2), we consider two renormalized solutions $g_i$, a renormalized function $\beta \in C_*^1(\mathbb{R})$ and we write

$$a \cdot \nabla_y \beta(\beta_\delta(g_1) + \beta_\delta(g_2)) = \beta'(\beta_\delta(g_1) + \beta_\delta(g_2))[(G_1 - bg_1)\beta'_\delta(g_1) + (G_2 - bg_2)\beta'_\delta(g_2)],$$

where we have added the two renormalized formulations (8.26) associated to $\beta_\delta(g_i)$ and renormalized once more the resulting solution using (1). Letting $\delta \to 0$, we immediately obtain

$$a \cdot \nabla_y \beta(g_1 + g_2) = \beta'(g_1 + g_2)[G_1 + G_2 - b(g_1 + g_2)]$$

in the distributional sense. For proving (3), we introduce the mollified sequence $(g_\epsilon)$ and $(\Phi_\epsilon)$ defined as in the statement of Lemma 8.5 so that

$$a \cdot \nabla g_\epsilon + bg_\epsilon = G_\epsilon, \quad a \cdot \nabla \Phi_\epsilon = c_\epsilon$$

with $G_\epsilon \to G$ and $c_\epsilon \to c$ in $L^1_{\text{loc}}(\bar{\mathcal{O}})$ as $\epsilon \to 0$. The smooth function $h_\epsilon := g_\epsilon e^{-\Phi_\epsilon}$ satisfies

$$a \cdot \nabla_y h_\epsilon + (b+c_\epsilon)h_\epsilon = G_\epsilon e^{-\Phi_\epsilon}$$

and then

$$a \cdot \nabla_y \beta(h_\epsilon) + \beta'(h_\epsilon)(b+c_\epsilon)h_\epsilon = \beta'(h_\epsilon)G_\epsilon e^{-\Phi_\epsilon}$$

for any $\beta \in C_*^1(\mathbb{R})$. Passing to the limit $\epsilon \to 0$, we obtain the renormalized formulation of (8.27). $\square$

We now generalize the trace result to the framework of renormalized solutions.

**Theorem 8.8.** Assume that $a \in W_{\text{loc}}^{1,1}(\bar{\mathcal{O}})$, $b, G \in L^1_{\text{loc}}(\bar{\mathcal{O}})$ and that $g \in L^1_{\text{loc}}(\bar{\mathcal{O}})$ is a renormalized solution to the transport equation (8.25). Then there exists a unique function

$$\gamma g \in L(\Sigma \setminus \Sigma_0; d\sigma_y)$$

which satisfies the renormalized Green formula

$$(8.28) \qquad \int_{\mathcal{O}} (\beta'(g)(G - bg) \varphi + \beta(g) \text{div}(a\varphi)) dy = \int_{\Sigma} \beta(\gamma g) \varphi a \cdot n \, d\sigma_y,$$

for any $\varphi \in C_c^1(\bar{\mathcal{O}})$ and any $\beta \in C_{\text{pw},*}^1(\mathbb{R})$.

**Proof of Theorem 8.8.** We fix $\beta_1: \mathbb{R} \to \mathbb{R}$ defined by $\beta_1(s) := s(1+s^2)^{-1/2}$, so that $\beta_1 \in C_b^1(\mathbb{R})$ and $\beta_1: \mathbb{R} \to (-1, 1)$ is a bijection. Since then $\beta_1(g) \in L^\infty(\mathcal{O})$ and $\beta'(g)(G - bg) \in L^1_{\text{loc}}(\bar{\mathcal{O}})$, we know from Theorem 8.3 that $\gamma\beta_1(g)$ is well defined in $L^\infty(\Sigma \setminus \Sigma_0)$ through the Green formula

$$\int_{\mathcal{O}} [\beta_1'(g)(G - bg) \varphi + \beta_1(g) \text{div}(a\varphi)] dy = \int_{\Sigma} \gamma \beta_1(g) \varphi a \cdot n \, d\sigma_y,$$

for any test function $\varphi \in C_c^1(\bar{\mathcal{O}})$. We set $\gamma g := \beta_1^{-1}(\gamma \beta_1(g)) \in L(\Sigma \setminus \Sigma_0; d\sigma_y)$, with the convention $\beta_1^{-1}(\pm 1) = \pm \infty$. For any $\beta \in C_{\text{pw},*}^1(\mathbb{R})$, we then deduce

$$\gamma \beta(g) = \gamma[(\beta \circ \beta_1^{-1})(\beta_1(g))] = \beta \circ \beta_1^{-1}(\gamma \beta_1(g)) = \beta(\gamma g),$$

where we have used the renormalization result stated in Theorem 8.3 and the chain rule (1) stated in Lemma 8.7 in the second equality and the very definition of $\gamma g$ in the third equality. In other words, the renormalized Green formula (8.28) holds. $\square$
## 8.1. The trace problem.

In this section, we are concerned with the trace problem associated to a (mainly stationary) transport equation for a general vector field $a: \mathcal{O} \to \mathbb{R}^D$ for which we only assume

$$ (8.5) \qquad a \in W_{\text{loc}}^{1,1}(\bar{\mathcal{O}}), $$

where we recall that $\mathcal{O} \subset \mathbb{R}^D$, $D \ge 1$, is a smooth open connected set. The regularity needed on the domain is formulated in the following way: we assume that there exists $n: \mathcal{O} \to \mathbb{R}^D$, $y \mapsto n(y)$ a vector field belonging to $W^{1,\infty}(\mathcal{O})$ and which coincides with the previously defined unit outgoing normal vector field on $\Sigma$ and satisfies $\|n\|_{L^\infty} = 1$. In that situation, it is well-known that the above vector field is the restriction of a vector field $a \in W_{\text{loc}}^{1,1}(\mathbb{R}^D)$ (where we abuse notations denoting the restriction and the extension in the same way). We also consider the associated differential equation

$$ (8.6) \qquad \frac{dY}{dt} = a(Y), \quad Y(0) = y, $$

and then define the characteristic flow $Y_t = Y(t, y)$, for any $y \in \mathcal{O}$, which is the solution to (8.6) on a maximal time interval $(t_-(y), t_+(y))$ where $t_-(y) < 0 < t_+(y)$ are defined by $t_- := -t_b$ and $t_+ := t_f$, the backward exit time is defined by

$$ (8.7) \qquad t_b(y) := \sup\{\tau > 0; Y_{-\tau}(y) \in \mathcal{O}, \forall t \in [0, \tau]\} \in (0, +\infty] $$

and the forward exit time is defined by

$$ (8.8) \qquad t_f(y) := \sup\{\tau > 0; Y_t(y) \in \mathcal{O}, \forall t \in [0, \tau]\} \in (0, +\infty]. $$

The real number $t_{\ell t}(y) := t_b(y) + t_f(y) \in (0, \infty]$ corresponds to the “life time” of the characteristic flow in $\mathcal{O}$ going by $y$. The construction of the flow $(Y_t)$ is classical when $a$ is a Lipschitz function and we refer to [138, Thm. II.3] for a more general situation which corresponds to the assumptions we will make in the present work (see also Lemma 8.14 below).

For a solution $g: \mathcal{O} \to \mathbb{R}$ to the transport equation

$$ (8.9) \qquad a \cdot \nabla_y g = G \quad \text{in } \mathcal{O}, $$

for a given source term $G: \mathcal{O} \to \mathbb{R}$, we wish to define the trace $\gamma g$ of $g$ on the boundary set $\Sigma$. Similarly, for a solution $g: (0, T) \times \mathcal{O} \to \mathbb{R}$, $T \in (0, +\infty]$, to the transport equation

$$ (8.10) \qquad \partial_t g + a \cdot \nabla_y g = G \quad \text{in } (0, T) \times \mathcal{O}, $$

for a given source term $G: (0, T) \times \mathcal{O} \to \mathbb{R}$, we wish to define the trace $\gamma g$ of $g$ on the boundary set $(0, T) \times \Sigma$. It is worth emphasizing that the trace will be in fact only defined out of the singular set $\Sigma_0$ and thus only on the boundary set $\Sigma \setminus \Sigma_0$.

We start by recalling several possible definitions of the trace of a function $g$ satisfying (8.9) when

$$ (8.11) \qquad a \in W_{\text{loc}}^{1,s}(\bar{\mathcal{O}}), \quad g \in L_{\text{loc}}^p(\bar{\mathcal{O}}), \quad G \in L_{\text{loc}}^q(\bar{\mathcal{O}}), \quad s, p, q \in [1, \infty]. $$

Here and below, we denote by $L(E)$ the Lebesgue space of measurable functions $g: E \to \bar{\mathbb{R}} := [-\infty, +\infty]$ with typically $E = \mathcal{O}$ or $E \subset \Sigma$, and by $L^0(E) = L^0(E, \mu) \subset L(E)$ the subset of almost everywhere finite measurable functions on a measurable space $(E, \mathcal{A}, \mu)$.

**Definition 8.1.** We say that a function $g$ on $\mathcal{O}$ satisfying (8.9) and (8.11) admits a trace if one of the following assertions holds true:

* **Extension of the restriction on the boundary.** There exists $\gamma g \in L_{\text{loc}}^r(\Sigma \setminus \Sigma_0)$, $r \in [1, \infty]$, such that

$$ g_{n|\Sigma \setminus \Sigma_0} \to \gamma g \quad \text{in} \quad L_{\text{loc}}^r(\Sigma \setminus \Sigma_0) $$

for any sequence $(g_n)$ satisfying

$$ (8.12) \qquad g_n \in C_c^1(\bar{\mathcal{O}}), \quad g_n \to g \quad \text{in} \quad L_{\text{loc}}^p(\bar{\mathcal{O}}), \quad a(y) \cdot \nabla_y g_n \to G \quad \text{in} \quad L_{\text{loc}}^q(\bar{\mathcal{O}}). $$

* **Characteristics.** There exists a measurable function $\gamma g$ on $\Sigma \setminus \Sigma_0$ such that for a.e. $y \in \mathcal{O}$ satisfying $t_-(y) > -\infty$, there holds

$$ (8.13) \qquad g(y) = \gamma g(Y(t_-(y), y)) + \int_{t_-(y)}^0 G(Y(t, y)) dt, $$
**Remark 8.9.** (1) We will see in Section 8.4 that under the same conditions as in Theorem 8.8 the information on $\gamma g$ can be slightly improved, in particular $\gamma g \in L^0(\Sigma \setminus \Sigma_0)$.

(2) Theorem 8.8 in particular holds when we assume $a \in W_{\text{loc}}^{1,p'}(\bar{O})$, $b \in L_{\text{loc}}^{p'}(\bar{O})$, $G \in L_{\text{loc}}^1(\bar{O})$ and $g \in L_{\text{loc}}^p(\bar{O})$ satisfy the transport equation (8.26) in the distributional sense. Indeed, in that situation one knows from the DiPerna-Lions renormalizing theory [139, Cor. II.2] that $g$ is also a renormalized solution to the transport equation (8.26) (in the above sense).

(3) Assuming more interior integrability on the functions $g$, $b$, $G$ and $a$, we may deduce more accurate information on $\gamma g$. A typical example, is that

$$ \int_{\Sigma \cap B_R} |\gamma g|^r (|a \cdot n| \wedge 1)^2 d\sigma_y < \infty, $$

for some $r \in [1, \infty)$ and any $R > 0$, under the additional assumption

$$ |g|^r (|\text{div}a| + |a \cdot \nabla T_1(a \cdot n)| + |b|) \in L_{\text{loc}}^1(\bar{O}), \quad |g|^{r-1}|G| \in L_{\text{loc}}^1(\bar{O}). $$

The proof follows by choosing $\varphi := T_1(a \cdot n)\chi$, $\chi \in C_c^1(\bar{O})$, $0 \le \chi \le 1$, and $\beta_k(s) = (|s| \wedge k)^r$ in the associated Green formula (8.16), and then to pass to the limit $k \to \infty$.

(4) Even more integrability on $\gamma_{\pm}g$ is available on one part of the boundary if additional integrability assumption is made on $\gamma_{\mp}g$ on the other part of the boundary. A typically example, is that

$$ \int_{\Sigma_{\pm} \cap B_R} |\gamma_{\pm} g|^r |a \cdot n| d\sigma_y < \infty, $$

under the additional assumption

$$ |g|^r (|\text{div}a| + |a| + |b|) \in L_{\text{loc}}^1(\bar{O}), \quad |g|^{r-1}|G| \in L_{\text{loc}}^1(\bar{O}), \quad |\gamma_{\mp} g|^r a \cdot n \in L_{\text{loc}}^1(\bar{\Sigma}_{\mp}). $$

The proof follows by choosing $\varphi \in C_c^1(\bar{O})$, $0 \le \varphi \le 1$, and $\beta_k(s) = (|s| \wedge k)^r$ in the associated Green formula (8.16), and then to pass to the limit $k \to \infty$.

(5) The results stated in Lemma 8.7, in Theorem 8.8 and in points (1), (2), (3) and (4) above may be straightforwardly adapted to the evolution transport equation (8.10). We refer to [271, 270, 272, 70] where such results are established in a slightly less general framework. Let us emphasize again that when $a \in W_{\text{loc}}^{1,1}([0,T] \times \bar{O})$ (what it is the case in the time independent case when a satisfies (8.5)) this extension is directly implied by Theorem 8.8 applied to the vector field $(1, a(t, y))$ in the open set $(0, T) \times \mathcal{O}$.

## 8.2. Well-posedness for the transport equation with given inflow at the boundary.

We deduce from the previous trace theorems and standard tools the well-posedness for the transport equation with several boundary conditions. In this section, we deal with the transport equation with given inflow at the boundary. We are first concerned with the stationary transport equation

$$ (8.29) \qquad \lambda g + a \cdot \nabla g + bg = G \quad \text{in } \mathcal{O}, \quad \gamma_- g = g \quad \text{on } \Sigma_-, $$

for a real number $\lambda \in \mathbb{R}$ large enough, a given source term $G: \mathcal{O} \to \mathbb{R}$ and a boundary term $g: \Sigma_- \to \mathbb{R}$. As we will see, our analysis also apply to the associated dual equation

$$ (8.30) \qquad \lambda\varphi - a \cdot \nabla\varphi + (b - \text{div}a)\varphi = \Phi \quad \text{in } \mathcal{D}'(\mathcal{O}), \quad \gamma_+\varphi = \psi \quad \text{on } \Sigma_+. $$

We will also consider the related evolution equation

$$ (8.31) \qquad \begin{cases} \frac{\partial g}{\partial t} + a \cdot \nabla g + bg = G & \text{on } (0, T) \times \mathcal{O}, \\ \gamma_- g = g & \text{on } (0, T) \times \Sigma_-, \quad g(0, \cdot) = g_0 & \text{on } \mathcal{O}, \end{cases} $$

with given source term $G: (0, T) \times \mathcal{O} \to \mathbb{R}$, boundary term $g: (0, T) \times \Sigma_- \to \mathbb{R}$ and initial datum $g_0: \mathcal{O} \to \mathbb{R}$.

A possible simple framework consists in imposing the following conditions

$$ (8.32) \qquad a \in W_{\text{loc}}^{1,1}(\bar{O}), \quad b \in L_{\text{loc}}^1(\bar{O}), $$

and

$$ (8.33) \qquad b_-, \text{div} a \in L^\infty(\mathcal{O}), \quad \frac{a}{\langle y \rangle \langle b_+ \rangle} \in L^1(\mathcal{O}) + L^\infty(\mathcal{O}). $$
and for a.e. $y \in \mathcal{O}$ satisfying $t_+(y) < \infty$, there holds

$$ (8.14) \qquad g(y) = \gamma g(Y(t_+(y), y)) - \int_0^{t_+(y)} G(Y(t, y)) dt. $$

• **Green formula.** There exists $\gamma g \in L_{\text{loc}}^r(\Sigma \setminus \Sigma_0)$, $r \in [1, \infty]$, such that

$$ (8.15) \qquad \int_{\mathcal{O}} (G\varphi + g \operatorname{div}(a\varphi)) dy = \int_{\Sigma} \gamma g \varphi a \cdot n d\sigma_y, $$
for any $\varphi \in C_c^1(\bar{\mathcal{O}} \setminus \Sigma_0)$.

• **Renormalized Green formula.** There exists a measurable function $\gamma g$ on $\Sigma \setminus \Sigma_0$ such that

$$ (8.16) \qquad \int_{\mathcal{O}} (\beta'(g) G \varphi + \beta(g) \operatorname{div}(a\varphi)) dy = \int_{\Sigma} \beta(\gamma g) \varphi a \cdot n d\sigma_y, $$
for any $\varphi \in C_c^1(\bar{\mathcal{O}})$ and any $\beta \in C^1(\mathbb{R})$ such that $\beta' \in L^\infty(\mathbb{R})$.

