数理解析研究所講究録 1475 巻 2006 年 155-166

155

A KINETIC APPROACH TO
A COMPARISON THEOREM
FOR DEGENERATEPARABOLIC EQUATIONS
早稲田大学・教育学部

小林 和夫 (Kazuo Kobayasi)

Department of Mathematics, School of Education, Waseda University
e-mail: kzokoba@waseda.jp

1. STATEMENT OF THE RESULT.

and $\in(0, +\infty]$ . Let
Let 0 be an open bounded subset ofthe outerward,
the boundary of ,denote the set $(0, T)$We
and the set $(0, T)$
unit normal to 0 at a pointconsider the following parabolic-hyperbolic problem:(1.1)
in$\mathrm{x}$$\Omega$an$\overline{x}\in$$\Omega$

an

$Q$

$T$

$\mathbb{R}^{d}$$\mathrm{n}(\overline{x})$$\Sigma$$\mathrm{x}$

an.

$Q$

$\partial_{t}u+\mathrm{d}\mathrm{i}\mathrm{v}A(u)-\triangle\beta(u)=0$

with the initial condition:$u(0, x)=u_{0}(x)$ ,and the boundary condition:$u(t,$x) $=u_{b}(t,$x),x$\in\Omega$

(1.2)

,

(t,$x)\in\Sigma$ ,(1.2)

and the function is nonwhere the flux function belongs todecreasing and Lipschitz continuous. This rnonotonicity assumptionof 4 allows us some degenerate diffusion cases which \’appear in many
interesting models, for example, filtration problems in porous media[2,5,8].
In the nondegenerate case (in which the function is strictly increasing), the problem (1.1) is of parabolic type and hence the existence and, theuniqueness of solutions are well known. In the case whereuniquenessproblem (1.1) being a nonlinear hyperbolic problem, the$A$$\beta$$C^{1}(\mathbb{R})$$\beta$$\beta’\equiv 0$156
Comparison theorem for degenerate PDEs

of weak solutions is not ensured, and one must consider a notion ofentropy solution, relying on the notion of boundary entropy-flux pairs
to recover uniqueness (see [11,16]). When is merely a nondecreas,
ing function, in the case of homogeneous boundary data, i.e.,
Carrillo [3] succeeded in proving the uniqueness of entropy solutions by
mainly using the dedoubling variable technique developed by Kruzkov[11]. The equivalence of entropy solutions and weak solutions is also
considered in [10]. In the case of nonhomogeneous boundary data exhave beenistence and uniqueness of entropy solutions toproved in [1,14,15]. The method used there is also the dedoubling vari$\beta$$u_{b}\equiv 0$$(\mathrm{L}1)-(1.3)$

able technique.
On the other hand Perthame $[12,17]$ proved the uniquness of entropy
solutions to the Cauchy problem of the conservation law (in which) by using the kinetic formulation which is inandand Tadmor [12], without relying on thetroduced by Lion
dedoubling variable technique. Imbert and Vovelle [9] developed analogous techniques for conservation law with boundary conditions, proved
the Comparison Theorem for entropy sub- and supersolutions, and applied their results to the BGK-like model. This technique was also
applied in [6] to study the parabolic approximation of a multidimensional conservation law with initial and boundary conditions.
The purpose of this note is to give a comparison result for theirsub- and supersolutions by using kinetic techniques. Although thecontractivity and, therefore, uniquness of entropy weak solutions has
been obtained, it would seem that any comparison theorem for thosesolutions is not proven.
According to [14] we introduce the definition of entropy sub- andsupersolution.
$\Omega=\mathbb{R}^{d}$$\beta’\equiv 0$$\mathrm{s},\mathrm{P}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{m}\mathrm{e}$$\mathrm{s}$$L^{1}$

Define$\mathrm{s}\mathrm{g}\mathrm{n}^{+}(r)=\{_{0}^{1}$$\mathrm{i}\mathrm{f}r\leq 0\mathrm{i}\mathrm{f}r>0,$,and$r^{\pm}=\mathrm{s}\mathrm{g}\mathrm{n}^{\pm}(r)r$and$\mathrm{s}\mathrm{g}\mathrm{n}^{-}(r)$$=\{\begin{array}{l}-1\mathrm{i}\mathrm{f}r<00\mathrm{i}\mathrm{f}r\geq 0\end{array}$.

Definition 1.1. A function u of $L^{1}(Q)$ is said to be a weak solution of
the problern (1.1) (1.3) if it satisfies:, $A(u)\in$$\beta(u)-\beta(u_{b})$-$L^{1}(Q)^{d}$$\in L^{2}(0,T; H_{0}^{1}(\Omega))$and$\oint_{Q}u\varphi_{t}+(A(u)-\nabla\beta(u))$.

$\nabla\varphi dxdt+\oint_{\Omega}u_{0}\varphi(0, x)dx=0$(1.1)

for any$\varphi\in C_{c}^{\infty}([0.T)\mathrm{x}$$\Omega)$.