**Remark 8.2.** (1) In order that the first definition makes sense, we implicitly assume that there exists at least one sequence $(g_n)$ which satisfies (8.12). That last fact corresponds to the density of $C_c^1(\bar{\mathcal{O}})$ in the Sobolev space $\{g \in L^p(\mathcal{O}); a(y) \cdot \nabla_y g \in L^q(\mathcal{O})\}$, which is true as we will see in Lemma 8.5 below under the regularity assumptions made on a and $\mathcal{O}$. It is worth emphasizing that the last convergence in (8.12) may require additional integrability assumption, typically $a \in W^{1,s}(\mathcal{O})$ with $1/r \ge 1/p + 1/s$. Such a definition has been introduced by Bardos in [37] for a $C^1$ vector field a. It is also the point of view adopted by Cessenat in [98, 99] in the case of the neutron operator, see also [125, chap. XXI] or Agoshkov [2, 3, 4].

(2) In order that the second definition makes sense, we implicitly assume that the set of points $y \in \mathcal{O}$ such that the characteristic $Y_t(y)$ hits the boundary on $\Sigma_0$ has zero measure in $\mathcal{O}$. It is indeed the case thanks to the Sard theorem under enough regularity assumption on a and $\mathcal{O}$, see [37, Prop. 2.3]. It is worth emphasizing that what we really need in order to write (8.13) and (8.14) is that $t \mapsto G(Y(t, y)) \in L^1(t_-(y), t_+(y))$ for a.e. $y \in \mathcal{O}$. We also mention that this characteristics description leads to a layer cake formula linking the integral of a function on the domain to the integral of its trace on the boundary. Such a definition has been widely used in kinetic theory for constructing DiPerna-Lions renormalized solution, see [137, 198, 18] and the references therein. For the classical kinetic operator this trace approach is developed by Arkeryd, Cercignani and co-authors in [85, 95, 16, 17] while for more general (but still regular) vector fields, the approach has been developed in [37, 347, 43, 183] and more recently by Arlotti et al. in [20, 21, 22, 23, 24].

(3) In order that the third definition makes sense, we need that $a \cdot n \gamma g \in L_{\text{loc}}^1(\Sigma)$ and $p \ge s'$. In some situation, this third definition is in some sense the weakest: it makes sense also when $\gamma g \in M_{\text{loc}}^1(\Sigma \setminus \Sigma_0)$ for instance and can be relevant under the weak assumption $a$, $\text{div} \in L_{\text{loc}}^{p'}(\bar{\mathcal{O}})$ as it is the case in the early works on weak solution to the Vlasov-Poisson equation in [321, 193, 1, 360]. It is also easier to handle than the two first definitions because of the way it connects the function $g$ and its trace.

(4) We will adopt the last definition which extends up to the boundary the renormalization technique introduced in [138]. It is more general and adapted to the weak regularity assumption made on the vector field $a$ than the two first definitions and we recover the third definition by just letting $\beta(s) \to s$ when the conditions of integrability make the limit well defined. Such a kind of definition has been introduced in [269, 271] for kinetic equations and in [70, 10] for transport equations.

We start with a trace result in a $L^\infty$ framework. We denote by $C_{\text{pw}}^1(\mathbb{R})$ the space of continuous functions $\beta: \mathbb{R} \to \mathbb{R}$ with piecewise continuous derivative.

**Theorem 8.3.** Assume that $g \in L^\infty(\mathcal{O})$, $a \in W_{\text{loc}}^{1,1}(\bar{\mathcal{O}})$ and $G \in L_{\text{loc}}^1(\bar{\mathcal{O}})$ satisfy the transport equation (8.9) in the distributional sense. Then, there exists a unique function

$$ \gamma g \in L^\infty(\Sigma \setminus \Sigma_0; d\sigma_y), \quad \|\gamma g\|_{L^\infty} \le \|g\|_{L^\infty}, $$

which satisfies the renormalized Green formula

$$ (8.17) \qquad \int_{\mathcal{O}} (\beta'(g) G \varphi + \beta(g) \operatorname{div}(a\varphi)) dy = \int_{\Sigma} \beta(\gamma g) \varphi a \cdot n d\sigma_y, $$
The first condition on $a$ is useful for the renormalization trick and the definition of the trace, the second condition is needed for the existence results in a $L^p$ framework when $p \neq \infty$ and the last condition is used for proving the uniqueness result. In order to be able to apply our results to more general (and realistic) situations, we rather consider the following situation. We assume that $a$ and $b$ satisfy (8.32), and defining

$$ (8.34) \qquad \bar{\varpi} = \bar{\varpi}_p := b - \frac{1}{p} \mathrm{div} a - a \cdot \frac{\nabla m}{m}, $$

for some smooth enough weight function $m: \bar{\mathcal{O}} \to (0, \infty)$ and some exponent $p \in [1, \infty]$, we assume

$$ (8.35) \qquad \bar{\varpi}_- \in L^\infty(\mathcal{O}), \quad b, \mathrm{div} a \in L^\infty_{\langle\bar{\varpi}_+\rangle^{-1}}(\mathcal{O}), \quad \frac{a}{\langle y \rangle} \in L^\infty_{\langle\bar{\varpi}_+\rangle^{-1}}(\mathcal{O}) + L^1(\mathcal{O}) $$

In the case $p=1$ and $p=\infty$, we will additionally assume $(\bar{\varpi}_q)_- \in L^\infty$ for any $q \in (1, \infty)$. It is worth emphasizing that condition (8.35) automatically holds when $m \equiv 1$ and $a, b$ satisfy (8.33). We also define the critical real number

$$ (8.36) \qquad \lambda_p^* = \lambda_p^*(a, b, m) := \|\bar{\varpi}_-\|_{L^\infty}, $$

and we may observe that

$$ (8.37) \qquad \lambda_{p'}^*(-a, b - \mathrm{div} a, m^{-1}) = \lambda_p^*(a, b, m), $$

what links up the primal and the dual problems. In order to shorten notations, we introduce the three weight functions

$$ (8.38) \qquad m_{\mathcal{O}} := m(\bar{\varpi}_+)^{1/p}, \quad \tilde{m}_{\mathcal{O}} := m(\bar{\varpi})^{-1/p'}, \quad m_{\Sigma} := m|a \cdot n|^{1/p}. $$

We start with a general discussion about a priori bounds, formal representation formulas and general stability results.

**A priori estimates.** Consider a solution $g$ to the stationary equation (8.29). For any renormalizing function $\beta: \mathbb{R} \to \mathbb{R}_+$ and any function $\varphi: \bar{\mathcal{O}} \to (0, \infty)$, we (at least) formally have

$$ \int_{\mathcal{O}} [(\lambda + b)g\beta'(g)\varphi - \beta(g)(\mathrm{div}(a\varphi))] + \int_{\Sigma_+} a \cdot n\beta(\gamma_+g)\varphi = \int_{\mathcal{O}} \beta'(g)G\varphi + \int_{\Sigma_-} |a \cdot n|\beta(\mathfrak{g})\varphi. $$

Choosing $\beta(s) := |s|^p, 1 \le p < \infty$, and $\varphi := m^p$, we get in particular

$$ (8.39) \qquad \int_{\mathcal{O}} |g|^p m^p (\lambda + \bar{\varpi}) + \frac{1}{p} \int_{\Sigma_+} |\gamma_+ g|^p m^p a \cdot n = \int_{\mathcal{O}} Gg |g|^{p-2} m^p + \frac{1}{p} \int_{\Sigma_-} |\mathfrak{g}|^p m^p |a \cdot n|. $$

For $p=1$ and any $\lambda > \lambda_1^*$, we get

$$ \int_{\mathcal{O}} |g|m\{\lambda - \lambda_1^* + \bar{\varpi}_+\} + \int_{\Sigma_+} |\gamma_+ g|m_{\Sigma} \le \int_{\mathcal{O}} |G|m + \int_{\Sigma_-} |\mathfrak{g}|m_{\Sigma}. $$

For $p \in (1, \infty)$, we split $G = G_1 + G_2$ and using the Young inequality, we have

$$ \int_{\mathcal{O}} Gg|g|^{p-2} m^p \le \varepsilon_1 \int g^p m^p + \varepsilon_2 \int g^p m_{\mathcal{O}}^p + \frac{1}{p} \frac{1}{(p'\varepsilon_1)^{p/p'}} \int G_1^p m^p + \frac{1}{p} \frac{1}{(p'\varepsilon_2)^{p/p'}} \int G_2^p \tilde{m}_{\mathcal{O}}^p, $$

for any $\varepsilon_i > 0$. For $\lambda > \lambda_p^*$, we choose $\varepsilon_1 := (\lambda - \lambda_p^*)/2$ and $\varepsilon_2 := 1/2$, we get

$$ \begin{aligned} & \frac{1}{2} \int_{\mathcal{O}} |g|^p m_{\mathcal{O}}^p + \frac{\lambda - \lambda_p^*}{2} \int_{\mathcal{O}} |g|^p m^p + \frac{1}{p} \int_{\Sigma_+} |\gamma_+ g|^p m_{\Sigma}^p \\ & \le \frac{2^{p-1}}{p(p')^{p/p'}} (\lambda - \lambda_p^*)^{1-p} \int_{\mathcal{O}} |G_1|^p m^p + \frac{1}{p(p'/2)^{p/p'}} \int_{\mathcal{O}} |G_2|^p \tilde{m}_{\mathcal{O}}^p + \frac{1}{p} \int_{\Sigma_-} |\mathfrak{g}|^p m_{\Sigma}^p. \end{aligned} $$

We thus deduce

$$ (8.40) \qquad \|g\|_{L_m^p} \le \frac{C_p}{\lambda - \lambda_p^*} \|G_1\|_{L_m^p} + \frac{C_p}{(\lambda - \lambda_p^*)^{1/p}} (\|\mathfrak{g}\|_{L_{m\Sigma}^p} + \|G_2\|_{L_{\tilde{m}\mathcal{O}}^p}) $$

and

$$ (8.41) \qquad \|g\|_{L_{\tilde{m}\mathcal{O}}^p} + \|\gamma_+ g\|_{L_{m\Sigma}^p} \le \frac{C_p}{(\lambda - \lambda_p^*)^{1/p'}} \|G_1\|_{L_{\tilde{m}\mathcal{O}}^p} + C_p (\|\mathfrak{g}\|_{L_{m\Sigma}^p} + \|G_2\|_{L_{\tilde{m}\mathcal{O}}^p}), $$
for any $\varphi \in C_c^1(\bar{O})$ and any $\beta \in C_{pw}^1(\mathbb{R})$. As a consequence, renormalization and trace operations commute:

$$ (8.18) \qquad \gamma \beta(g) = \beta(\gamma g), \quad \forall \beta \in C_{pw}^1(\mathbb{R}). $$

**Remark 8.4.** (1) Because of the very general assumption (8.5) made on the vector field $a: \mathcal{O} \to \mathbb{R}^D$ which is exactly the one made in the DiPerna-Lions theory for transport equation in the whole space developed in [138], the above trace result slightly improves the similar trace result established by Boyer in [70, Thm. 3.1], where an additional assumption $a \cdot n \in L^\zeta(\partial\mathcal{O})$, $\zeta > 1$, is made.

(2) An alternative approach has been developed by Ambrosio and co-authors by assuming weaker bound on Da but stronger bound on $a$. More precisely, denoting by $M_\infty$ the set of vector fields $a \in L^\infty(\mathcal{O})$ such that $\text{diva} \in M^1(\mathcal{O})$, it is established in [10, Prop. 3.2] that there exists a linear and bounded mapping $\text{Tr}: M_\infty(\mathcal{O}) \to L^\infty(\partial\mathcal{O})$ such that $\text{Tra} = n \cdot a|_{\partial\mathcal{O}}$ when $a \in C^1(\bar{O})$. The proof relies on Ambrosio's extension to a BV framework in [8] of the famous DiPerna-Lions improvement [138, Lem. II.1] of Freidrichs' type Lemma on the estimate of the commutator between directional derivative and convolution (see Lemma 8.5 below). Moreover, it is also established in [10] (see in particular [10, Thm. 4.2]) that

$$ \text{Tr}(a\beta(g)) = \beta\left(\frac{\text{Tr}(ag)}{\text{Tra}}\right)\text{Tr}(a), \quad \forall \beta \in C^1(\mathbb{R}), $$

for any $a \in BV(\mathcal{O}) \cap L^\infty(\mathcal{O})$ and $g \in L^\infty(\mathcal{O})$ such that $ag \in M_\infty$. The above formula is then nothing but (8.18) when $a \in W^{1,1}(\mathcal{O}) \cap L^\infty(\mathcal{O})$.

Before coming to the proof of Theorem 8.3, we state one technical but fundamental result. We define the mollifier $(\rho_\varepsilon)_{\varepsilon>0}$ by

$$ (8.19) \qquad \rho_\varepsilon(z) = \frac{1}{\varepsilon^d} \rho(z/\varepsilon), \quad 0 \le \rho \in \mathcal{D}(\mathbb{R}^d), \quad \text{supp} \rho \subset B_1, \quad \int_{\mathbb{R}^d} \rho(z) dz = 1, $$

and for any $u \in L_{\text{loc}}^1(\bar{O})$, $v_\varepsilon \in C_c(\mathbb{R}^D)$, $\text{supp} v_\varepsilon \subset B_\varepsilon$, we introduce the convolution-translation function $u *_\varepsilon v_\varepsilon$ defined by

$$ (8.20) \qquad (u *_\varepsilon v_\varepsilon)(y) := \int_{\mathcal{O}} u(z) v_\varepsilon(y - 2\varepsilon n(y) - z) dz. $$

**Lemma 8.5.** For $g \in L_{\text{loc}}^p(\bar{O})$, $p \in [1, \infty]$, $a \in W_{\text{loc}}^{1,p'}(\bar{O})$ and $G \in L_{\text{loc}}^1(\bar{O})$ satisfying (8.9) in the distributional sense, the sequence $(g_\varepsilon)$ defined by $g_\varepsilon := g *_\varepsilon \rho_\varepsilon$ satisfies

$$ g_\varepsilon \in W_{\text{loc}}^{1,\infty}(\bar{O}), \quad G_\varepsilon := a \cdot \nabla g_\varepsilon \to a \cdot \nabla g \text{ in } L_{\text{loc}}^1(\bar{O}), $$

as $\varepsilon \to 0$, and

$$ g_\varepsilon \to g \text{ in } L_{\text{loc}}^p(\bar{O}), \quad \text{if } p < \infty, $$

$$ g_\varepsilon \to g \text{ in } L_{\text{loc}}^1(\bar{O}), \quad (g_\varepsilon) \text{ bounded in } L_{\text{loc}}^\infty(\bar{O}), \quad \text{if } p = \infty. $$

We skip the proof of Lemma 8.5 since it follows by just repeating the proofs of [138, Lem. II.1], [270, Lem. 1] or [70, Lem. 3.1].