157
K. Kobayasi. is said to be an entropy subsolution
Definition 1.2. Letof (1.1) (1.3) if it is a weak solution and satisfies:$u\in L^{\infty}(Q)$$u$-$\int_{Q}(u-\kappa)^{+}\partial_{t}\varphi+(F^{+}(u, \kappa)-\nabla(\beta(u)-\beta(\kappa))^{+})\cdot\nabla\varphi dxdt$(1.5)$+ \int_{\Omega}(u_{0}-\kappa)^{+}\varphi(0, x)dx+M\int_{\Sigma}(u_{b}-\kappa)^{+}\varphi d\sigma dt\geq 0$such thatand any$\beta(\kappa))\varphi=0a.e$. on I.
is said to be an entropy supersolution if (1.7) is replaced byfor any$\mathrm{s}\mathrm{g}\mathrm{n}^{+}(\beta(u_{b})-$$\mathbb{R}^{d})^{+}$$\varphi\in C_{c}^{\infty}([\mathrm{O}, T)\mathrm{x}$$\kappa\in \mathbb{R}$$u$.

$\int_{Q}(u-\kappa)^{-}\partial_{t}\varphi+(F^{-}(u, \kappa)-\nabla(\beta(u)-\beta(\kappa))^{-})$$\nabla\varphi dxdt$(1.6)$+ \int_{\Omega}(u_{0}-\kappa)^{-}\varphi(0, x)dx+M\int_{\Sigma}(u_{b}-\kappa)^{-}\varphi d\sigma dt\geq 0$

for any$\kappa\in \mathbb{R}$$\beta(\kappa))\varphi=0a.e$functions inand any$\varphi\in C_{c}^{\infty}([0, T)$. on .Here$\Sigma$$C_{c}^{\infty}([0, T)$$\mathrm{x}$$\mathrm{x}$$C_{c}^{\infty}([0, T)\mathrm{x}$$\mathbb{R}^{d})$$\mathbb{R}^{d})^{+}$$\mathbb{R}^{d})^{+}$.

such thatis the set of nonnegative$\mathrm{s}\mathrm{g}\mathrm{n}^{-}(\beta(u_{b})$-We also set$M= \sup\{|\mathrm{A}^{t}(r)|;|r|\underline{<}\max\{||u_{0}||_{L(\Omega)}\infty, ||u_{b}||_{L^{\varpi}(\Sigma)}\}$

(1.7)

and
$L=1\leq i\leq N\mathrm{m}\mathrm{a}\mathrm{x}||\triangle_{\overline{x}}h_{i}(\overline{T_{i}x})||_{L}\infty(\Sigma_{\lambda_{i}})$

.

(1.8)

We are now in a position to state the main theorem which obviously
contractive property for entropy solutions
extends the$L^{1}$

Theorem Assume that the following conditions hold:is ,whose boundary
is a bounded open subset of(A1)
is a nondecreasing Lipschitz continuousand :unction.
.
and
(A2)
$u\in
L^{\infty}(Q)$be an entropy subsolution of (1.1) (1.3) with dataet,and let be an entropy supersolution of (1.1) (1.3) with data
. Then we have$\partial\Omega$$\mathbb{R}^{d}$$\Omega$$A\in C^{1}(\mathbb{R}, \mathbb{R})$$\beta$$C^{2}$$\mathbb{R}arrow \mathbb{R}$$f$$u_{0}\in L^{\infty}(\Omega)$$u_{b}\in L^{\infty}(\Sigma)$$L$$u_{0}$-$u_{b})$-$\tilde{u}$$\{$$\tilde{u}_{0},\tilde{u}_{b})$$\frac{1}{T}\int_{0}^{T}\oint_{\Omega}(u(t,x)-\tilde{u}(t,x))^{+}dxdt$$\leq\int_{\Omega}(u_{0}(x)-\tilde{u}_{0}(x))^{+}dx+M\int_{0}^{T}\int_{\partial\Omega}(u_{b}(t, x)-\tilde{u}_{b}(t, x))^{+}d\sigma dt$(1.9)$+ \frac{L}{2}\int_{0}^{T}\int_{\partial\Omega}(\beta(u_{b}(t, x))-\beta(\tilde{u}_{b}(t, x)))^{+}d\sigma dt$.

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Comparison theorem for degenerate PDEs

2. SKETCH OF Proof.
The semi-Kruzkov entropies are the convex functions defined by$\eta_{k}^{\pm}(r)=(r-k)^{\pm}$ ,,while the corresponding entropy flux are the function defined by$k\in \mathbb{R}$$\mathcal{F}^{\pm}(r, k)=\mathrm{s}\mathrm{g}\mathrm{n}^{\pm}(r-k)(A(r)-A(k))$

For a function $u\in L^{\infty}(Q)$ and$\xi\in \mathbb{R}$.

we set$f_{\pm}(t, x, \xi)=\mathrm{s}\mathrm{g}\mathrm{n}^{\pm}(u(t, x)-\xi)$.