**Proof of Theorem 8.3.** Let us fix $\chi \in \mathcal{D}(\bar{O})$ such that $0 \le \chi \le 1$ and denote $R > 0$ a real number such that $\text{supp} \chi \subset B_R$. We observe that $\chi\text{sign}(a \cdot n) \in L^1(\Sigma)$. From Gagliardo trace theorem [169, Teor. 1.II], there exists $\psi \in W^{1,1}(\mathcal{O})$ such that $\gamma\psi = \chi\text{sign}(a \cdot n)$ and $\text{supp} \psi \subset B_R$. Denoting $T_1: \mathbb{R} \to [-1, 1]$ the truncation function which is odd and is defined by $T_1(\sigma) = \sigma \wedge 1$ for any $\sigma \ge 0$, we see that $\gamma T_1(\psi) = T_1(\gamma\psi) = \gamma\psi$, and thus we may assume $\psi \in L^\infty(\mathcal{O})$ up to replacing $\psi$ by $T_1(\psi)$. As a consequence, there exists a sequence $(\psi_k)$ of $W^{1,\infty}(\mathcal{O})$ such that $\psi_k \to \psi$ in $W^{1,1}(\mathcal{O})$, with $(\psi_k)$ bounded in $L^\infty(\mathcal{O})$, $\text{supp}\psi_k \subset B_R$, and $\gamma\psi_k \to \chi\text{sign}(a \cdot n)$ in $L^1(\Sigma)$, with $(\gamma\psi_k)$ bounded in $L^\infty(\Sigma)$.
for some numerical constant $C_p \in (0, \infty)$ and any $p \in (1, \infty)$ and also for $p = 1$ because of the previous estimate. Finally, for $\lambda > \lambda_\infty^*$ and $\alpha \in (0, \lambda - \lambda_\infty^*)$, we may proceed exactly as above, but throwing also away the contribution of $\varpi_+$, and we may thus first write

$$ (8.42) \qquad \left( \lambda - \|\varpi_-\|_{L^\infty} - \frac{\alpha^{p'}}{p'} \right) \|g\|_{L_m^p(\mathcal{O})}^p \le \frac{1}{p} \left( \|G/\alpha\|_{L_m^p(\mathcal{O})}^p + \|g|a \cdot n|^{1/p}\|_{L_m^p(\Sigma_-)}^p \right), $$

for any $p \in (1, \infty)$ large enough in such a way that the coefficient in front of $\|g\|_{L_m^p(\mathcal{O})}^p$ is positive. Taking the power $1/p$ in both sides and passing first to the limit $p \nearrow \infty$ and next to the limit $\alpha \searrow \lambda - \lambda_\infty^*$, we end with

$$ (8.43) \qquad \|g\|_{L_m^\infty(\mathcal{O})} \le \max \left( \frac{1}{\lambda - \lambda_\infty^*} \|G\|_{L_m^\infty(\mathcal{O})} + \|g\|_{L_m^\infty(\Sigma_-)} \right). $$

Consider now a solution to the evolution equation (8.31). For any renormalizing function $\beta : \mathbb{R} \to \mathbb{R}_+$ and any test function $\varphi : [0, \infty) \times \bar{\mathcal{O}} \to (0, \infty)$, we (at least) formally have

$$ (8.44) \qquad \begin{aligned} & \int_0^t \int_{\mathcal{O}} (\beta'(g)G\varphi - \beta'(g)g\varphi + \beta(g)[\partial_t\varphi + \operatorname{div}(a\varphi)]) dyds \\ & = \left[ \int_{\mathcal{O}} \beta(g(s,y))\varphi(s,y)dy \right]_0^t + \int_0^t \int_{\Sigma} \beta(\gamma g)\varphi a \cdot n d\sigma_y ds. \end{aligned} $$

Choosing $\beta(s) := |s|^p$, $1 \le p < \infty$, and $\varphi(t, y) := m^p(y)$, we get in particular

$$ \begin{aligned} & \int_{\mathcal{O}} |g(t)|^p m^p + \int_0^t \int_{\Sigma_+} a \cdot n |\gamma_+ g|^p m^p + p \int_0^t \int_{\mathcal{O}} |g|^p m^p \varpi \\ & = \int_{\mathcal{O}} |g_0|^p m^p + p \int_0^t \int_{\mathcal{O}} g |g|^{p-2} G m^p + \int_0^t \int_{\Sigma_-} |g|^p m^p |a \cdot n|. \end{aligned} $$

Using the Young inequality

$$ p \int_{\mathcal{O}} g |g|^{p-2} G m^p \le \frac{p}{p'} \int_{\mathcal{O}} |g|^p m_{\mathcal{O}}^p + \int_{\mathcal{O}} |G|^p \tilde{m}_{\mathcal{O}}^p $$

and the Gronwall lemma, we then deduce

$$ (8.45) \qquad \begin{aligned} & \|g(t)\|_{L_m^p}^p + \int_0^t e^{p\kappa(t-s)} (\|g_s\|_{L_m^p}^p + \|\gamma_+g_s\|_{L_m^{\Sigma}}^p) ds \\ & \le e^{p\kappa t} \|g_0\|_{L_m^p}^p + \int_0^t e^{p\kappa(t-s)} (\|G_s\|_{L_{\tilde{m}\mathcal{O}}^p}^p + \|g_s\|_{L_m^p}^p) ds, \quad \forall t \ge 0, \end{aligned} $$

with $\kappa := \|\langle \varpi_- \rangle\|_{L^\infty}$. Passing to the limit $p \to \infty$, we also have

$$ (8.46) \qquad \max(\|g(t)\|_{L_m^\infty}, \|\gamma_+g(t)\|_{L_m^\infty}) \le e^{\kappa t} \max(\|g_0\|_{L_m^\infty}, \sup_{[0,t]}(\|G_s\|_{L_m^\infty} + \|g_s\|_{L_m^\infty})), $$

for any $t \ge 0$.

**Representation formulas.** In a smooth functions framework or still formally, one classically knows that the solution $g$ to the evolution transport equation (8.31) is given by

$$ (8.47) \qquad g(t, y) = g_0(Y_{-t}(y))e^{-\int_0^t b(Y_{s-t}(y))ds} \mathbf{1}_{t<t_b} + \mathfrak{g}(t-t_b, y_b)e^{-\int_0^{t_b} b(Y_{s-t_b}(y))ds} \mathbf{1}_{t>t_b} \\ \qquad \qquad + \int_0^{t_b'} G(s, Y_{s-t_b'}(y))e^{-\int_0^{t_b'} b(Y_{-\tau}(y))d\tau} du, $$

where we recall that the characteristics $Y$ and the backward exit time $t_b$ are defined in (8.6)-(8.7) and we denote $t_b' := \min(t, t_b)$. Similarly, the solution $g$ to the stationary transport equation (8.29) is given by

$$ (8.48) \qquad g(y) = \mathfrak{g}(y_i(y))e^{-\int_0^{t_b} b(Y_{-s}(y))ds} + \int_0^{t_b} G(Y_{-s}(y))e^{-\int_0^s b(Y_{-\tau}(y))d\tau} du. $$

Alternatively, we may define a semigroup $S_b$ (say on $L^\infty(\mathcal{O})$) by

$$ (8.49) \qquad (S_b(t)f_0)(y) := \begin{cases} f_0(Y_{-t}(y)) \exp(-\int_0^t b(Y_{\tau-t}(y))d\tau), & \text{if } t \in (0, t_b(y)), \\ 0 & \text{otherwise.} \end{cases} $$
Let us then consider the sequences $(g_\epsilon)$ and $(G_\epsilon)$ defined in Lemma 8.5. The classical Green formula for Lipschitz functions writes

$$
\begin{align*}
& \int_{\Sigma} (g_{\epsilon}|_{\Sigma} - g_{\epsilon'|\Sigma})^2 |a \cdot n| \chi \, d\sigma_y \\
&= \int_{\Sigma} (g_{\epsilon}|_{\Sigma} - g_{\epsilon'|\Sigma})^2 a \cdot n \psi_k \, d\sigma_y + \int_{\Sigma} (g_{\epsilon}|_{\Sigma} - g_{\epsilon'|\Sigma})^2 [|a \cdot n| \chi - a \cdot n \psi_k] \, d\sigma_y \\
&= \int_{\mathcal{O}} [2(g_{\epsilon} - g_{\epsilon'}) (G_{\epsilon} - G_{\epsilon'}) \psi_k \, dy + (g_{\epsilon} - g_{\epsilon'})^2 \mathrm{div}(a\psi_k)] \, dy \\
&\quad + \int_{\Sigma} (g_{\epsilon}|_{\Sigma} - g_{\epsilon'|\Sigma})^2 [|a \cdot n| \chi - a \cdot n \psi_k] \, d\sigma_y \\
&\le 4 \|\psi_k\|_{L^\infty} \|g\|_{L^\infty} \|G_\epsilon - G_{\epsilon'}\|_{L^1(B_R)} + \|\psi_k\|_{W^{1,\infty}} \int_{B_R} (|a| + |\mathrm{div}a|)(g_\epsilon - g_{\epsilon'})^2 \, dy \\
&\quad + 2 \|g\|_{L^\infty}^2 \|(a \cdot n)\gamma\psi_k - \chi|a \cdot n\|_{L^1(\Sigma)},
\end{align*}
$$

for any $\epsilon > 0$ and $k \ge 1$. We deduce that $(g_{\epsilon}|_{\Sigma})$ is a Cauchy sequence in $L^2(|a \cdot n| \chi d\sigma)$. From the fact that $(g_{\epsilon})$ is bounded in $L^{\infty}(\mathcal{O})$, we deduce that the sequence $(\gamma g_{\epsilon})$ is also bounded in $L^{\infty}(\Sigma)$. As a consequence, there exists a function $\gamma g \in L^{\infty}(\Sigma)$ such that $\gamma g_{\epsilon} \to \gamma g$ in $L^2(|a \cdot n| \chi d\sigma)$. Next, we may write the Green formula

$$
\int_{\mathcal{O}} [G_{\epsilon} \varphi + g_{\epsilon} \mathrm{div}(a\varphi)] \, dy = \int_{\Sigma} \gamma g_{\epsilon} \varphi a \cdot n \, d\sigma_y,
$$

for any test function $\varphi \in C_c^1(\bar{\mathcal{O}})$, and we may pass to the limit as $\epsilon \to 0$. We deduce that the Green formula

$$
(8.21) \qquad \int_{\mathcal{O}} (G \varphi + g \operatorname{div}(a\varphi)) \, dy = \int_{\Sigma} \gamma g \varphi a(y) \cdot n(y) \, d\sigma_y,
$$

holds for any $\varphi \in C_c^1(\bar{\mathcal{O}})$. That clearly uniquely defines the trace function $\gamma g$ on $\Sigma \setminus \Sigma_0$.

Now, on the one hand, from the DiPerna-Lions renormalizing theory [138, proof of Corollary II.1], we know that $\beta(g) \in L^\infty(\mathcal{O})$ satisfies the transport equation

$$
(8.22) \qquad a(y) \cdot \nabla_y \beta(g) = \beta'(g)G \quad \text{in } \mathcal{D}'(\mathcal{O}),
$$

for any renormalizing function $\beta \in \operatorname{Lip}(\mathbb{R})$ and any test function $\varphi \in C_c^1(\mathcal{O})$. Using the already established trace result, we know that there exists $\gamma\beta(g) \in L^\infty(\Sigma \setminus \Sigma_0)$ such that

$$
(8.23) \qquad \int_{\mathcal{O}} [\beta'(g)G\varphi + \beta(g)\mathrm{div}(a\varphi)] \, dy = \int_{\Sigma} \gamma\beta(g) \varphi a \cdot n \, d\sigma_y,
$$

for any test function $\varphi \in C_c^1(\bar{\mathcal{O}})$. On the other hand, from the classical Green formula for Lipschitz functions and because $\beta(g_\epsilon)|_\Sigma = \beta(g_{\epsilon_\epsilon|\Sigma})$, we have

$$
\int_{\mathcal{O}} [\beta'(g_{\epsilon}) G_{\epsilon} \varphi + \beta(g_{\epsilon}) \mathrm{div}(a\varphi)] \, dy = \int_{\Sigma} \beta(g_{\epsilon}|_{\Sigma}) \varphi a \cdot n \, d\sigma_y,
$$

for any renormalizing function $\beta \in \operatorname{Lip}(\mathbb{R})$ and any test function $\varphi \in C_c^1(\bar{\mathcal{O}})$. Using that

$$
\beta'(g_{\epsilon})G_{\epsilon} \rightarrow \beta'(g)G, \quad \beta(g_{\epsilon}) \rightarrow \beta(g), \quad \beta(g_{\epsilon}|_{\Sigma}) \rightarrow \beta(\gamma g)
$$

respectively in $L_{\mathrm{loc}}^1(\bar{\mathcal{O}})$ and in $L_{\mathrm{loc}}^1(\Sigma)$, and that the two last sequences are bounded in $L^\infty$, we may pass to the limit $\epsilon \to 0$ in the last Green formula, and we thus get

$$
\int_{\mathcal{O}} [\beta'(g)G \varphi + \beta(g)\mathrm{div}(a\varphi)] \, dy = \int_{\Sigma} \beta(\gamma g) \varphi a \cdot n \, d\sigma_y.
$$

Together with (8.23) and by uniqueness of the trace function, we conclude to $\gamma\beta(g) = \beta(\gamma g)$. $\square$

Let us state several variants of the preceding trace result. For the transport evolution equation (8.10) a first possible trace result writes as follows.

**Theorem 8.6.** Assume that $g \in L^\infty((0, T) \times \mathcal{O})$, $a \in L^1(0, T; W_{\mathrm{loc}}^{1,1}(\bar{\mathcal{O}}))$ and $G \in L_{\mathrm{loc}}^1([0, T] \times \bar{\mathcal{O}})$ satisfy the evolution transport equation (8.10) in the distributional sense. Then,

$g \in C([0, T]; L_{\mathrm{loc}}^{1}(\bar{\mathcal{O}}))$
Given $f_0: \mathcal{O} \to \mathbb{R}$, the function $f(t, y) := (S_b(t)f_0)(y)$ is thus a solution to the evolution equation
$$ \partial_t f + a \cdot \nabla f + bf = 0 \text{ in } (0, \infty) \times \mathcal{O}, \quad \gamma_-f = 0 \text{ on } (0, \infty) \times \Sigma_- $$

For $G, \tilde{g}: \mathcal{O} \to \mathbb{R}$, we next define

$$ (8.50) \qquad g := \tilde{g} + \int_0^\infty e^{-\lambda t} S_b(t) \tilde{G} dt, $$

with $\tilde{G} := G - \lambda \tilde{g} + a \cdot \nabla_x \tilde{g} - b \tilde{g}$. By construction, it is a solution to the stationary transport equation (8.29).

**Stability.** We present some stability and continuity results. Generalizing slightly [272, Definitions 2.6 and 3.1], we say that a sequence $(g_n)$ of $L(E)$ converges in the renormalized sense to $g$, we note $g_n \xrightarrow{r} g$, if for any $\delta \in \Delta$, $\Delta \subset (0, 1]$, $0 \in \bar{\Delta}$, there exists $\bar{\beta}_\delta \in L^\infty(E)$ such that

$$ (8.51) \qquad \beta_{\delta}(g_n) \to \bar{\beta}_{\delta} * \sigma(L^{\infty}, L^1) \text{ as } n \to \infty \text{ and } \beta_1(\bar{\beta}_{\delta}) \to \beta_1(g) L_{\text{loc}}^{1}(\bar{\mathcal{O}}) \text{ as } \delta \to 0. $$

We may observe that in particular $g_n \xrightarrow{r} g$ weakly $L^1(E)$ or $g_n \to g$ a.e. in $E$ implies $g_n \xrightarrow{r} g$. We refer to [272] and the references therein for more material about the subject.

**Proposition 8.10.** Let us consider four sequences $(g_k)$ of $L_{\text{loc}}^1(\bar{\mathcal{O}})$, $(a_k)$ of $W_{\text{loc}}^{1,1}(\bar{\mathcal{O}})$, $(b_k)$ and $(G_k)$ of $L_{\text{loc}}^1(\bar{\mathcal{O}})$ such that

$$ a_k \cdot \nabla \beta(g_k) + b_k \beta'(g_k) g_k = \beta'(g_k) G_k \quad \text{in } \mathcal{D}'(\mathcal{O}), $$

for any $k \ge 1$ and any $\beta \in C_*^1(\mathbb{R})$ and four functions $g \in L_{\text{loc}}^1(\bar{\mathcal{O}})$, $a \in W_{\text{loc}}^{1,1}(\bar{\mathcal{O}})$, $b, G \in L_{\text{loc}}^1(\bar{\mathcal{O}})$ such that $a_k \to a$ in $W_{\text{loc}}^{1,1}(\bar{\mathcal{O}})$ and $b_k \to b$, $G_k \to G$ in $L_{\text{loc}}^1(\bar{\mathcal{O}})$. Let us denote by $\Sigma_0$ the boundary singular subset associated to $a$.

(1) If $g_k \to g$ a.e. in $\mathcal{O}$ then $g$ satisfies (8.26) for any $\beta \in C_*^1(\mathbb{R})$ and, up to the extraction of a subsequence, $\gamma g_k \to \gamma g$ a.e. on $\Sigma \setminus \Sigma_0$.

(2) If $g_k \xrightarrow{r} g$ weakly in $L_{\text{loc}}^1(\bar{\mathcal{O}})$ then $g$ satisfies (8.26) and, up to the extraction of a subsequence, $\gamma g_k \to \gamma g$ on $\Sigma \setminus \Sigma_0$.

**Remark 8.11.** Because of Remark 8.9-(5) and the time independence made on $a$ and $b$ in (8.35), exactly the same stability result holds for the evolution equation (8.31) as a consequence of Proposition 8.10.

*Proof of Proposition 8.10.* We split the proof into two steps.

**Step 1.** We establish (1). We fix $\beta_1$ as in the proof of Theorem 8.8 and we write the Green formula

$$ \int_{\mathcal{O}} [\beta'_{1}(g_{k}) G_{k} \varphi - \beta'_{1}(g_{k}) g_{k} b_{k} \varphi + \beta_{1}(g_{k}) \operatorname{div}(a_{k} \varphi)] dy = \int_{\Sigma} \beta_{1}(\gamma g_{k}) \varphi a_{k} \cdot n \, d\sigma_{y}, $$

for any test function $\varphi \in C_c^1(\bar{\mathcal{O}})$. There exists $\bar{\beta}_1 \in L^\infty(\Sigma \setminus \Sigma_0)$ and a subsequence $(g_{n_k})$ such that $\beta_1(\gamma g_{n_k}) \to \bar{\beta}_1$ weakly $\sigma(L^\infty, L^1)$. Passing to the limit in the above equation, we get

$$ \int_{\mathcal{O}} [\beta'_{1}(g) G \varphi - \beta'_{1}(g) g b \varphi + \beta_{1}(g) \operatorname{div}(a \varphi)] dy = \int_{\Sigma} \bar{\beta}_{1} \varphi a \cdot n \, d\sigma_{y}. $$

From Lemma 8.7 and Theorem 8.8, we deduce that $\bar{\beta}_1 = \beta_1(\gamma g)$, so that $\beta_1(\gamma g_n) \to \beta_1(\gamma g)$ weakly $\sigma(L^\infty, L^1)$. Fixing now $\beta_2 := \beta_1^2 \in C_*^1(\mathbb{R})$ and repeating the same argument, we get $\beta_2(\gamma g_n) \to \beta_2(\gamma g)$ weakly $\sigma(L^\infty, L^1)$. We then immediately deduce that

$$ (\beta_1(\gamma g_n) - \beta_1(\gamma g))^2 \to 0 \quad \text{weakly } \sigma(L^\infty, L^1), $$

so that $\beta_1(\gamma g_n) \to \beta_1(\gamma g)$ in $L_{\text{loc}}^1(\Sigma \setminus \Sigma_0)$. We conclude by using that $\beta_1$ is one-to-one.