. Thus, we can
We assume that is abounded open subset infind a finite open coverof and a partition of unity
on subordinate tosuch that, for $i\geq 1$ , up to a change of
coordinates represented by an orthogonal matrix , the setisthe epigraph of a
, that is to say::function$C^{2}$$\Omega$$\mathbb{R}^{d}$$\{B_{i}\}_{i=0}^{N}$$\overline{\Omega}$$\overline{\Omega}$$\{\lambda_{i}\}_{i=0}^{N}$$\{B_{i}\}_{i=0}^{N}$$\Omega\cap B_{i}$$T_{i}$$C^{2}$$h_{i}$$\mathbb{R}^{d-1}arrow \mathbb{R}$$\Omega_{\lambda:}\cap B_{i}=\{x\in B_{i;}(T_{i}x)_{d}>h_{i}(\overline{T_{i}x})\}$

and$\partial\Omega_{\lambda_{i}}=\partial\Omega\cap B_{i}=\{x\in B_{i;}(T_{i}x)_{d}=h_{i}(\overline{T_{i}x})$

,

where
and $\overline{x}=(x_{1r}\cdots, x_{d-1})$ . For simplicity wewill drop the index and we suppose that the change of coordinates
is trivial: $1_{i}^{f}’=Id$ . We also write.
,. We denote byandthe outward unit normal toat a pointofand byda(x\overline ) the $(d-1)-$ dimensional area element in:$x=(\mathrm{x}, x_{d})\in \mathbb{R}^{d}$$\mathrm{i}$$Q_{\lambda}=(0, T)\mathrm{x}$$\partial\Omega_{\lambda}$$\Theta_{\lambda}=(0, T)$$\Pi_{\lambda}=\{\overline{x};x\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\lambda)\cap\Omega\}$$\mathrm{n}(\overline{x})$$\Sigma_{\lambda}=(0, T)\mathrm{x}$$\mathrm{n}\mathrm{A}$$\mathrm{x}$$\Pi_{\lambda}$$(\overline{x}, h(\overline{x}))$$\Omega_{\lambda}$$\partial\Omega_{\lambda}$$\partial\Omega_{\lambda}$$\mathrm{n}(\overline{x})=(1+|\nabla_{\overline{x}}h(\overline{x})|^{2})^{-1/2}(\nabla_{\overline{x}}h(\overline{x}), -1)$da$(_{\mathrm{X}}^{rightarrow})$$=(1+|\nabla_{\overline{x}}h(\overline{x})|^{2})^{1/2}d\overline{x}$,.

To regularize the functions, for small , $s>0$ let us consider a smoothfunction:such that,for $ps\leq r\leq s$ and. Then, for $>0$ and, we setand.
For simplicity, we wii also use the following notations:$\rho$$\theta_{\rho,s}$$\mathbb{R}arrow \mathbb{R}^{+}$$s^{-1}$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\theta_{\rho,s}\subset[\rho s/2, (1+\rho)s]$$\int_{\mathbb{R}}\theta_{\rho,s}(r)dr=1$$(\epsilon_{1}, \cdots, \epsilon_{d})\in(\mathbb{R}^{+})^{d}$$\nu$$\gamma_{\rho,\epsilon}(x)=\Pi_{i=1}^{d}\theta_{\rho,\epsilon}(:x_{i})$$\theta_{\rho,\nu}(t)\gamma_{\rho,\epsilon}(x)$$\mathrm{n}_{1}=\sqrt{1+|\nabla_{\overline{x}}h(\overline{x})|^{2}}$,for$\mathrm{n}$$\overline{x}_{r}=(\overline{x}, h(\overline{x})+r)$$\overline{x}=(x_{1}, \cdots, x_{d-1}))$$\theta_{\rho,s}(r)=$$\epsilon=$$\gamma_{\rho,\nu,\epsilon}(t, x)=$

159
K. Kobayasi, i.e.,
denotes the restriction of toand, where is a function on $[0, T)$. Moreover we set
A is an element of the partition of unity$s \vee t=\max\{s, t\}$ andA $t= \min\{s, t\}$ .
The proof of the theorem will follow from the following three lemmas
whose proofs will be given in the forthcoming paper.
$\psi^{\lambda}$

stands for$\psi\lambda$

and$\Sigma \mathrm{x}$$\psi$$\overline{\psi}$$\mathbb{R}_{\xi}$$\mathbb{R}^{d+1}$$\mathrm{x}$$\psi$$\overline{\psi}(t,\overline{x}, \xi)=\psi(t,\overline{x}, h(\overline{x}, \xi))$$\{\lambda_{i}\}_{i=0}^{N}$$s$

and letLemma 2.1. Let be an entropy subsolution with data
. Then we have:
A be an element of the partition of unity
such that(a) There exists$(u_{0}, u_{b})$$u$$\{\lambda_{i}\}_{i=0}^{N}$$f_{+}^{\tau 0}\in L^{\infty}(\Omega \mathrm{x} \mathbb{R})$$\lim_{sarrow+0}\int_{\Omega \mathrm{x}\mathbb{R}}[\frac{1}{s}\int_{0}^{s}f_{+}(t, x, \xi)dt]\phi$$dxd \xi=\int_{\Omega \mathrm{x}\mathbb{R}}f_{+}^{\tau 0}\phi$$dxd\xi$(2.1)

for any$\phi$$\in C_{c}^{\infty}(\Omega \mathrm{x}\mathbb{R})$(b) For anyof$\psi$.

$\in C_{c}^{\infty}([0,T)\mathrm{x}$$\frac{1}{s}\int_{0}^{\mathrm{S}}f_{+}(t,\overline{x}_{f}, \xi)dr$as$sarrow+\mathrm{O}$$\mathbb{R}^{d+1})^{+}$

inand any weak* cluster point, we have$f_{+}^{\tau}$$L^{\infty}(\mathrm{O}-_{\lambda}><\mathbb{R})$$\int_{Q_{\lambda}\mathrm{x}\mathbb{R}}(f_{+}(\partial_{t}+a\cdot\nabla)\psi^{\lambda}-\beta’\nabla f_{+}\cdot\nabla\psi^{\lambda})dtdxd\xi$$+ \int_{\Omega \mathrm{x}1\mathrm{R}}f_{+}^{\tau_{0}}\psi^{\lambda}(0, x)dxd\xi+\int_{\Theta_{\lambda\}}<\mathbb{R}}\beta’(\nabla_{\overline{x}}h(\overline{x})\cdot\nabla_{\overline{x}}f_{+}^{b})\overline{\psi^{\lambda}}dtd\overline{x}d\xi$(2.2)$+ \int_{\Theta_{\lambda}\mathrm{x}\mathbb{R}}(-\mathrm{n}_{1}\cdot a)f_{+}^{\tau}\overline{\psi^{\lambda}}dtd\overline{x}d\xi$$\geq\int_{Q_{\lambda}\mathrm{x}\mathbb{R}}\partial_{\xi}\psi^{\lambda}d(m_{+}+n_{+})$.

andLemma 2.2. There exist families of probability measureson , called Young measuers, supported in $(-\infty, ||u||_{L}-]$ andand, respectively, and nonnegative functions
such thatdefined on$\{\nu_{x}^{\tau_{0}}\}_{x\in\Omega}$$\{\tilde{\nu}_{x}^{\tau_{0}}\}_{x\in\Omega}$$\mathbb{R}_{\xi}$$m_{+}^{0}(x, \xi)$$[-||\tilde{u}||_{L}\infty, \infty))$$\tilde{m}_{-}^{0}(x, \xi))$$\Omega \mathrm{x}\mathbb{R}_{\xi}$$m_{+}^{0},\tilde{m}_{-}^{0}\in C(\mathbb{R}_{\xi};w- \mathrm{A}4^{+}(\Omega))$$\lim_{\xiarrow\infty}m_{+}^{0}(x, \xi)=\lim_{\xiarrow-\infty}\tilde{m}_{-}^{0}(x, \xi)=0$,

for $a.e$ .

$x\in\Omega$,$f_{+}^{\tau 0}(x, \xi)=\nu_{x}^{\tau 0}([\xi, \infty))=\partial_{\xi}m_{+}^{0}(x, \xi)+\mathrm{s}\mathrm{g}\mathrm{n}^{+}(u_{0}(x)-\xi)$

(2.3)

and
$\tilde{f}_{-}^{\tau 0}(x, \xi)=-\tilde{\nu}_{x}^{\tau 0}((-\infty, \xi])=\partial_{\xi}\tilde{m}_{-}^{0}(x, \xi)+$ $\mathrm{s}\mathrm{g}\mathrm{n}^{-}(\tilde{u}_{0}(x)-\zeta-))$.

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Comparison theorem for degenerate PDEs

Lemma 2.3. Let A be an element of the partition of unityand letandbe weak* cluster point ofandrespectively }as $s\prec+0_{2}$ in. Thereexist Young measuresandon , supported inand, respectively, and nonnegative functions
andsuch thatdefined on$\{\lambda_{i}\}_{i=0}^{N}$$\tilde{f}_{-}^{\tau}$$\frac{1}{s}\int_{0}^{S}f_{+}(t,\overline{x}_{r}, \xi)dr$$f_{+}^{\tau}$$\frac{1}{s}\int_{0}^{\mathrm{S}}\tilde{f}_{-}$$(t,\overline{x}_{r},\xi)dr_{f}$$L^{\infty}(\Theta_{\lambda}\mathrm{x} \mathbb{R})$$\{\nu_{t,y}^{\tau}\}_{(t,y)\in\Sigma}$$\{\tilde{\nu}_{t,y}^{\tau}\}_{(t,y)\in\Sigma})$$\mathbb{R}_{\xi}$$(-\infty, ||u||_{L}\infty]$$[-||\tilde{u}||_{L}\infty, \infty))$$m_{+}^{b}(t, y, \xi)$$\tilde{m}_{-}^{b}(t, y, \xi))$$\Sigma$$\mathrm{x}\mathbb{R}_{\xi}$for$\lim_{\xiarrow\infty}m_{+}^{b}(t, y, \xi)=\lim_{\xiarrow-\infty}\tilde{m}_{-}^{b}(t, y, \xi)=0$$f_{+}^{\tau}(t, y, \xi)=\nu_{t,y}^{\tau}([\xi_{2}\infty))$,$0.\mathrm{e}$. $(t, y)\in\Sigma.$ ,,$\tilde{f}_{-}^{\tau}=-\tilde{\nu}_{t,y}^{\tau}((-\infty, \xi])$(2.4)$(-a\cdot \mathrm{n}_{1})f_{+}^{\tau}=\partial_{\xi}m_{+}^{b}+M\mathrm{s}\mathrm{g}\mathrm{n}^{+}(u_{b}-\xi)$$(-a\cdot \mathrm{n}_{1})\tilde{f}_{-}^{\tau}=\partial_{\xi}\tilde{m}_{-}^{b}+M\mathrm{s}\mathrm{g}\mathrm{n}^{-}(\tilde{u}_{b}-\xi)$