**Step 2.** We establish (2). We fix $\beta_\delta$ as defined just before the statement of Lemma 8.7 and we write the Green formula

$$ \int_{\mathcal{O}} [\beta'_{\delta}(g_k) G_k \varphi + \beta_{\delta}(g_k) \operatorname{div}(a\varphi)] dy = \int_{\Sigma} \beta_{\delta}(\gamma g_k) \varphi a \cdot n \, d\sigma_y, $$

for any test function $\varphi \in C_c^1(\bar{\mathcal{O}})$. There exist $\bar{\beta}_\delta, \tilde{\beta}_\delta, \bar{\beta}'_\delta \in L^\infty(\mathcal{O})$, $\gamma\bar{\beta}_\delta \in L^\infty(\Sigma \setminus \Sigma_0)$ and a subsequence $(g_{n_k})$ such that $\beta_\delta(g_{n_k}) \to \bar{\beta}_\delta$, $g_{n_k}\beta'_\delta(g_{n_k}) \to \tilde{\beta}_\delta$, $\beta'_\delta(g_{n_k}) \to \bar{\beta}'_\delta$ and $\beta_\delta(\gamma g_{n_k}) \to \gamma\bar{\beta}_\delta$ weakly $\sigma(L^\infty, L^1)$. Passing to the limit in the above equation, we get

$$ a \cdot \nabla_y \bar{\beta}_\delta + b \tilde{\beta}_\delta = \bar{\beta}'_\delta G \quad \text{in } \mathcal{D}'(\mathcal{O}), \quad \gamma \bar{\beta}_\delta = \bar{\gamma} \bar{\beta}_\delta \quad \text{on } \Sigma \setminus \Sigma_0. $$
and there exists a unique function

$$ \gamma g \in L^{\infty}((0, T) \times \Sigma \setminus \Sigma_0; dt \otimes d\sigma_y), \quad \|\gamma g\|_{L^{\infty}} \le \|g\|_{L^{\infty}}, $$

which satisfies the renormalized Green formula

$$ (8.24) \qquad \int_{t_0}^{t_1} \int_{\mathcal{O}} (\beta'(g) G \varphi + \beta(g) [\partial_t \varphi + \operatorname{div}(a\varphi)]) dydt \\ = \left[ \int_{\mathcal{O}} \beta(g(t, \cdot)) \varphi dy \right]_{t_0}^{t_1} + \int_{t_0}^{t_1} \int_{\Sigma} \beta(\gamma g) \varphi a \cdot n \, d\sigma_y dt, $$

for any $\varphi \in C_c^1([0, T] \times \bar{\mathcal{O}})$, any $t_0, t_1 \in [0, T]$ and any $\beta \in C_{\text{pw}}^1(\mathbb{R})$. In particular renormalization and trace operations commute: (8.18) holds.

We skip the proof of Theorem 8.6 which is very similar to the proof Theorem 8.3 using the slight modifications that one can find in [270, Thm. 2] or [70, Thm. 3.1]. Under the slightly more regularity assumption $a \in W_{\text{loc}}^{1,1}([0, T] \times \bar{\mathcal{O}})$, Theorem 8.6 is a direct corollary of Theorem 8.3 applied to the for field $(1, a(t, y))$ on the open set $(0, T) \times \mathcal{O}$.

For some additional function $b: \mathcal{O} \to \mathbb{R}$, another possible variant is the following trace result for the stationary transport equation

$$ (8.25) \qquad a \cdot \nabla_y g + bg = G \quad \text{in } \mathcal{O} $$

in the renormalized framework as introduced by DiPerna and Lions in [138]. Assuming $a \in W_{\text{loc}}^{1,1}(\bar{\mathcal{O}})$, $b, G \in L_{\text{loc}}^1(\bar{\mathcal{O}})$, we say that $g \in L_{\text{loc}}^1(\bar{\mathcal{O}})$ is a renormalized solution to the transport equation (8.25) if

$$ (8.26) \qquad a \cdot \nabla_y \beta(g) + b\beta'(g)g = \beta'(g)G, $$

in the distributional sense for any renormalizing function $\beta \in C_*^1(\mathbb{R})$ the set of $C^1(\mathbb{R})$ functions such that $\beta$ admits some finite limits in $\pm\infty$ and $s \mapsto \langle s \rangle \beta'(s)$ is bounded on $\mathbb{R}$, in particular $C_*^1(\mathbb{R}) \subset C_b^1(\mathbb{R})$. We also denote by $\beta \in C_{\text{pw},*}^1(\mathbb{R})$ the $C^1$ piecewise variant of $C_*^1(\mathbb{R})$. We will repeatedly use the family of functions $\beta_\delta \in C_*^1(\mathbb{R})$ defined by $\beta_\delta(s) := s/(1 + \delta s^2)^{1/2}$ for any $\delta \in (0, 1]$. We observe that $\beta'_\delta(s) = (1 + \delta s^2)^{-3/2}$, so that $s\beta'(s) \to 0$ as $s \to \pm\infty$.

Let us start formulating some basic facts on renormalized solutions to equation (8.25).

**Lemma 8.7.** Assume $a \in W_{\text{loc}}^{1,1}(\bar{\mathcal{O}})$, $b, G \in L_{\text{loc}}^1(\bar{\mathcal{O}})$.

(1) If $g \in L_{\text{loc}}^1(\bar{\mathcal{O}})$ and $\alpha(g)$ satisfies equation (8.26) for one renormalizing function $\alpha: \mathbb{R} \to (-1, 1)$ which is bijective and belongs to $C_{\text{pw},*}^1(\mathbb{R})$ then $\beta(g)$ satisfies equation (8.26) for any renormalizing function $\beta \in C_{\text{pw},*}^1(\mathbb{R})$.

(2) If $g_1, g_2 \in L_{\text{loc}}^1(\bar{\mathcal{O}})$ are two renormalized solutions to the transport equations

$$ a \cdot \nabla_y g_i + bg_i = G_i \in L_{\text{loc}}^1(\bar{\mathcal{O}}), $$

then $g := g_1 + g_2$ is a renormalized solution to the transport equation (8.25) with $G := G_1 + G_2$.

(3) If $g$ is a renormalized solution to the transport equation (8.25) and $\Phi, c \in L^{\infty}(\mathcal{O})$ satisfy

$$ a \cdot \nabla_y \Phi = c $$

in the distributional sense, then $h := ge^{-\Phi}$ satisfies

$$ (8.27) \qquad a \cdot \nabla_y h + (b+c)h = Ge^{-\Phi} $$

in the renormalized sense.

**Proof of Lemma 8.7.** We briefly sketch the proof and for more details we refer to [138], in particular to [138, Lem. II.2]. It is worth mentioning that only the case $b \in L^{\infty}(\mathcal{O})$ is considered in [138], but it readily extends to our framework. Assertion (1) is just a consequence of the chain rule $\beta'(s) = (\beta \circ \alpha^{-1})'(\alpha(s))\alpha'(s)$ for smooth enough solutions and thus for any solution thanks to Lemma 8.5 (see the proof of [138, Cor. II.1]) and to a standard approximation procedure in order to deal with piecewise $C^1$ functions. In order to establish (2), we consider two renormalized solutions $g_i$, a renormalized function $\beta \in C_*^1(\mathbb{R})$ and we write

$$ a \cdot \nabla_y \beta(\beta_\delta(g_1) + \beta_\delta(g_2)) = \beta'(\beta_\delta(g_1) + \beta_\delta(g_2))[(G_1 - bg_1)\beta'_\delta(g_1) + (G_2 - bg_2)\beta'_\delta(g_2)], $$
From the fact that $(g_k)$ is locally uniformly integrable, we classically deduce that

$$ \bar{\beta}_{\delta}, \tilde{\beta}_{\delta} \to g \text{ in } L_{\text{loc}}^{1}(\mathcal{O}), \quad \bar{\beta}'_{\delta} G \to G \text{ in } L_{\text{loc}}^{1}(\mathcal{O}), $$

as $\delta \to 0$. More precisely, the two first convergences come from the elementary inequalities

$$ \forall M > 0, \exists c_M > 0, |s - \beta_{\delta}(s)| \le |s - s\beta'_{\delta}(s)| \le c_M \delta + |s|1_{|s|>M}, $$

for any $s \in \mathbb{R}$, $\delta \in (0, 1)$, and the last convergence comes from the convexity inequality $\bar{\beta}'_{\delta} \ge \beta'_{\delta}(g)$ and the elementary inequalities

$$ \forall M > 0, \exists c_M > 0, 0 \le 1 - \beta'_{\delta}(s) \le c_M \delta + 1_{|s|>M}, \forall s \in \mathbb{R}, \forall \delta \in (0, 1). $$

From Step 1, we deduce that $g$ satisfies (8.26) and that, up to the extraction of a subsequence, $\bar{\gamma}\beta_{\delta} = \gamma\bar{\beta}_{\delta} \to \gamma g$ a.e. on $\Sigma \setminus \Sigma_0$. Using the Cantor diagonal process, we obtain that there exist two sequences $(\delta_m)$ and $(g_{n_k})$ such that $\delta_m \searrow 0$ and $\gamma g_{n_k} \xrightarrow{r} \gamma g$ in the renormalized sense associated to $(\delta_m)$. $\square$

**Existence.** We establish two existence results of solutions to the transport equation (8.29).

**Lemma 8.12 (Existence in $L_m^\infty$).** We assume that $a$ and $b$ satisfy (8.35) with $p = \infty$ and some weight function $m: \bar{\mathcal{O}} \to [1, \infty)$. For any $\lambda > \lambda_\infty^*$ and any given functions $G \in L_m^\infty(\mathcal{O})$ and $g \in L_m^\infty(\Sigma_0)$, there exists $g \in L_m^\infty(\mathcal{O})$ solution to (8.29) in the distributional sense. This solution satisfies (8.29) in the renormalized sense, the weak maximum principle, namely

$$ (8.52) \qquad g \ge 0 \text{ in } \mathcal{O} \text{ if } g \ge 0 \text{ on } \Sigma_0 \text{ and } G \ge 0 \text{ in } \mathcal{O}, $$

and the $L_m^\infty$ estimate (8.43).

**Proof of Lemma 8.12.** The proof follows [270, Lem. 3] using [37, Thm. 2.3]; we only sketch it. Under the stronger regularity assumption $a, b \in C_b^1(\bar{\mathcal{O}})$, $G \in C_c^1(\mathcal{O})$, $g := \tilde{g}|_{\Sigma_0}$ with $\tilde{g} \in C_c^1(\bar{\mathcal{O}})$, both definitions (8.48) and (8.50) provide a classical (and thus also renormalized) solution $g$ to (8.29). In such a situation, we may justify the computations made in the above a priori estimates paragraph and we conclude that $g$ satisfies the $L^\infty$ estimate (8.43). In the general case for $a, b, g$ and $G$, we introduce some sequences $(a^\epsilon)$, $(b^\epsilon)$, $(g^\epsilon)$ and $(G^\epsilon)$ of regular and approximating functions so that we may apply the first step above. In that way, we build a sequence $(g^\epsilon)$ of renormalized solutions to the approximated problem which is uniformly bounded and thus converges (up to the extraction of a subsequence) in the weakly $\ast\sigma(L^\infty, L^1)$ sense to a function $g \in L^\infty(\mathcal{O})$ satisfying (8.46). We then immediately conclude by passing to the limit $\epsilon \to 0$ thanks to Proposition 8.10. $\square$

We give a first version of an existence result in a $L^p$ framework with strong assumption on the boundary condition.

**Lemma 8.13 (Existence in $L^p$).** We assume that $a$ and $b$ satisfy (8.35) for some $p \in [1, \infty)$ and some weight function $m: \bar{\mathcal{O}} \to [1, \infty)$. For any $\lambda > \lambda_p^*$, $G \in L_m^p(\mathcal{O})$ and $g \in L_m^p(\Sigma_0)$, there exists $g \in L_m^p(\mathcal{O})$ a renormalized solution to the transport equation (8.29). This one satisfies (8.40), (8.52) and $\gamma_+g \in L_m^p(\Sigma_+)$.

**Proof of Lemma 8.13.** We argue similarly as during the proof of Lemma 8.12. When $g = \tilde{g}|_{\Sigma_0}$ with $\tilde{g}, a, b, G$ smooth and with compact support, the classical solution built above satisfies (8.40), and thus

$$ (8.53) \qquad \|g\|_{L_m^p(\mathcal{O})} \lesssim \|G\|_{L_m^p} + \|g m\|_{L^\infty} \|a\|_{L^1(\text{supp}g)}^{1/p}. $$

For $p > 1$, and under the general conditions (8.35) on $a$ and $b$, but still assuming $g = \tilde{g}|_{\Sigma_0}$ and $G, \tilde{g} \in C_c^1(\bar{\mathcal{O}})$, we may introduce two sequences $(a_\epsilon)$ and $(b_\epsilon)$ of smooth functions approximating $a$ and $b$. Since the resulting solution $g_\epsilon$ satisfies (8.53), so that the sequence $(g_\epsilon)$ is bounded in $L_m^p$, we may argue with the same (compactness) argument as in the proof of Lemma 8.12. We then conclude to the existence of a (renormalized) solution $g \in L_m^p$ to the transport equation (8.29) satisfying (8.40). Still for $p > 1$, but assuming $G \in L_m^p$ and $g \in L_m^p(\Sigma_0)$, we may introduce two sequences $(G_\epsilon)$ and $(g^\epsilon)$ of smooth functions approximating $G$ and $g$. Thanks to (8.40), the associated sequence of solutions $(g_\epsilon)$ is bounded (and better it is a Cauchy sequence) in $L_m^p(\mathcal{O})$ and we conclude again to the existence of a (renormalized) solution $g \in L_m^p(\mathcal{O})$ to the transport equation (8.29) satisfying (8.40). Finally, in the case $p = 1$ and $\lambda > \lambda_1^*$, we may find $q > 1$ small enough such that $\lambda > \lambda_q^*$. For $G, g \in L^1 \cap L^q$, the last step imply the existence of a renormalized
where we have added the two renormalized formulations (8.26) associated to $\beta_\delta(g_i)$ and renormalized once more the resulting solution using (1). Letting $\delta \to 0$, we immediately obtain

$$a \cdot \nabla_y \beta(g_1 + g_2) = \beta'(g_1 + g_2)[G_1 + G_2 - b(g_1 + g_2)]$$

in the distributional sense. For proving (3), we introduce the mollified sequence $(g_\epsilon)$ and $(\Phi_\epsilon)$ defined as in the statement of Lemma 8.5 so that

$$a \cdot \nabla g_{\epsilon} + bg_{\epsilon} = G_{\epsilon}, \quad a \cdot \nabla \Phi_{\epsilon} = c_{\epsilon}$$

with $G_\epsilon \to G$ and $c_\epsilon \to c$ in $L_{\text{loc}}^1(\bar{O})$ as $\epsilon \to 0$. The smooth function $h_\epsilon := g_\epsilon e^{-\Phi_\epsilon}$ satisfies

$$a \cdot \nabla_y h_\epsilon + (b + c_\epsilon)h_\epsilon = G_\epsilon e^{-\Phi_\epsilon}$$

and then

$$a \cdot \nabla_y \beta(h_\epsilon) + \beta'(h_\epsilon)(b + c_\epsilon)h_\epsilon = \beta'(h_\epsilon)G_\epsilon e^{-\Phi_\epsilon}$$

for any $\beta \in C_*^1(\mathbb{R})$. Passing to the limit $\epsilon \to 0$, we obtain the renormalized formulation of (8.27). $\square$

We now generalize the trace result to the framework of renormalized solutions.