,

(2.5)

$\int_{\Theta_{\lambda}}m_{+}^{b}(t,\overline{x}_{0}, \xi)\overline{\varphi}^{\lambda}(t,\overline{x}_{0})dtd\overline{x}\geq 0$

for any$\overline{\varphi}\in C(\Sigma)^{+}$

andsatisfying$\mathrm{s}\mathrm{g}\mathrm{n}^{+}(\beta(u_{b})-\beta(\xi))\overline{\varphi}=0a.e$. on I$\int_{\Theta_{\lambda}}\tilde{m}_{-}^{b}(t,\overline{x}_{0}, \xi)\overline{\varphi}^{\lambda}(t,\overline{x}_{0})dtd\overline{x}\geq 0$

for any$\overline{\varphi}\in C(\Sigma)^{+}$

satisfying. on I.

$\mathrm{s}\mathrm{g}\mathrm{n}^{-}(\beta(\tilde{u}_{b})-\beta(\xi))\overline{\varphi}=0a.e$

We continue the proof of Theorem. Let ,andbe the functions defined for as above.
denotes the time kinetic traces anda cluster point of space kinetic traces associated with . The corresponding ones associated with will be denoted byand, respectively. We set for,$f_{+}$$u$$m_{+}$$n_{+}$$f_{+}^{\tau_{0}}$$f_{+}^{\tau}$$u$$\tilde{u}$$\tilde{f}_{-},$$n\sim-,$ $m\sim-,$$f-\tau_{0}$$\tilde{f}_{-}^{\tau}$$(t,\overline{x}, \xi)\in\Theta_{\lambda}\mathrm{x}$$F_{+}(t,\overline{x}, \xi)=-\mathrm{n}_{1}(\overline{x}_{0})$.

$\mathbb{R}$$a(\xi)f_{+}^{\tau}(t,\overline{x}_{0}, \xi)+\beta’(\xi)\nabla_{\overline{x}}h(\overline{x})\cdot\nabla_{\overline{x}}f_{+}^{b}(t,\overline{x}_{0}, \xi)$and$\tilde{F}_{-}(t,\overline{x}, \xi)=-\mathrm{n}_{1}(\overline{x}_{0})\cdot a(\xi)\tilde{f}_{-}^{\tau}(t,\overline{x}_{0}, \xi)+\beta’(\xi)\nabla_{\overline{x}}h(\overline{x})\cdot\nabla_{\overline{x}}\tilde{f}_{-}^{b}(t,\overline{x}_{0}, \xi)$where$\tilde{f}_{-}^{b}=$$\mathrm{s}\mathrm{g}\mathrm{n}^{-}(\tilde{u}_{b}-\xi)$. For$\rho$,$f_{+}^{\rho,\nu,\epsilon}=(f_{+}\mathrm{x}1_{Q_{\lambda}})*\gamma_{\rho,\nu,\epsilon}$

$\nu$

,

$F_{+}^{\beta,tJ\xi}’=(F_{+}\cross 1_{\Sigma_{\lambda}})*\gamma_{\rho,\nu,\epsilon}$and$\in \mathbb{R}_{+}$

and$f_{+}^{\tau 0\rho,\epsilon}=(f_{+}^{\tau_{0}}\mathrm{x} 1_{\Omega_{\lambda}})*\gamma_{\rho,\epsilon}$

,

, set

$\epsilon=(\overline{\epsilon}, \epsilon_{d})\in \mathbb{R}_{+}^{d}$,$m^{\rho\nu,\epsilon}\dotplus=(m_{+}\mathrm{x} 1_{Q_{\lambda}})*\gamma_{\rho,\nu,\epsilon}$$n^{\rho\nu,\epsilon}\dotplus=(n_{+}\mathrm{x}1_{Q_{\lambda}})*\gamma_{\rho,\nu,\epsilon}$.

181
K, Kobayasi, etc., their regularizations
As for, etc.
similarlyaredefined in the same manner as above, but with different
parameters , , . Let
and apply (2.2) inLemma 2.1 to the test function, whereis defined by$\overline{f}_{-},\tilde{f}_{-}^{\tau 0},$$F\sim-$$\tilde{f}_{-}^{\eta,\mu,\delta},\tilde{f}_{-}^{\tau_{0}\eta,\delta},\tilde{F}_{-}^{\eta,\mu,\delta}$$\psi\in C_{c}^{\infty}([0, T)\mathrm{x}\mathbb{R}^{d+1})^{+}$$\delta$$\mu$$\eta$$\psi^{\lambda}*\check{\gamma}_{\rho,\nu,\epsilon}$$\check{\gamma}_{\rho,\nu,\epsilon}(t, x, \xi)=\gamma_{\rho,\nu,\epsilon}(-t, -x, -\xi)$$\check{\gamma}_{\rho,\nu,\epsilon}$:$\oint_{\mathbb{R}^{d+2}}($ $f_{+}^{\rho,\nu,\epsilon}(\partial_{t}+a\cdot\nabla)\psi^{\lambda}-\beta’\nabla f_{+}^{\rho,\nu,\epsilon}$$+(f_{+}^{\tau 0\rho,\epsilon}\theta_{\rho,\nu}+F_{+}^{\rho,\nu,\epsilon})\psi^{\lambda})$$\geq\int_{\mathbb{R}^{d+2}}\partial_{\xi}\psi^{\lambda}d(m\dotplus^{\nu,\epsilon}+n^{\rho\nu,\epsilon}\dotplus)\beta$

.

(2.6)

$\nabla\psi^{\lambda}$

didtdx.

On the other hand we can regularize the equation satisfied by,
the same method and obtain for same$f\sim-$by$\psi’ \mathrm{s}$$- \int_{\mathbb{R}^{d+2}}($$\tilde{f}_{-}^{\eta,\mu,\delta}(\partial_{t}+a\cdot\nabla)\psi^{\lambda}+\beta’\nabla\tilde{f}_{-}^{\eta,\mu,\mathit{5}}\cdot\nabla\psi^{\lambda}$$+(\tilde{f}_{-}^{\tau_{0}\eta,\delta}\theta_{\eta,\mu}+\tilde{F}_{-}^{\eta,\mu,\mathit{5}})\psi^{\lambda})$$\geq-\int_{\mathbb{R}^{d+2}}\partial_{\xi}\psi^{\lambda}d(\overline{m}_{-}^{\eta,\mu,\delta}+\tilde{n}_{-}^{\eta,\mu,\delta})$(2.7)

didtdx.

. Apply (2.6)
Now let us fix a test function, andtoand (2.7) to
add the two equations together. After integrating by parts the left handside of the resultant inequality, we obtain$\varphi(t, x)\in C_{c}^{\infty}([0, T)\mathrm{x}\mathbb{R}^{d})^{+}$$\psi=-\tilde{f}_{-}^{\eta,\mu,\delta}(t, x, \xi)\varphi(t, x)$$\psi=f_{+}^{\rho,\nu,\epsilon}(t, x, \xi)\varphi(t, x)$$I_{\mathbb{R}^{d+2}}(-f_{+}^{\rho,\nu,\epsilon}\tilde{f}_{-}^{\eta,\mu,\delta}(\partial_{t}+a\cdot\nabla+\beta’\triangle+2\beta’\nabla f_{+}^{\rho,\nu,\epsilon}\cdot\nabla\tilde{f}_{-}^{\eta,\mu,\delta})\varphi^{\lambda}d\xi dtdx$$- \int_{\mathbb{R}^{d+2}}(f_{+}^{\tau 0\rho,\epsilon}\theta_{\rho,\nu}\tilde{f}_{-}^{\eta,\mu,\delta}+\tilde{f}_{-}^{\tau 0\eta,\delta}\theta_{\eta,\mu}f_{+}^{\rho,\nu,\epsilon}$$+F_{+}^{\rho,\nu,\epsilon}\tilde{f}_{-}^{\eta,\mu,\delta}+\tilde{F}_{-}^{\eta,\mu,\delta}f_{+}^{\rho,\nu,\epsilon})\varphi^{\lambda}d\xi dtdx$$\geq-\int_{\mathbb{R}^{d+2}}\partial_{\xi}\tilde{f}_{-\varphi^{\lambda}d(m\dotplus+n\dotplus)-}^{\eta,\mu,\delta p\nu,\epsilon p\nu,\epsilon}\oint_{\mathbb{R}^{d+2}}\partial_{\xi}f_{+}^{\rho,\nu,\epsilon}\varphi^{\lambda}d(\tilde{m}^{\eta\mu,\delta\eta\mu,\delta}\dotplus+\tilde{n}\dotplus)$

Notice that if
hence
we haveto see that$\xi\in F_{7}$and. Similarly,. On the other hand, it is easyand. Noting also thatandare nonnegativ

then %$(\mathrm{t}, x, \xi)=\mathrm{s}\mathrm{g}\mathrm{n}^{+}(\beta(u(t, x))-\beta(\xi))$$\nabla f_{+}^{p,\nu,\epsilon}=[\delta(\xi-u)\nabla\beta(u)]^{\rho,\nu,\epsilon}\equiv\delta(\xi-u)\mathrm{x}$$1_{Q}]*\gamma_{\rho,\nu,\epsilon}$$\nabla\tilde{f}_{-}^{\eta,\mu,\delta}=[\delta(\xi-\tilde{u})\nabla\beta(\tilde{u})]^{\eta,\mu,\delta}$$\partial_{\xi}f_{+}^{\rho,nu,\epsilon}=-\delta(\xi-u)^{p,\nu,\epsilon}\equiv-[\delta(\xi-u)\mathrm{x}1_{Q}]*\gamma_{\rho,\nu,\epsilon}$$\partial_{\xi}\tilde{f}^{\eta,\mu,\delta}=-\delta(\xi-\tilde{u})^{\eta,\mu,\delta}$$m_{+}$$\tilde{m}_{-}$

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Comparison theorem for degenerate PDEs

measures, we have.