**Theorem 8.8.** Assume that $a \in W_{\text{loc}}^{1,1}(\bar{O})$, $b, G \in L_{\text{loc}}^1(\bar{O})$ and that $g \in L_{\text{loc}}^1(\bar{O})$ is a renormalized solution to the transport equation (8.25). Then there exists a unique function

$$\gamma g \in L(\Sigma \setminus \Sigma_0; d\sigma_y)$$

which satisfies the renormalized Green formula

$$(8.28) \qquad \int_{\mathcal{O}} (\beta'(g)(G - bg) \varphi + \beta(g) \operatorname{div}(a\varphi)) dy = \int_{\Sigma} \beta(\gamma g) \varphi a \cdot n d\sigma_y,$$

for any $\varphi \in C_c^1(\bar{O})$ and any $\beta \in C_{\text{pw},*}^1(\mathbb{R})$.

*Proof of Theorem 8.8.* We fix $\beta_1: \mathbb{R} \to \mathbb{R}$ defined by $\beta_1(s) := s(1 + s^2)^{-1/2}$, so that $\beta_1 \in C_b^1(\mathbb{R})$ and $\beta_1: \mathbb{R} \to (-1, 1)$ is a bijection. Since then $\beta_1(g) \in L^\infty(\mathcal{O})$ and $\beta'(g)(G - bg) \in L_{\text{loc}}^1(\bar{O})$, we know from Theorem 8.3 that $\gamma\beta_1(g)$ is well defined in $L^\infty(\Sigma \setminus \Sigma_0)$ through the Green formula

$$\int_{\mathcal{O}} [\beta'_{1}(g)(G - bg) \varphi + \beta_{1}(g) \operatorname{div}(a\varphi)] dy = \int_{\Sigma} \gamma \beta_{1}(g) \varphi a \cdot n d\sigma_{y},$$

for any test function $\varphi \in C_c^1(\bar{O})$. We set $\gamma g := \beta_1^{-1}(\gamma \beta_1(g)) \in L(\Sigma \setminus \Sigma_0; d\sigma_y)$, with the convention $\beta_1^{-1}(\pm 1) = \pm \infty$. For any $\beta \in C_{\text{pw},*}^1(\mathbb{R})$, we then deduce

$$\gamma \beta(g) = \gamma[(\beta \circ \beta_1^{-1})(\beta_1(g))] = \beta \circ \beta_1^{-1}(\gamma \beta_1(g)) = \beta(\gamma g),$$

where we have used the renormalization result stated in Theorem 8.3 and the chain rule (1) stated in Lemma 8.7 in the second equality and the very definition of $\gamma g$ in the third equality. In other words, the renormalized Green formula (8.28) holds. $\square$

**Remark 8.9.** (1) We will see in Section 8.4 that under the same conditions as in Theorem 8.8 the information on $\gamma g$ can be slightly improved, in particular $\gamma g \in L^0(\Sigma \setminus \Sigma_0)$.

(2) Theorem 8.8 in particular holds when we assume $a \in W_{\text{loc}}^{1,p'}(\bar{O})$, $b \in L_{\text{loc}}^{p'}(\bar{O})$, $G \in L_{\text{loc}}^1(\bar{O})$ and $g \in L_{\text{loc}}^p(\bar{O})$ satisfy the transport equation (8.26) in the distributional sense. Indeed, in that situation one knows from the DiPerna-Lions renormalizing theory [138, Cor. II.2] that $g$ is also a renormalized solution to the transport equation (8.26) (in the above sense).

(3) Assuming more interior integrability on the functions $g$, $b$, $G$ and $a$, we may deduce more accurate information on $\gamma g$. A typical example, is that

$$\int_{\Sigma \cap B_R} |\gamma g|^r (|a \cdot n| \wedge 1)^2 d\sigma_y < \infty,$$

for some $r \in [1, \infty)$ and any $R > 0$, under the additional assumption

$$|g|^r (|\operatorname{div} a| + |a \cdot \nabla T_1(a \cdot n)| + |b|) \in L_{\text{loc}}^1(\bar{O}), \quad |g|^{r-1} |G| \in L_{\text{loc}}^1(\bar{O}).$$

The proof follows by choosing $\varphi := T_1(a \cdot n)\chi$, $\chi \in C_c^1(\bar{O})$, $0 \le \chi \le 1$, and $\beta_k(s) = (|s| \wedge k)^r$ in the associated Green formula (8.16), and then to pass to the limit $k \to \infty$.
solution $g \in L_{m_o}^q(\mathcal{O})$ to the transport equation (8.29). Renormalizing the equation, we deduce that $g$ satisfies (8.40) for $p = 1$. When $G, g \in L^1$, we introduce two sequences $(G_\epsilon)$ and $(g^\epsilon)$ of $L^1 \cap L^q$ functions approximating $G$ and $g$, and using (8.40) for $p = 1$, we deduce that the resulting sequence $(g_\epsilon)$ is a Cauchy sequence in $L_{m_o}^1(\mathcal{O})$. We easily conclude again. Finally $\gamma_+g \in L_{m_\Sigma}^p(\Sigma_+)$ from (8.41) (see also Remark 8.9-(4)). $\square$

**Uniqueness.** We present now a uniqueness result.

**Lemma 8.14 (Uniqueness).** We assume that $a$ and $b$ satisfy (8.35) for some exponent $p \in [1, \infty]$ and some weight function $m: \bar{\mathcal{O}} \to [1, \infty)$ as well as for $p = 1$ and $m \equiv 1$. We additionally assume diva $\in L_{\text{loc}}^\infty(\bar{\mathcal{O}})$ (what is automatically true under assumption (8.33)). With obvious notations, for any $\lambda > \max(\lambda_p^*(m), \lambda_1^*(1))$, and any solution $g \in L_{m_o}^p(\mathcal{O})$ to the transport equation

$$ (8.54) \qquad \lambda g + a \cdot \nabla g + bg = 0 \quad \text{in } \mathcal{D}'(\mathcal{O}), \quad \gamma_-g = 0 \quad \text{on } \Sigma_-, $$

we have $g \equiv 0$.

**Proof of Lemma 8.14.** We main follow the proof of [139, Cor. II.1]. We fix $\beta \in W^{1,\infty}(\mathbb{R})$, $\beta(0) = 0$, in such a way that $\beta(g) \in L_{m_o}^p \cap L^\infty$ is a solution to

$$ (\lambda + b)g\beta'(g) + a \cdot \nabla\beta(g) = 0 \quad \text{in } \mathcal{D}'(\mathcal{O}), \quad \gamma_-\beta(g) = 0 \quad \text{on } \Sigma_- $$

For any $\psi \in C_c(\mathcal{O})$ and any $\lambda > \lambda_{m,p}^*$, we solve in $L_{m^{-1}(\omega_+)^{1/p'}}^{p'} \cap L^\infty$ the dual problem

$$ (8.55) \qquad \lambda\varphi - a \cdot \nabla\varphi + (b - \text{diva})\varphi = \psi \quad \text{in } \mathcal{D}'(\mathcal{O}), \quad \gamma_+\varphi = 0 \quad \text{on } \Sigma_+, $$

thanks to Lemma 8.12 and Lemma 8.13, where we observe that, because of (8.37), the necessary condition on $\lambda$ in these results is precisely the one made here. For $\chi \in C_c^1(\mathbb{R}^D)$, $\mathbf{1}_{B_1} \le \chi \le \mathbf{1}_{B_2}$, and $R > 0$, we define $\chi_R(x) := \chi(x/R)$. Using the Green formula (8.21), we have

$$ 0 = \int_{\mathcal{O}} ((\lambda + b)\varphi - \psi)\beta(g)\chi_R - \int_{\mathcal{O}} (\lambda + b)\varphi g\beta'(g)\chi_R + \int_{\mathcal{O}} \varphi\beta(g) \frac{a}{R} \cdot (\nabla\chi)_R. $$

Because on the one hand $\varphi\beta(g) \in L_{\langle\omega_+\rangle}^1 \cap L^\infty$ and on the other hand $a/R \cdot (\nabla\chi)_R \to 0$ a.e. and is bounded in $L_{\langle\omega_+\rangle^{-1}}^\infty + L^1$ we deduce that the last term vanishes when $R \to \infty$. Using also that $b\varphi g \in L^1$ thanks to (8.35), we may pass to the limit $R \to \infty$ in the above equation and we get

$$ 0 = \int_{\mathcal{O}} ((\lambda + b)\varphi - \psi)\beta(g) - \int_{\mathcal{O}} (\lambda + b)\varphi g\beta'(g). $$

We take $\beta := \beta_\delta$ and we observe that $\langle b \rangle |\varpi||\beta_\delta(g) - g\beta_\delta'(g)| \le \langle b \rangle |\varphi g| \in L^1(\mathcal{O})$. We may then pass to the limit $\delta \to 0$ in the last equation, and we get

$$ 0 = - \int_{\mathcal{O}} \psi g, \quad \forall \psi \in C_c(\mathcal{O}), $$

from which we conclude that $g \equiv 0$. $\square$

We come to the time dependent transport equation by formulating a general continuity result.

**Proposition 8.15.** Assume that $a \in W_{\text{loc}}^{1,1}(\bar{\mathcal{O}})$, $b \in L_{\text{loc}}^1(\bar{\mathcal{O}})$, $G \in L_{\text{loc}}^1([0, T] \times \bar{\mathcal{O}})$. Any renormalized solution $g \in L_{\text{loc}}^1([0, T] \times \bar{\mathcal{O}})$ to the first equation in (8.31), meaning

$$ \frac{\partial}{\partial t}\beta(g) + a \cdot \nabla\beta(g) + \beta'(g)bg = \beta'(g)G \quad \text{in } \mathcal{D}'((0, T) \times \mathcal{O}), $$

for any renormalizing function $\beta \in C_*^1(\mathbb{R})$, satisfies $g \in C([0, T]; L^0(\mathcal{O}))$, meaning that $\beta(g) \in C([0, T]; L_{\text{loc}}^1(\bar{\mathcal{O}}))$ for any $\beta \in C_b(\mathbb{R})$.

**Proof of Proposition 8.15.** The proof is a variant of the proof of [139, Thm. II.3] and we just allude it. Because $\beta(g) \in L^\infty((0, T) \times \mathcal{O})$ is a solution to the transport equation with source term $\beta'(g)G - \beta'(g)bg \in L_{\text{loc}}^1([0, T] \times \bar{\mathcal{O}})$, we have $\beta(g) \in C([0, T]; \mathcal{D}'(\mathcal{O}))$ for any $\beta \in C_*^1(\mathbb{R})$. Fixing $\beta_0 \in C_*^1(\mathbb{R})$ strictly increasing, we deduce that $\beta_0(g), \beta_0(g)^2 \in C([0, T]; \mathcal{D}'(\mathcal{O}))$, so that $\beta_0(g) \in C([0, T]; L_{\text{loc}}^2(\bar{\mathcal{O}}))$, and the conclusion. $\square$

We consider now the time dependent transport equation (8.31).
(4) Even more integrability on $\gamma_{\pm}g$ is available on one part of the boundary if additional integrability assumption is made on $\gamma_{\mp}g$ on the other part of the boundary. A typically example, is that

$$ \int_{\Sigma_{\pm} \cap B_R} |\gamma_{\pm} g|^r |a \cdot n| d\sigma_y < \infty, $$

under the additional assumption

$$ |g|^r (|\operatorname{div} a| + |a| + |b|) \in L_{\text{loc}}^1(\bar{O}), \quad |g|^{r-1} |G| \in L_{\text{loc}}^1(\bar{O}), \quad |\gamma_{\mp} g|^r |a \cdot n| \in L_{\text{loc}}^1(\bar{\Sigma}_{\mp}). $$

The proof follows by choosing $\varphi \in C_c^1(\bar{O})$, $0 \le \varphi \le 1$, and $\beta_k(s) = (|s| \wedge k)^r$ in the associated Green formula (8.16), and then to pass to the limit $k \to \infty$.

(5) The results stated in Lemma 8.7, in Theorem 8.8 and in points (1), (2), (3) and (4) above may be straightforwardly adapted to the evolution transport equation (8.10). We refer to [270, 269, 271, 70] where such results are established in a slightly less general framework. Let us emphasize again that when $a \in W_{\text{loc}}^{1,1}([0,T] \times \bar{O})$ (what it is the case in the time independent case when a satisfies (8.5)) this extension is directly implied by Theorem 8.8 applied to the vector field $(1, a(t, y))$ in the open set $(0, T) \times \mathcal{O}$.

## 8.2. Well-posedness for the transport equation with given inflow at the boundary.

We deduce from the previous trace theorems and standard tools the well-posedness for the transport equation with several boundary conditions. In this section, we deal with the transport equation with given inflow at the boundary. We are first concerned with the stationary transport equation

$$ (8.29) \qquad \lambda g + a \cdot \nabla g + bg = G \quad \text{in } \mathcal{O}, \quad \gamma_{-}g = g \quad \text{on } \Sigma_{-}, $$

for a real number $\lambda \in \mathbb{R}$ large enough, a given source term $G : \mathcal{O} \to \mathbb{R}$ and a boundary term $g : \Sigma_{-} \to \mathbb{R}$. As we will see, our analysis also apply to the associated dual equation

$$ (8.30) \qquad \lambda\varphi - a \cdot \nabla\varphi + (b - \operatorname{div} a)\varphi = \Phi \quad \text{in } \mathcal{D}'(\mathcal{O}), \quad \gamma_{+}\varphi = \psi \quad \text{on } \Sigma_{+}. $$

We will also consider the related evolution equation

$$ (8.31) \qquad \begin{cases} \frac{\partial g}{\partial t} + a \cdot \nabla g + bg = G & \text{on } (0, T) \times \mathcal{O}, \\ \gamma_{-}g = g & \text{on } (0, T) \times \Sigma_{-}, \\ g(0, \cdot) = g_0 & \text{on } \mathcal{O}, \end{cases} $$

with given source term $G : (0, T) \times \mathcal{O} \to \mathbb{R}$, boundary term $g : (0, T) \times \Sigma_{-} \to \mathbb{R}$ and initial datum $g_0 : \mathcal{O} \to \mathbb{R}$.

A possible simple framework consists in imposing the following conditions

$$ (8.32) \qquad a \in W_{\text{loc}}^{1,1}(\bar{O}), \quad b \in L_{\text{loc}}^1(\bar{O}), $$

and

$$ (8.33) \qquad b_{-}, \operatorname{div} a \in L^{\infty}(\mathcal{O}), \quad \frac{a}{\langle y \rangle \langle b_{+} \rangle} \in L^{1}(\mathcal{O}) + L^{\infty}(\mathcal{O}). $$

The first condition on $a$ is useful for the renormalization trick and the definition of the trace, the second condition is needed for the existence results in a $L^p$ framework when $p \ne \infty$ and the last condition is used for proving the uniqueness result. In order to be able to apply our results to more general (and realistic) situations, we rather consider the following situation. We assume that $a$ and $b$ satisfy (8.32), and defining

$$ (8.34) \qquad \bar{\varpi} = \bar{\varpi}_p := b - \frac{1}{p}\operatorname{div} a - a \cdot \frac{\nabla m}{m}, $$

for some smooth enough weight function $m : \bar{O} \to (0, \infty)$ and some exponent $p \in [1, \infty]$, we assume

$$ (8.35) \qquad \bar{\varpi}_{-} \in L^{\infty}(\mathcal{O}), \quad b, \operatorname{div} a \in L_{\langle \bar{\varpi}_{+} \rangle}^{\infty}(\mathcal{O}), \quad \frac{a}{\langle y \rangle} \in L_{\langle \bar{\varpi}_{+} \rangle}^{\infty}(\mathcal{O}) + L^{1}(\mathcal{O}) $$

In the case $p = 1$ and $p = \infty$, we will additionally assume $(\bar{\varpi}_q)_- \in L^\infty$ for any $q \in (1, \infty)$. It is worth emphasizing that condition (8.35) automatically holds when $m \equiv 1$ and $a, b$ satisfy (8.33). We also define the critical real number

$$ (8.36) \qquad \lambda_p^* = \lambda_p^*(a, b, m) := \|\bar{\varpi}_-\|_{L^\infty}, $$
**Proposition 8.16** (Renormalized solutions). We assume (8.35) for some $p \in [1, \infty]$ and some weight function $m$. For any $g_0 \in L_m^p(\mathcal{O})$, $G \in L_{\bar{m}_\infty}^p((0, T) \times \mathcal{O})$, $\mathfrak{g} \in L_{m_\Sigma}^p((0, T) \times \Sigma)$, there exists a unique $g \in C([0, T]; L_{\text{loc}}^1(\bar{\mathcal{O}}))$ satisfying the estimate (8.45) or (8.46) and being a solution to the transport equation (8.31) in the renormalized sense, namely

$$ (8.56) \qquad \begin{cases} \frac{\partial \beta(g)}{\partial t} + a \cdot \nabla \beta(g) + \beta'(g)bg = \beta'(g)G & \text{on } (0, T) \times \mathcal{O}, \\ \gamma_{-} \beta(g) = \beta(\mathfrak{g}) & \text{on } (0, T) \times \Sigma_{-}, \quad \beta(g)(0, \cdot) = \beta(g_0) & \text{on } \mathcal{O}, \end{cases} $$

for any $\beta \in C_{\text{pw},*}^1$. Furthermore, $g \in C([0, T]; L_m^p)$ when $p \in [1, \infty)$ and $g \ge 0$ if $g_0, G, \mathfrak{g} \ge 0$.