$\int_{\mathbb{R}^{d+2}}(-f_{+}^{\rho,\nu,\epsilon}\tilde{f}_{-}^{\eta,\mu,\delta}(\partial_{t}+a$$\nabla+\beta’\triangle$$+2\beta’[\delta(\xi-u)\nabla\beta(u)]^{\rho,\nu,\epsilon}[\delta(\xi-\tilde{u})\nabla\beta(\overline{u})]^{\eta,\mu,\delta})\varphi^{\lambda}d\xi dtdx$$- \int_{\mathbb{R}^{\mathrm{d}+2}}(f_{+}^{\tau 0\rho,\epsilon}\theta_{\rho,\nu}\tilde{f}_{-}^{\eta,\mu,\delta}+\tilde{f}_{-}^{\tau 0\eta,\delta}\theta_{\eta,\mu}f_{+}^{\rho,\nu,\epsilon}$$+F_{+}^{\rho,\nu,\epsilon}\tilde{f}_{-}^{\eta,\mu,\delta}+\tilde{F}_{-}^{\eta,\mu,\delta}f_{+}^{\rho,\nu,\epsilon})\varphi^{\lambda}d\xi dtdx$$\geq\int_{\mathbb{R}^{d+2}}\delta(\xi-\tilde{u})^{\eta,\mu,\delta}\varphi^{\lambda}dn^{\rho\nu,\epsilon}\dotplus+\int_{\mathbb{R}^{d+2}}\delta(\xi-u)^{\rho,\nu,\epsilon}\varphi^{\lambda}d\tilde{n}_{-}^{\eta,\mu,\delta}$

Let successively , ,$\eta$

$\mu$

a and

go to +0:$\delta_{d}$$\oint_{Q_{\lambda}\mathrm{x}\mathbb{R}}(-f_{+}^{\rho,\nu,\epsilon}\tilde{f}_{-}(\partial_{t}+a\cdot$$\nabla+\beta’\triangle$$+2\beta’[\delta(\xi-u)\nabla\beta(u)]^{\rho,\nu,\epsilon}\delta(\xi-\tilde{u})\nabla\beta(\tilde{u}))\varphi^{\lambda}d\xi dtdx$$- \int_{Q_{\lambda}\mathrm{x}\mathbb{R}}(f_{+}^{\tau_{0\beta,\mathcal{E}}}\theta_{p,\nu}\tilde{f}_{-}+F_{+}^{\rho,\nu,\epsilon}\tilde{f}_{-})\varphi^{\lambda}d\xi dtdx$.

$\geq\int_{Q_{\lambda}\mathrm{x}\mathbb{R}}\delta(\xi-\tilde{u})\varphi^{\lambda}dn^{p\nu,\epsilon}\dotplus\dagger\int_{Q_{\lambda}\mathrm{x}\mathbb{R}}\delta(\xi-u)^{\rho,\nu,\epsilon}\varphi^{\lambda}d\tilde{n}_{-}$

Here we used the fact that regularized functions equal zero at $t=0$
and go to +0 andand at the boundary. Then, let successively ,use (2.2) in Lemma 2.1 to obtain$p$$\int_{Q_{\lambda}\mathrm{x}\mathbb{R}}($$-f_{+}\tilde{f}_{-}(\partial_{t}+a$.

$\nu,\overline{\epsilon}$$\epsilon_{d}$(2.8)$\nabla+\beta’\Delta$$+2\beta’\mathit{5}(\xi-u)\delta(\xi-\tilde{u})\nabla\beta(u)$.

$\nabla\beta(\tilde{u}))\varphi^{\lambda}d\xi dtdx$$- \oint_{\Omega_{\lambda}\mathrm{x}\mathbb{R}}f_{+}^{\tau 0}\tilde{f}_{-}^{\tau_{0}}\varphi^{\lambda}(0, \cdot)d\xi dx+\int_{\Sigma_{\lambda}\cross \mathbb{R}}((\mathrm{n}_{1}\cdot a)f_{+}^{\tau}\tilde{f}_{-}^{\mathcal{T}}$$-\beta’(\nabla_{\tilde{x}}h\cdot\nabla_{\overline{x}}f_{+}^{b})\tilde{f}_{-}^{b})\overline{\varphi}^{\lambda}d\xi dtd\tilde{x}$$\geq\oint_{Q_{\lambda}\mathrm{x}\mathbb{R}}\delta(\xi-\tilde{u})\varphi^{\lambda}dn_{+}+\int_{Q_{\lambda}\mathrm{x}\mathbb{R}}\delta(\xi-u)\varphi^{\lambda}d\tilde{n}_{-}$

Next, let successively ,and go to +0 and then let successively
and for somego to +0: For any weak’ cluster point
,
and
$p$

$\mu,\overline{\delta}$$\eta$.

$\nu,\overline{\epsilon}$$\epsilon_{d}$$\tilde{f}_{-}^{\tau}$$\delta_{d}$183
K. Kobayasi

weak’ cluster point$f_{+}^{\tau}$, we have$-f_{+}\tilde{f}_{-}(\partial_{t}+a$$\int_{Q_{\lambda}\mathrm{x}\mathbb{R}}($

.

(2.9)

$\nabla+\beta’\triangle$.

$+$ $2\beta’\delta(\xi-u)\delta(\xi-\tilde{u})\nabla\beta(u)$$\nabla\beta(\tilde{u}))\varphi^{\lambda}d\xi dtdx$$- \int_{\Omega_{\lambda}\mathrm{x}\mathbb{R}}f_{+}^{\tau_{0}}\tilde{f}_{-}^{\tau 0}\varphi^{\lambda}(0, \cdot)d\xi dx$$+ \int_{\Sigma_{\lambda}\cross \mathbb{R}}$($(\mathrm{n}_{1}.