**Remark 8.17.** (1) The above result extends some previous results due to Bardos in [37, Chap. III], Boyer in [70, Thm. 4.1] and Crippa et al in [120, Thm. 1.1] and [119, Thm. 1.1], where the cases $p=2$ or $p=\infty$ are considered with always the additional assumption $a \in L^\infty$ (in the last paper however the present $W^{1,1}$ bound on $a$ is relaxed into a BV condition) by adapting the Di Perna-Lions theory developed in [139, Sec. II].

(2) We immediately deduce from the above result and Lemma 8.7-(2) a weak maximum principle: $g_1 \le g_2$ if $g_i$ is renormalized solution to the transport equation (8.31) associated to the data $g_{0i}$, $G_i$, $\mathfrak{g}_i$ such that $g_{01} \le g_{02}$, $G_{01} \le G_{02}$, $\mathfrak{g}_{01} \le \mathfrak{g}_{02}$.

**Proof of Proposition 8.16.** We proceed similarly as during the proof of Lemma 8.12.

**Step 1. Characteristics.** We assume first $a \in C^1(\mathbb{R}^D)$, $g_0 \in C_c(\mathcal{O})$, $b \in C_b(\bar{\mathcal{O}})$, $\mathfrak{g} \in C_c((0, T) \times \Sigma_0)$, $G \in C_c^1((0, T) \times \mathcal{O})$. We use the characteristics representation (8.47). We may verify that $\bar{g}$ both satisfies the transport equation in the renormalized sense and the boundary conditions in (8.56) and we may justify the computations leading to the a priori estimates (8.45) and (8.46).

**Step 2. Existence.** In the general case, we define some regularized sequence $(a_\epsilon)$, $(g_{0,\epsilon})$, $(b_\epsilon)$ ($\mathfrak{g}_\epsilon$), $(G_\epsilon)$ and thanks to the first step we deduce the existence of an associated function $g_\epsilon \in C([0, T]; L_m^p)$ satisfies both the equation (8.56) in the renormalized sense and the a priori estimates (8.45) or (8.46). When $p > 1$, the sequence $(g_\epsilon)$ is bounded in $L^\infty(0, T; L_m^p)$ and (up to the extraction of a subsequence) we may pass to the limit $\epsilon \to 0$ using Proposition 8.10-(2) and Remark 8.11. We have established the existence of a renormalized solution to the transport equation which satisfies the estimate (8.45) or (8.46). When $p = 1$, we may for instance proceed in the following way by first assuming $0 \le g_0 \in L_m^1$, $0 \le G \in L_m^1((0, T) \times \mathcal{O})$, $0 \le \mathfrak{g} \in L_{m_\Sigma}^1((0, T) \times \Sigma)$. We may thus consider some nonnegative approximating sequences $(g_{0,\epsilon})$ in $L_m^p \cap L_m^1$, $G_\epsilon \in L_{\bar{m}_\infty}^p \cap L_m^1$, $\mathfrak{g}_\epsilon \in L_{m_\Sigma}^p \cap L_{m_\Sigma}^1$ such that $g_{0,\epsilon} \nearrow g_0$, $G_\epsilon \nearrow G$ and $\mathfrak{g}_\epsilon \nearrow \mathfrak{g}$. The same construction as above implies the existence of $0 \le g_\epsilon \in L^\infty(0, T; L_m^1 \cap L_m^p)$ renormalized solution to the transport equation associated to these data and such that $(g_\epsilon)$ is increasing and uniformly bounded in $L^\infty(0, T; L_m^1)$ thanks to the a priori $L_m^1$ estimate (8.45). There thus exists $0 \le g \in L^\infty(0, T; L_m^1)$ such that $g_\epsilon \nearrow g$, and we get that $g$ is a renormalized solution to the transport equation by using again Proposition 8.10-(2) and Remark 8.11. We remove the nonnegative condition on $g_0$, $G$ and $\mathfrak{g}$ by introducing the positive and negative parts of each function, using the preceding step in order to prove the existence of two solutions $0 \le g_\pm \in L^\infty(0, T; L_m^1)$ associated respectively to $(g_{0+}, G_+, \mathfrak{g}_+)$ and $(g_{0-}, G_-, \mathfrak{g}_-)$, and finally defining $g := g_+ - g_-$ which is a renormalized solution to the transport equation thanks to Lemma 8.7 and Remark 8.9-(5).

**Step 3. Continuity.** From Proposition 8.15, we already know that $g \in C([0, T]; L^0(\mathcal{O}))$. Together with the a priori estimate (8.45) or (8.46), we also have $g \in C([0, T]; L_{\text{loc}}^1(\mathcal{O}))$ when $p > 1$. When $p \in [1, \infty)$, we may improve the above continuity properties by arguing in the following way. We define $\tilde{g} := gm$ and we observe that it is a solution to the transport equation

$$ \partial_t \tilde{g} + a \cdot \nabla \tilde{g} + b \tilde{g} = \tilde{G}, \quad \gamma \tilde{g} = \tilde{\mathfrak{g}}, \quad \tilde{g}(0) = \tilde{g}_0, $$

with $\tilde{b} := b - a \cdot \nabla m/m$, $\tilde{G} := gm$, $\tilde{\mathfrak{g}} := gm$ and $\tilde{g}_0 := g_0m$. We write the associated renormalized equation (8.44) for the renormalizing function $\beta_M(s) := (|s| \wedge M)^p$, $M > 0$, and the test function $\varphi := \chi_R$, with $\chi \in C_c^1(\mathbb{R}^d)$, $\mathbf{1}_{B_1} \le \chi \le \mathbf{1}_{B_2}$ and $\chi_R(y) := \chi(y/R)$. Observing in particular that

$$ \int_0^t \int_{\mathcal{O}} \beta_M(\tilde{g}) a \cdot \nabla \chi_R \to 0 \text{ as } R \to \infty, $$
and we may observe that

$$ (8.37) \qquad \lambda_{p'}^*(-a, b - \operatorname{div} a, m^{-1}) = \lambda_p^*(a, b, m), $$

what links up the primal and the dual problems. In order to shorten notations, we introduce the three weight functions

$$ (8.38) \qquad m_O := m(\bar{\omega}_+)^{1/p}, \quad \tilde{m}_O := m(\bar{\omega})^{-1/p'}, \quad m_\Sigma := m|a \cdot n|^{1/p}. $$

We start with a general discussion about a priori bounds, formal representation formulas and general stability results.

**A priori estimates.** Consider a solution $g$ to the stationary equation (8.29). For any renormalizing function $\beta : \mathbb{R} \to \mathbb{R}_+$ and any function $\varphi : \bar{\mathcal{O}} \to (0, \infty)$, we (at least) formally have

$$ \int_{\mathcal{O}} [(\lambda + b)g\beta'(g)\varphi - \beta(g)(\operatorname{div}(a\varphi))] + \int_{\Sigma_+} a \cdot n\beta(\gamma_+g)\varphi = \int_{\mathcal{O}} \beta'(g)G\varphi + \int_{\Sigma_-} |a \cdot n|\beta(\mathfrak{g})\varphi. $$

Choosing $\beta(s) := |s|^p$, $1 \le p < \infty$, and $\varphi := m^p$, we get in particular

$$ (8.39) \qquad \int_{\mathcal{O}} |g|^p m^p (\lambda + \bar{\omega}) + \frac{1}{p} \int_{\Sigma_+} |\gamma_+ g|^p m^p a \cdot n = \int_{\mathcal{O}} G g |g|^{p-2} m^p + \frac{1}{p} \int_{\Sigma_-} |\mathfrak{g}|^p m^p |a \cdot n|. $$

For $p=1$ and any $\lambda > \lambda_1^*$, we get

$$ \int_{\mathcal{O}} |g| m \{ \lambda - \lambda_1^* + \bar{\omega}_+ \} + \int_{\Sigma_+} |\gamma_+ g| m_{\Sigma} \le \int_{\mathcal{O}} |G| m + \int_{\Sigma_-} |\mathfrak{g}| m_{\Sigma}. $$

For $p \in (1, \infty)$, we split $G = G_1 + G_2$ and using the Young inequality, we have

$$ \int_{\mathcal{O}} G g |g|^{p-2} m^p \le \varepsilon_1 \int g^p m^p + \varepsilon_2 \int g^p m_{\mathcal{O}}^p + \frac{1}{p} \frac{1}{(p'\varepsilon_1)^{p/p'}} \int G_1^p m^p + \frac{1}{p} \frac{1}{(p'\varepsilon_2)^{p/p'}} \int G_2^p \tilde{m}_{\mathcal{O}}^p, $$

for any $\varepsilon_i > 0$. For $\lambda > \lambda_p^*$, we choose $\varepsilon_1 := (\lambda - \lambda_p^*)/2$ and $\varepsilon_2 := 1/2$, we get

$$ \begin{aligned} & \frac{1}{2} \int_{\mathcal{O}} |g|^p m_{\mathcal{O}}^p + \frac{\lambda - \lambda_p^*}{2} \int_{\mathcal{O}} |g|^p m^p + \frac{1}{p} \int_{\Sigma_+} |\gamma_+ g|^p m_{\Sigma}^p \\ & \le \frac{2^{p-1}}{p(p')^{p/p'}} (\lambda - \lambda_p^*)^{-1-p} \int_{\mathcal{O}} |G_1|^p m^p + \frac{1}{p(p'/2)^{p/p'}} \int_{\mathcal{O}} |G_2|^p \tilde{m}_{\mathcal{O}}^p + \frac{1}{p} \int_{\Sigma_-} |\mathfrak{g}|^p m_{\Sigma}^p. \end{aligned} $$

We thus deduce

$$ (8.40) \qquad \|g\|_{L_m^p} \le \frac{C_p}{\lambda - \lambda_p^*} \|G_1\|_{L_m^p} + \frac{C_p}{(\lambda - \lambda_p^*)^{1/p}} (\|\mathfrak{g}\|_{L_{m\Sigma}^p} + \|G_2\|_{L_{\tilde{m}\mathcal{O}}^p}) $$

and

$$ (8.41) \qquad \|g\|_{L_{m\mathcal{O}}^p} + \|\gamma_+ g\|_{L_{m\Sigma}^p} \le \frac{C_p}{(\lambda - \lambda_p^*)^{1/p'}} \|G_1\|_{L_m^p} + C_p (\|\mathfrak{g}\|_{L_{m\Sigma}^p} + \|G_2\|_{L_{\tilde{m}\mathcal{O}}^p}), $$

for some numerical constant $C_p \in (0, \infty)$ and any $p \in (1, \infty)$ and also for $p=1$ because of the previous estimate. Finally, for $\lambda > \lambda_\infty^*$ and $\alpha \in (0, \lambda - \lambda_\infty^*)$, we may proceed exactly as above, but throwing also away the contribution of $\bar{\omega}_+$, and we may thus first write

$$ (8.42) \qquad \left( \lambda - \|\bar{\omega}_-\|_{L^\infty} - \frac{\alpha^{p'}}{p'} \right) \|g\|_{L_m^p(\mathcal{O})}^p \le \frac{1}{p} \left( \|G/\alpha\|_{L_m^p(\mathcal{O})}^p + \|\mathfrak{g}|a \cdot n|^{1/p} \|_{L_m^p(\Sigma_-)}^p \right), $$

for any $p \in (1, \infty)$ large enough in such a way that the coefficient in front of $\|g\|_{L_m^p(\mathcal{O})}^p$ is positive. Taking the power $1/p$ in both sides and passing first to the limit $p \nearrow \infty$ and next to the limit $\alpha \nearrow \lambda - \lambda_\infty^*$, we end with

$$ (8.43) \qquad \|g\|_{L_m^\infty(\mathcal{O})} \le \max \left( \frac{1}{\lambda - \lambda_\infty^*} \|G\|_{L_m^\infty(\mathcal{O})}, \|\mathfrak{g}\|_{L_m^\infty(\Sigma_-)} \right). $$
because of (8.45) and (8.35) by arguing as in the proof of Lemma 8.14, we may pass to the limit
in the associated renormalized equation as $R \to \infty$, and we obtain

$$
\begin{aligned}
\left[ \int_{\mathcal{O}} \beta_M(\tilde{g}) dy \right]_0^t &= \int_0^t \int_{\mathcal{O}} \beta'_M(\tilde{g}) \tilde{G} dy ds - \int_0^t \int_{\Sigma} \beta_M(\gamma \tilde{g}) a \cdot n d\sigma_y ds \\
&\quad + \int_0^t \int_{\mathcal{O}} \beta_M(\tilde{g}) \mathbf{1}_{\tilde{g}>M} \mathrm{div} a dy ds - \int_0^t \int_{\mathcal{O}} p \beta_M(\tilde{g}) \mathbf{1}_{\tilde{g}\le M} \varpi dy ds.
\end{aligned}
$$

Using again (8.45) and (8.35), we may next pass to the limit as $M \to \infty$ in the above equation,
and we get

$$
\frac{d}{dt} \int_{\mathcal{O}} |\tilde{g}|^p = -p \int_{\mathcal{O}} |\tilde{g}|^p \varpi + \int_{\mathcal{O}} p \tilde{G} \tilde{g} |\tilde{g}|^{p-2} + \int_{\Sigma} |\gamma \tilde{g}|^p a \cdot n \in L^1(0,T).
$$

We deduce that $t \mapsto \|g(t)\|_{L_m^p} = \|\tilde{g}(t)\|_{L^p}$ is continuous. Consider then $t \in [0, T]$ and $t_k \to t$, so
that in particular $\|g_{t_k}\|_{L_m^p} \to \|g_t\|_{L_m^p}$ as $k \to \infty$. On the other hand, we have yet established that
$\|\beta_0(g_{t_k}) - \beta_0(g_t)\|_{L^1(\mathcal{O}\cap B_R)} \to 0$ as $k \to \infty$ for any $R > 0$. There exists thus a subsequence $(g_{t_{k'}})$
such that $g_{t_{k'}} \to g_t$ a.e. on $\mathcal{O}$. Thanks to Brézis-Lieb theorem [72], we deduce that $g_{t_{k'}} \to g_t$ in $L_m^p$
and it is the whole sequence which converges by uniqueness of the limit. We have thus established
$g \in C([0, T]; L_m^p)$ when $p \in [1, \infty)$.

Step 4. Uniqueness. Because of Lemma 8.7 and Remark 8.9-(5), we just have to prove that $g \equiv 0$ if $g$ is a renormalized solution associated to vanishing data $g_0 = 0$, $G = 0$ and $g = 0$. When $p \in [1, \infty)$, the previous step implies that

$$
\frac{d}{dt} \int_{\mathcal{O}} |g|^p m^p = \int_{\mathcal{O}} |g|^p m^p \varpi \in L^1(0, T), \quad \int_{\mathcal{O}} |g(0)|^p m^p = 0,
$$

and together with the Gronwall lemma, we deduce that $g = 0$. The case $p = \infty$ may be tackled
thanks to a duality argument exactly as in the proof of Lemma 8.14. $\square$

**Corollary 8.18.** The semigroup $S_b$ defined by (8.49) extends to a positive semigroup of contractions in $L_m^p$.

Proof of Corollary 8.18. We just apply Proposition 8.16 with $G = g = 0$. When $p \in [1, \infty)$, we define in that way a mapping $L_m^p \to C(\mathbb{R}_+; L_m^p)$, $g_0 \mapsto g$, where $g$ denotes the unique renormalized solution. Defining then $S(t)g_0 := g(t)$ we have built a strongly continuous semigroup in $L_m^p$. The case $p = \infty$ is identical, except the fact that the semigroup is only weak $\ast\sigma(L_m^\infty, L_{m-1}^1)$ continuous. The positivity has been established in Proposition 8.16 and the contraction property comes from the estimates (8.45) and (8.46). $\square$

**Remark 8.19.** It is worth emphasizing that in Bardos [37] the semigroup is defined by its representation formula for smooth data and by Hille-Yosida theory for $L^2$ data. Here we proceed in another way, by rather following [139, 271, 261].

**8.3. Optimal weighted trace theorem and transport equation with reflection at the boundary.** We define the functions $\tau^{\pm}$ as the solutions to

$$
(8.57) \qquad \lambda_0 \tau^{\pm} \mp a \cdot \nabla \tau^{\pm} = 1 \quad \text{in } \mathcal{D}'(\mathcal{O}), \quad \gamma_{\pm} \tau^{\pm} = 0 \quad \text{on } \Sigma_{\pm},
$$

with $\lambda_0 := 1 + \|\mathrm{div}a\|_{L^\infty}$.

**Lemma 8.20.** Each of the two equations (8.57) has a unique solution $\tau^{\pm} \in L^{\infty}(\mathcal{O})$ and

$0 < \tau^{\pm} \le 1 \text{ a.e. in } \mathcal{O}, \quad 0 < \gamma_{\pm} \tau^{\pm} \le 1 \text{ a.e. on } \Sigma_{\mp}.$

Proof of Lemma 8.20. We follow a similar proof as in [70, Prop. 5.1] (see also [271, Sec. 5]). We only deals with $\tau^{-}$ since the case of $\tau^{+}$ can be handled in the same way. The existence of $\tau^{-} \in L^{\infty}$, its non negativity and the upperbound are consequences of Lemma 8.12 while the uniqueness is ensured by Lemma 8.14. In order to prove the strict positivity we argue as follows. We first fix $A \in \mathcal{O}$, $|A| \in (0, \infty)$ and we solve

$$
\lambda_0 \varphi - \operatorname{div}(a\varphi) = \mathbf{1}_A \quad \text{in } \mathcal{D}'(\mathcal{O}), \quad \gamma_+ \varphi = 0 \quad \text{on } \Sigma_+,
$$
Consider now a solution to the evolution equation (8.31). For any renormalizing function $\beta: \mathbb{R} \to \mathbb{R}_+$ and any test function $\varphi: [0, \infty) \times \mathcal{O} \to (0, \infty)$, we (at least) formally have

$$ (8.44) \qquad \begin{aligned} & \int_0^t \int_{\mathcal{O}} (\beta'(g)G\varphi - \beta'(g)g\varphi + \beta(g)[\partial_t\varphi + \text{div}(a\varphi)])dyds \\ &= \left[ \int_{\mathcal{O}} \beta(g(s,y))\varphi(s,y)dy \right]_0^t + \int_0^t \int_{\Sigma} \beta(\gamma g)\varphi a \cdot n d\sigma_y ds. \end{aligned} $$

Choosing $\beta(s) := |s|^p$, $1 \le p < \infty$, and $\varphi(t, y) := m^p(y)$, we get in particular

$$ \begin{aligned} & \int_{\mathcal{O}} |g(t)|^p m^p + \int_0^t \int_{\Sigma_+} a \cdot n |\gamma_+ g|^p m^p + p \int_0^t \int_{\mathcal{O}} |g|^p m^p \varpi \\ &= \int_{\mathcal{O}} |g_0|^p m^p + p \int_0^t \int_{\mathcal{O}} g |g|^{p-2} G m^p + \int_0^t \int_{\Sigma_-} |g|^p m^p |a \cdot n|. \end{aligned} $$

Using the Young inequality

$$ p \int_{\mathcal{O}} g |g|^{p-2} G m^p \le \frac{p}{p'} \int_{\mathcal{O}} |g|^p m_{\mathcal{O}}^p + \int_{\mathcal{O}} |G|^p \tilde{m}_{\mathcal{O}}^p $$

and the Gronwall lemma, we then deduce

$$ (8.45) \qquad \begin{aligned} & \|g(t)\|_{L_m^p}^p + \int_0^t e^{p\kappa(t-s)} (\|g_s\|_{L_m^p}^p + \|\gamma_+g_s\|_{L_m^{\infty}}^p) ds \\ & \le e^{p\kappa t} \|g_0\|_{L_m^p}^p + \int_0^t e^{p\kappa(t-s)} (\|G_s\|_{L_m^p}^p + \|g_s\|_{L_m^{\infty}}^p) ds, \quad \forall t \ge 0, \end{aligned} $$

with $\kappa := \|\langle \varpi_- \rangle\|_{L^\infty}$. Passing to the limit $p \to \infty$, we also have

$$ (8.46) \qquad \max(\|g(t)\|_{L_m^\infty}, \|\gamma_+g(t)\|_{L_m^\infty}) \le e^{\kappa t} \max(\|g_0\|_{L_m^\infty}, \sup_{[0,t]}(\|G_s\|_{L_m^\infty} + \|g_s\|_{L_m^\infty})), $$

for any $t \ge 0$.

**Representation formulas.** In a smooth functions framework or still formally, one classically knows that the solution $g$ to the evolution transport equation (8.31) is given by

$$ (8.47) \qquad \begin{aligned} g(t,y) = & g_0(Y_{-t}(y))e^{-\int_0^t b(Y_{s-t}(y))ds} \mathbf{1}_{t<t_b} + \mathfrak{g}(t-t_b, y_b)e^{-\int_0^{t_b} b(Y_{s-t_b}(y))ds} \mathbf{1}_{t>t_b} \\ & + \int_0^{t_b} G(s, Y_{s-t_b'}(y))e^{-\int_0^{t_b-s} b(Y_{-\tau}(y))d\tau} du, \end{aligned} $$

where we recall that the characteristics $Y$ and the backward exit time $t_b$ are defined in (8.6)-(8.7) and we denote $t_b' := \min(t, t_b)$. Similarly, the solution $g$ to the stationary transport equation (8.29) is given by

$$ (8.48) \qquad g(y) = \mathfrak{g}(y_i(y))e^{-\int_0^{t_b} b(Y_{-s}(y))ds} + \int_0^{t_b} G(Y_{-s}(y))e^{-\int_0^s b(Y_{-\tau}(y))d\tau} du. $$

Alternatively, we may define a semigroup $S_b$ (say on $L^\infty(\mathcal{O})$) by

$$ (8.49) \qquad (S_b(t)f_0)(y) := \begin{cases} f_0(Y_{-t}(y)) \exp(-\int_0^t b(Y_{\tau-t}(y))d\tau), & \text{if } t \in (0, t_b(y)), \\ 0 & \text{otherwise.} \end{cases} $$

Given $f_0: \mathcal{O} \to \mathbb{R}$, the function $f(t, y) := (S_b(t)f_0)(y)$ is thus a solution to the evolution equation

$$ \partial_t f + a \cdot \nabla f + bf = 0 \text{ in } (0, \infty) \times \mathcal{O}, \quad \gamma_-f = 0 \text{ on } (0, \infty) \times \Sigma_- $$

For $G, \tilde{g}: \mathcal{O} \to \mathbb{R}$, we next define

$$ (8.50) \qquad g := \tilde{g} + \int_0^\infty e^{-\lambda t} S_b(t) \tilde{G} dt, $$

with $\tilde{G} := G - \lambda \tilde{g} + a \cdot \nabla_x \tilde{g} - b \tilde{g}$. By construction, it is a solution to the stationary transport equation (8.29).
for which there exists a unique solution $\varphi \in L^1(\mathcal{O})$ thanks to Lemma 8.12 and Lemma 8.14, which furthermore satisfies $\varphi \ge 0$ and $\varphi \ne 0$. We observe that $\tau^-\varphi \in L^1(\mathcal{O})$ satisfies

$$ \mathrm{div}(a\tau^{-}\varphi) = \varphi - \tau \mathbf{1}_{A} \quad \text{in } \mathcal{O}, \quad \gamma(\tau^{-}\varphi) = 0 \quad \text{on } \Sigma \setminus \Sigma_{0}. $$

Thanks to the Green formula, first written for $\beta_{\delta}(\tau^{-}\varphi)$ and next passing to the limit $\delta \to 0$, we deduce

$$ 0 = \int_{\Sigma} \gamma(\tau^{-}\varphi) a \cdot n d\sigma_y = \int_{\mathcal{O}} \mathrm{div}(a\tau^{-}\varphi) dy = \int_{\mathcal{O}} \varphi - \int_{A} \tau_{-} dy, $$

so that the last integral does not vanish. This being true for any $A \subset \mathcal{O}$, we get $\tau^{-} > 0$ a.e. on $\mathcal{O}$. For $A \subset \Sigma_{+}$ such that

$$ 0 < \int_{A} (a \cdot n)_{+} d\sigma_y < \infty, $$

we solve

$$ \lambda_0\varphi - \mathrm{div}(a\varphi) = 0 \quad \text{in } \mathcal{D}'(\mathcal{O}), \quad \gamma_+\varphi = \mathbf{1}_A \quad \text{on } \Sigma_+, $$

thanks to Lemma 8.12 and Lemma 8.14, and we get a unique solution $0 \le \varphi \in L^1(\mathcal{O})$ such that $\varphi \ne 0$. The Green formula again implies

$$ \int_A \gamma \tau^{-} (a \cdot n)_{+} d\sigma_y = \int_{\mathcal{O}} \mathrm{div}(a\tau^{-}\varphi) dy = \int_{\mathcal{O}} \varphi, $$

so that the first integral does not vanish. This being true for any $A \subset \Sigma_+$, we conclude that $\gamma_+\tau^- > 0$ a.e. on $\Sigma_+$. $\square$

**Lemma 8.21 (Optimal weight).** We assume that a satisfies (8.33) as well as $a \in W_{\mathrm{loc}}^{1,p'}(\mathcal{O})$ for some $1 \le p < \infty$. For any $g \in L^p(\mathcal{O})$ satisfying (8.9) in the distributional sense with $G \in L^p(\mathcal{O})$, the associated trace function $\gamma g$ defined in Theorem 8.8 satisfies

$$ \gamma g \in L^p(\Sigma, |n \cdot a| \tau d\sigma). $$

**Proof of Lemma 8.21.** One fixes $\beta_M(z) = (|z| \wedge M)^p$. From the DiPerna-Lions renormalizing theory, we have

$$ a \cdot \nabla (\beta_M(g) \tau^+) = \beta'_M(g) G \tau^+ + \beta_M(g) (\tau^+ - 1) \quad \text{in } \mathcal{D}'(\mathcal{O}). $$

Because $\beta_M(g_\epsilon) \tau^+ \in L^1(\mathcal{O}) \cap L^\infty(\mathcal{O})$ and $a/\langle y \rangle \in L^1 + L^\infty$, we may use the Green formula (8.21) with $\phi \equiv 1$, and we get

$$ \int_{\Sigma_{-}} \beta_M(\gamma g) \tau^{+} |n \cdot a| d\sigma = \int_{\mathcal{O}} \{ (\mathrm{div}a)\beta_M(g) \tau^{+} - \beta'_{M}(g) G \tau^{+} + \beta_M(g)(\tau^{+} - 1) \} \\ \lesssim \|g\|_{L^p}^{p-1} \{ \|g\|_{L^p} + \|G\|_{L^p} \}. $$

Passing to the limit $M \to \infty$, we obtain $\gamma_- g \in L^p(\Sigma_-, |n \cdot a| \tau d\sigma)$. In a very same way, we prove $\gamma_+ g \in L^p(\Sigma_+, |n \cdot a| \tau d\sigma)$. $\square$

We give now a second version of an existence result in a $L^p$ framework with optimal assumption on the boundary condition in the sense that it is reverse with respect to Lemma 8.21. That also a posteriori justifies that Lemma 8.21 provides the optimal trace result in term of weight function on the boundary.

**Lemma 8.22 (Existence in $L^p$ - optimal assumption).** We make the same assumption on a, b and p as in Lemma 8.13. For any $\lambda > \lambda_{a,b,p} + 1/p$ and any given functions $G \in L^p(\mathcal{O})$ and $g \in L^p(\Sigma_-, \tau|a \cdot n|d\sigma)$, there exists $g \in L^p(\mathcal{O})$ solution to (8.29).

**Proof of Lemma 8.22.** We only sketch the proof in the case of equation (8.29), arguing along the lines of Lemma 8.12. We start with an a priori estimate. Observing that

$$ \mathrm{div}(a\tau^+|g|^p) = (\mathrm{div}a)\tau^+g^p + (\tau^+ - 1)g^p + p\tau^+(Gg|g|^{p-2} - b|g|^p - \lambda|g|^p), $$

we have

$$ \int_{\mathcal{O}} |g|^p \{1 + p\tau^+(\lambda + b - \frac{1}{p}\mathrm{div}a - \frac{1}{p}\} = \int_{\Sigma_{-}} |\gamma_{-}g|^p \tau |a \cdot n| d\sigma + \int_{\mathcal{O}} Gg|g|^{p-2}\tau^+. $$
**Stability.** We present some stability and continuity results. Generalizing slightly [271, Definitions 2.6 and 3.1], we say that a sequence $(g_n)$ of $L(E)$ converges in the renormalized sense to $g$, we note $g_n \xrightarrow{r} g$, if for any $\delta \in \Delta$, $\Delta \subset (0, 1]$, $0 \in \bar{\Delta}$, there exists $\bar{\beta}_\delta \in L^\infty(E)$ such that

$$ (8.51) \qquad \beta_{\delta}(g_n) \to \bar{\beta}_{\delta} * \sigma(L^{\infty}, L^1) \text{ as } n \to \infty \quad \text{and} \quad \beta_1(\bar{\beta}_{\delta}) \to \beta_1(g) L_{\text{loc}}^{1}(\bar{O}) \text{ as } \delta \to 0. $$

We may observe that in particular $g_n \xrightarrow{r} g$ weakly $L^1(E)$ or $g_n \to g$ a.e. in $E$ implies $g_n \xrightarrow{r} g$, and that when defined, the renormalized limit $g$ is unique. We refer to [271] and the references therein for more material about the subject.

**Proposition 8.10.** Let us consider four sequences $(g_k)$ of $L_{\text{loc}}^1(\bar{O})$, $(a_k)$ of $W_{\text{loc}}^{1,1}(\bar{O})$, $(b_k)$ and $(G_k)$ of $L_{\text{loc}}^1(\bar{O})$ such that

$$ a_k \cdot \nabla \beta(g_k) + b_k \beta'(g_k) g_k = \beta'(g_k) G_k \quad \text{in } \mathcal{D}'(\mathcal{O}), $$

for any $k \ge 1$ and any $\beta \in C_*^1(\mathbb{R})$ and four functions $g \in L_{\text{loc}}^1(\bar{O})$, $a \in W_{\text{loc}}^{1,1}(\bar{O})$, $b, G \in L_{\text{loc}}^1(\bar{O})$ such that $a_k \to a$ in $W_{\text{loc}}^{1,1}(\bar{O})$ and $b_k \to b$, $G_k \to G$ in $L_{\text{loc}}^1(\bar{O})$. Let us denote by $\Sigma_0$ the boundary singular subset associated to $a$.

(1) If $g_k \to g$ a.e. in $\mathcal{O}$ then $g$ satisfies (8.26) for any $\beta \in C_*^1(\mathbb{R})$ and, up to the extraction of a subsequence, $\gamma g_k \to \gamma g$ a.e. on $\Sigma \setminus \Sigma_0$.

(2) If $g_k \to g$ weakly in $L_{\text{loc}}^1(\bar{O})$ then $g$ satisfies (8.26) and, up to the extraction of a subsequence, $\gamma g_k \xrightarrow{r} \gamma g$ on $\Sigma \setminus \Sigma_0$.

**Remark 8.11.** Because of Remark 8.9-(5) and the time independence made on a and b in (8.35), exactly the same stability result holds for the evolution equation (8.31) as a consequence of Proposition 8.10.

*Proof of Proposition 8.10.* We split the proof into two steps.

**Step 1.** We establish (1). We fix $\beta_1$ as in the proof of Theorem 8.8 and we write the Green formula

$$ \int_{\mathcal{O}} [\beta'_{1}(g_{k})G_{k} \varphi - \beta'_{1}(g_{k})g_{k}b_{k} \varphi + \beta_{1}(g_{k}) \operatorname{div}(a_{k}\varphi)] dy = \int_{\Sigma} \beta_{1}(\gamma g_{k}) \varphi a_{k} \cdot n \, d\sigma_{y}, $$

for any test function $\varphi \in C_c^1(\bar{O})$. There exists $\bar{\beta}_1 \in L^\infty(\Sigma \setminus \Sigma_0)$ and a subsequence $(g_{n_k})$ such that $\beta_1(\gamma g_{n_k}) \to \bar{\beta}_1$ weakly $\sigma(L^\infty, L^1)$. Passing to the limit in the above equation, we get

$$ \int_{\mathcal{O}} [\beta'_{1}(g)G \varphi - \beta'_{1}(g)gb \varphi + \beta_{1}(g) \operatorname{div}(a\varphi)] dy = \int_{\Sigma} \bar{\beta}_{1} \varphi a \cdot n \, d\sigma_{y}. $$

From Lemma 8.7 and Theorem 8.8, we deduce that $\bar{\beta}_1 = \beta_1(\gamma g)$, so that $\beta_1(\gamma g_n) \to \beta_1(\gamma g)$ weakly $\sigma(L^\infty, L^1)$. Fixing now $\beta_2 := \beta_1^2 \in C_*^1(\mathbb{R})$ and repeating the same argument, we get $\beta_2(\gamma g_n) \to \beta_2(\gamma g)$ weakly $\sigma(L^\infty, L^1)$. We then immediately deduce that

$$ (\beta_1(\gamma g_n) - \beta_1(\gamma g))^2 \to 0 \quad \text{weakly } \sigma(L^\infty, L^1), $$

so that $\beta_1(\gamma g_n) \to \beta_1(\gamma g)$ in $L_{\text{loc}}^1(\Sigma \setminus \Sigma_0)$. We conclude by using that $\beta_1$ is one-to-one.

**Step 2.** We establish (2). We fix $\beta_\delta$ as defined just before the statement of Lemma 8.7 and we write the Green formula

$$ \int_{\mathcal{O}} [\beta'_{\delta}(g_k)G_k \varphi + \beta_{\delta}(g_k) \operatorname{div}(a\varphi)] dy = \int_{\Sigma} \beta_{\delta}(\gamma g_k) \varphi a \cdot n \, d\sigma_y, $$

for any test function $\varphi \in C_c^1(\bar{O})$. There exist $\overline{\beta}_\delta, \tilde{\beta}_\delta, \overline{\beta}'_\delta \in L^\infty(\mathcal{O})$, $\overline{\gamma\beta}_\delta \in L^\infty(\Sigma \setminus \Sigma_0)$ and a subsequence $(g_{n_k})$ such that $\beta_\delta(g_{n_k}) \to \overline{\beta}_\delta$, $g_{n_k}\beta'_\delta(g_{n_k}) \to \tilde{\beta}_\delta$, $\beta'_\delta(g_{n_k}) \to \overline{\beta}'_\delta$ and $\beta_\delta(\gamma g_{n_k}) \to \overline{\gamma\beta}_\delta$ weakly $\sigma(L^\infty, L^1)$. Passing to the limit in the above equation, we get

$$ a \cdot \nabla_y \overline{\beta}_\delta + b \tilde{\beta}_\delta = \overline{\beta}'_\delta G \quad \text{in } \mathcal{D}'(\mathcal{O}), \quad \overline{\gamma\beta}_\delta = \overline{\gamma\beta}'_\delta \quad \text{on } \Sigma \setminus \Sigma_0. $$

From the fact that $(g_k)$ is locally uniformly integrable, we classically deduce that

$$ \overline{\beta}_\delta, \tilde{\beta}_\delta \to g \quad \text{in } L_{\text{loc}}^1(\mathcal{O}), \quad \overline{\beta}'_\delta G \to G \quad \text{in } L_{\text{loc}}^1(\mathcal{O}), $$

as $\delta \to 0$. More precisely, the two first convergences come from the elementary inequalities

$$ \forall M > 0, \exists c_M > 0, \quad |s - \beta_{\delta}(s)| \le |s - s\beta'_{\delta}(s)| \le c_M \delta + |s|\mathbf{1}_{|s|>M}, $$
Using the condition on $\lambda$, the property $0 \le \tau^+ \le 1$ and the Young inequality, we deduce

$$ \frac{1}{p} \int_{\mathcal{O}} |g|^p \le \int_{\Sigma_{-}} |\gamma_{-} - g|^p \tau |a \cdot n| d\sigma + \frac{1}{p} \int_{\mathcal{O}} |G|^p. $$

We conclude in a similar way as in the proof of Lemma 8.13.

<figure><img src="image_5.png" /></figure>

We consider now the time dependent transport equation with positive abstract kernels

$$ (8.58) \qquad \begin{cases} \frac{\partial g}{\partial t} + a \cdot \nabla g + bg = \mathcal{H}[g] + G \quad \text{on } (0, T) \times \mathcal{O}, \\ \gamma_{-}g = \mathcal{R}[g, \gamma_{+}g] + \mathfrak{g} \quad \text{on } (0, T) \times \Sigma_{-}, \quad g(0, \cdot) = \mathfrak{g}_{0} \quad \text{on } \mathcal{O}, \end{cases} $$

with notations introduced at the beginning of the Section. We will work in a weighted Lebesgue space $L_m^p$ with the same conditions on $p, m, a$ and $b$ as introduced at the beginning of Section 8.2. On the other hand, we assume

$$ (8.59) \qquad \mathcal{H}: L_{m\mathcal{O}}^{p}(\mathcal{O}) \to L_{\tilde{m}\mathcal{O}}^{p}(\mathcal{O}) \text{ linear and positive,} $$

$$ (8.60) \qquad \mathcal{R}: L_{m\mathcal{O}}^{p}(\mathcal{O}) \times L_{m\Sigma}^{p}(\Sigma_{+}) \to L_{m\Sigma}^{p}(\Sigma_{-}) \text{ linear and positive in each variable,} $$

where we recall that the weight functions $m_{\mathcal{O}}$, $\tilde{m}_{\mathcal{O}}$ and $m_{\Sigma}$ have been defined in (8.38). More precisely, recalling that $\mathcal{R} = \mathcal{R}_{\mathcal{O}} + \mathcal{R}_{\Sigma}$ with $\mathcal{R}_{\mathcal{O}}$ and $\mathcal{R}_{\Sigma}$ defined by (8.4), we assume

$$ (8.61) \qquad \|\mathcal{H}[g]\|_{L_{\tilde{m}\mathcal{O}}^{p}} \le \alpha_{\mathcal{X}} \|g\|_{L_{m\mathcal{O}}^{p}} + M_{\mathcal{X}} \|g\|_{L_{m}^{p}}^{p}, $$

$$ (8.62) \qquad \|\mathcal{R}_{\mathcal{O}}[g]\|_{L_{m\Sigma}^p}^p \le \alpha_{\mathcal{R}} \|g\|_{L_{m\Sigma}^p}^p + M_{\mathcal{R}} \|g\|_{L_m^p}^p, \quad \|\mathcal{R}_{\Sigma}[h]\|_{L_{m\Sigma}^p}^p \le \beta_{\mathcal{R}} \|h\|_{L_{m\Sigma}^p}^p, $$

with $\alpha_{\mathcal{X}}$, $\alpha_{\mathcal{R}}$, $\beta_{\mathcal{R}} \in [0, 1]$, $M_{\mathcal{X}}$, $M_{\mathcal{R}} \ge 0$ and

$$ (8.63) \qquad \vartheta_{\mathcal{O}} := (1 - \alpha_{\mathcal{R}} - \alpha_{\mathcal{X}})/2 > 0, \quad \vartheta_{\Sigma} := 1 - \beta_{\mathcal{R}} \ge 0. $$

Let us emphasize that when $p = 1$, the assumption (8.61) is equivalent to the Lyapunov type condition

$$ \mathcal{H}^*[m] \le \alpha_{\mathcal{X}} \varpi_{+}m + M_{\mathcal{X}} m. $$

**Proposition 8.23.** We assume that $a, b, \mathcal{H}$ and $\mathcal{R}$ satisfy the conditions (8.35), (8.59), (8.60), (8.61), (8.62), and (8.63) for some weight function $m: \mathcal{O} \to [1, \infty)$ and some exponent $p \in [1, \infty)$. We consider some data $g_0 \in L_m^p(\mathcal{O})$, $G \in L_m^p((0, T) \times \mathcal{O})$ and $\mathfrak{g} \in L_{m\Sigma}^p((0, T) \times \Sigma_+)$ with either
(1) $\beta_{\mathcal{R}} \in [0, 1]$;

or $\beta_{\mathcal{R}} = 1$. In the latter case, we assume that $\mathfrak{g} = 0$ and we make one of the following additional structural assumptions

(2) there exist an exponent $p_0 \in [1, p]$ and a weight function $m_0$ such that $\mathcal{H}$ and $\mathcal{R}$ satisfy (8.59) in $L_{m_0}^{p_0}$ and (8.60) in $L_{m_0\Sigma}^{p_0}(\mathcal{O}) \times L_{m_0\Sigma}^{p_0}(\Sigma_+)$, with obvious definitions for the weight functions $m_{0\mathcal{O}}$ and $m_{0\Sigma}$, and with $L_m^p \subset L_{m_0}^{p_0}$, $L_{m\Sigma}^p \subset L_{m_0\Sigma}^{p_0}$, $L_{\tilde{m}\Sigma}^p \subset L_{\tilde{m}_0\Sigma}^{p_0}$, where $\tilde{m}_{\Sigma} := m(\tau^+ a \cdot n)^{2/p}$;

(3) $p = 1$ and $\mathcal{R}_{\Sigma}$ is diffusive, namely $\mathcal{R}_{\Sigma}^*[\tau^+ m_{\Sigma}] \ge c_{\Sigma} m_{\Sigma}$ a.e. on $\Sigma_+$ with $c_{\Sigma} > 0$. In the above three cases, there exists a unique solution $g \in L^{\infty}(0, T; L_m^p(\mathcal{O})) \cap C([0, T]; L_{m_0}^{p_0}(\mathcal{O}))$ satisfying the transport equation (8.58) in the renormalized sense as well as $g \in L_{m_0}^{p_0}(0, T; L_{m_0\Sigma}^{p_0}(\mathcal{O}))$ and $\gamma g \in L_{m_0}^{p_0}(0, T; L_{m_0\Sigma}^{p_0}(\Sigma))$, with $p_0 = p$ and $m_0 = m$ in the first and the third cases.

**Remark 8.24.** (1) The above result extends some previous results initiated by Bardos in [37, Chap. III] and Beals et al in [43, Thm. 1&7], where however only the kinetic case were considered. We refer to Section 10 for a discussion about that important model.

(2) When $\beta_{\mathcal{R}} = 1$, the existence part of the above result still holds (without any additional structural assumption).

(3) Similarly as observed in Remark 8.17, a weak maximum principle holds: $g_1 \le g_2$ if $g_1$ and $g_2$ are the renormalized solutions to two transport equations (8.58) such that (with obvious notations) $b_1 \ge b_2$, $\mathcal{H}_1 \le \mathcal{H}_2$, $\mathcal{R}_1 \le \mathcal{R}_2$, $g_{01} \le g_{02}$, $G_{01} \le G_{02}$ and $\mathfrak{g}_{01} \le \mathfrak{g}_{02}$. That is an immediate consequence of the way we build the solutions $g_i$ thanks to the iterative scheme we present in Step 2 of the proof of Proposition 8.23.

(4) Another immediate consequence of the iterative way of building the solution, together with the fact that the characteristics representation (8.47) is the very first step of the construction, is the validity of the Duhamel formula

$$ S_{\mathcal{L}} = S_{\mathcal{B}} + S_{\mathcal{B}}A * S_{\mathcal{L}} $$
for any $s \in \mathbb{R}$, $\delta \in (0,1)$, and the last convergence comes from the convexity inequality $\bar{\beta}'_{\delta} \ge \beta'_{\delta}(g)$ and the elementary inequalities

$$ \forall M > 0, \exists c_M > 0, \ 0 \le 1 - \beta'_{\delta}(s) \le c_M \delta + \mathbf{1}_{|s|>M}, \forall s \in \mathbb{R}, \forall \delta \in (0,1). $$

From Step 1, we deduce that $g$ satisfies (8.26) and that, up to the extraction of a subsequence, $\bar{\gamma}\beta_{\delta} = \gamma\bar{\beta}_{\delta} \to \gamma g$ a.e. on $\Sigma \setminus \Sigma_0$. Using the Cantor diagonal process, we obtain that there exist two sequences $(\delta_m)$ and $(g_{n_k})$ such that $\delta_m \searrow 0$ and $\gamma g_{n_k} \xrightarrow{r} \gamma g$ in the renormalized sense associated to $(\delta_m)$. $\square$

**Existence.** We establish two existence results of solutions to the transport equation (8.29).

**Lemma 8.12 (Existence in $L_m^\infty$).** We assume that $a$ and $b$ satisfy (8.35) with $p = \infty$ and some weight function $m: \mathcal{O} \to [1, \infty)$. For any $\lambda > \lambda_\infty^*$ and any given functions $G \in L_m^\infty(\mathcal{O})$ and $g \in L_m^\infty(\Sigma_0)$, there exists $g \in L_m^\infty(\mathcal{O})$ solution to (8.29) in the distributional sense. This solution satisfies (8.29) in the renormalized sense, the weak maximum principle, namely

$$ (8.52) \qquad g \ge 0 \text{ in } \mathcal{O} \text{ if } g \ge 0 \text{ on } \Sigma_0 \text{ and } G \ge 0 \text{ in } \mathcal{O}, \\ \text{and the } L_m^\infty \text{ estimate (8.43).} $$

**Proof of Lemma 8.12.** The proof follows [269, Lem. 3] using [37, Thm. 2.3]; we only sketch it. Under the stronger regularity assumption $a, b \in C_b^1(\bar{\mathcal{O}})$, $G \in C_c^1(\mathcal{O})$, $g := \tilde{g}|_{\Sigma_0}$ with $\tilde{g} \in C_c^1(\bar{\mathcal{O}})$, both definitions (8.48) and (8.50) provide a classical (and thus also renormalized) solution $g$ to (8.29). In such a situation, we may justify the computations made in the above a priori estimates paragraph and we conclude that $g$ satisfies the $L^\infty$ estimate (8.43). In the general case for $a, b, g$ and $G$, we introduce some sequences $(a^\epsilon)$, $(b^\epsilon)$, $(g^\epsilon)$ and $(G^\epsilon)$ of regular and approximating functions so that we may apply the first step above. In that way, we build a sequence $(g^\epsilon)$ of renormalized solutions to the approximated problem which is uniformly bounded and thus converges (up to the extraction of a subsequence) in the weakly $\ast\sigma(L^\infty, L^1)$ sense to a function $g \in L^\infty(\mathcal{O})$ satisfying (8.46). We then conclude by passing to the limit $\epsilon \to 0$ thanks to Proposition 8.10-(2). $\square$

We give a first version of an existence result in a $L^p$ framework with strong assumption on the boundary condition.

**Lemma 8.13 (Existence in $L_{m\mathcal{O}}^p$).** We assume that $a$ and $b$ satisfy (8.35) for some $p \in [1, \infty)$ and some weight function $m: \bar{\mathcal{O}} \to [1, \infty)$. For any $\lambda > \lambda_p^*$, $G \in L_{\bar{m}\mathcal{O}}^p(\mathcal{O})$ and $g \in L_{m\Sigma}^p(\Sigma_0)$, there exists $g \in L_{m\mathcal{O}}^p(\mathcal{O})$ a renormalized solution to the transport equation (8.29). This one satisfies (8.40), (8.52) and $\gamma_+g \in L_{m\Sigma}^p(\Sigma_+)$.

**Proof of Lemma 8.13.** We argue similarly as during the proof of Lemma 8.12. When $g = \tilde{g}|_{\Sigma_0}$ with $\tilde{g}, a, b, G$ smooth and with compact support, the classical solution built above satisfies (8.40), and thus

$$ (8.53) \qquad \|g\|_{L_{m\mathcal{O}}^p(\mathcal{O})} \lesssim \|G\|_{L_{\bar{m}\mathcal{O}}^p} + \|gm\|_{L^\infty} \|a\|_{L^1(\text{supp}g)}^{1/p}. $$

For $p > 1$, and under the general conditions (8.35) on $a$ and $b$, but still assuming $g = \tilde{g}|_{\Sigma_0}$ and $G, \tilde{g} \in C_c^1(\bar{\mathcal{O}})$, we may introduce two sequences $(a_\epsilon)$ and $(b_\epsilon)$ of smooth functions approximating $a$ and $b$. Since the resulting solution $g_\epsilon$ satisfies (8.53), so that the sequence $(g_\epsilon)$ is bounded in $L_{m\mathcal{O}}^p$, we may argue with the same (compactness) argument as in the proof of Lemma 8.12. We then conclude to the existence of a (renormalized) solution $g \in L_{m\mathcal{O}}^p$ to the transport equation (8.29) satisfying (8.40). Still for $p > 1$, but assuming $G \in L_{\bar{m}\mathcal{O}}^p$ and $g \in L_{m\Sigma}^p(\Sigma_0)$, we may introduce two sequences $(G_\epsilon)$ and $(g_\epsilon)$ of smooth functions approximating $G$ and $g$. Thanks to (8.40), the associated sequence of solutions $(g_\epsilon)$ is bounded (and better it is a Cauchy sequence) in $L_{m\mathcal{O}}^p(\mathcal{O})$ and we conclude again to the existence of a (renormalized) solution $g \in L_{m\mathcal{O}}^p(\mathcal{O})$ to the transport equation (8.29) satisfying (8.40). Finally, in the case $p = 1$ and $\lambda > \lambda_1^*$, we may find $q > 1$ small enough such that $\lambda > \lambda_q^*$. For $G, g \in L^1 \cap L^q$, the last step imply the existence of a renormalized solution $g \in L_{m\mathcal{O}}^q(\mathcal{O})$ to the transport equation (8.29). Renormalizing the equation, we deduce that $g$ satisfies (8.40) for $p = 1$. When $G, g \in L^1$, we introduce two sequences $(G_\epsilon)$ and $(g_\epsilon)$ of $L^1 \cap L^q$ functions approximating $G$ and $g$, and using (8.40) for $p = 1$, we deduce that the resulting sequence $(g_\epsilon)$ is a Cauchy sequence in $L_{m\mathcal{O}}^1(\mathcal{O})$. We easily conclude again. Finally $\gamma_+g \in L_{m\Sigma}^p(\Sigma_+)$ from (8.41) (see also Remark 8.9-(4)). $\square$