)a)f_{+}^{\tau}\tilde{f}_{-}^{\tau}-\beta’(\nabla_{\overline{x}}h\cdot\nabla_{\overline{x}}\tilde{f}_{-}^{b})f_{+}^{b}$$\geq\int_{Q_{\lambda}\mathrm{x}\mathbb{R}}\delta(\xi-\tilde{u})\varphi^{\lambda}dn_{+}+\int_{Q_{\lambda}\mathrm{x}\mathbb{R}}\delta(\xi-u)\varphi^{\lambda}d\tilde{n}_{-}$$\overline{\varphi}^{\lambda}d\xi dtd\overline{x}$.

Adding (2.8) and (2.9) yields$\nabla+\beta’\triangle$$\int_{Q_{\lambda}\mathrm{x}\mathbb{R}}(-f_{+}\tilde{f}_{-}(\partial_{t}+a\cdot$$+2\beta^{l}\delta(\xi-u)\delta(\xi-\tilde{u})\nabla\beta(u)$.

$\nabla\beta(\tilde{u}))\varphi^{\lambda}d\xi dtdx$$- \int_{\Omega_{\lambda}\mathrm{x}\mathbb{R}}f_{+}^{\tau 0}\tilde{f}_{-}^{\tau 0}\varphi^{\lambda}(0, \cdot)d\xi dx$$+ \int_{\Sigma_{\lambda}\cross \mathbb{R}}$($( \mathrm{n}_{1}\cdot a)f_{+}^{\tau}\tilde{f}_{-}^{\tau}-\frac{1}{2}\beta’\nabla_{\overline{x}}h$.

$\nabla_{\overline{x}}(f_{+}^{b}\tilde{f}_{-}^{b})$$\geq\int_{Q_{\lambda}\mathrm{x}\mathbb{R}}(\delta(\xi-\tilde{u})n_{+}+\delta(\xi-u)\tilde{n}_{-})\varphi^{\lambda}d\xi dtdx$

for some weak* cluster point$\mathrm{s}$$f_{+}^{\tau}$

and$\tilde{f}_{-}^{\tau}$$2\beta’(\xi)\delta(\xi-u)\delta(\xi-\tilde{u})\nabla\beta(u)$)$\overline{\varphi}^{\lambda}d\xi dtd\overline{x}$.

. Since.

$\nabla\beta(\tilde{u})$$\leq 1_{F}(\xi)\delta(\xi-u)\delta(\xi-\tilde{u})(|\nabla\beta(u)|^{2}+|\nabla\beta(\tilde{u})|^{2})$$=\delta(\xi-\tilde{u})n_{+}(t, x,\xi)+\delta(\xi-u)\tilde{n}_{-}(t, x, \xi)$,we arrive at(2.10)$- \int_{Q_{\lambda}\mathrm{x}\mathbb{R}}f_{+}\tilde{f}_{-}(\partial_{t}+a\cdot\nabla+\beta’\triangle)\varphi^{\lambda}d\xi dtdx$$\geq\int_{\Omega_{\lambda}\mathrm{x}\mathbb{R}}f_{+}^{\tau 0}\tilde{f}_{-}^{\tau_{0}}\varphi^{\lambda}(0, \cdot)d\xi dx-\int_{\Sigma_{\lambda}\mathrm{x}\mathbb{R}}((\mathrm{n}_{1}\cdot a)f_{+}^{\tau}\tilde{f}_{-}^{\tau}$$- \frac{1}{2}\beta’\nabla_{\overline{x}}$A.

$\nabla_{\overline{x}}(f_{+}^{b}\tilde{f}_{-}^{b}))\overline{\varphi}^{\lambda}d\xi dtd\overline{x}$.

184Comparison theorem for degenerate PDEs

We compute each term of (2.26), Firstly,$- \int_{Q_{\lambda}\mathrm{x}\mathbb{R}}f_{+}\tilde{f}_{-}(\partial_{t}+a.

(2.11)\nabla+\beta’\triangle)\varphi^{\lambda}d\xi dtdx$$= \int_{Q_{\lambda}}((u-\tilde{u})^{+}+\mathcal{F}^{+}(u,\tilde{u})\nabla\varphi^{\lambda}+(\beta(u)-\beta(\tilde{u}))^{+}\triangle\varphi^{\lambda})$dtdx.

Secondly, by virtue of Lemma 2.2 and by using integration by parts one

can calculate:$\int_{\mathbb{R}}f_{+}^{\tau 0}\tilde{f}_{-}^{\tau_{0}}d\xi$$= \int_{-\infty}^{\tilde{u}_{0}}\nu_{x}^{\tau_{0}}([\xi, \infty))\partial_{\xi}\tilde{m}_{-}^{0}d\xi-\int_{\tilde{u}0}^{u\mathrm{o}\vee\tilde{u}0}\nu_{x}^{\tau 0}([\xi, \infty))\tilde{\nu}_{x}^{\tau_{0}}((-\infty, \xi])d\xi$$- \oint_{u\mathrm{o}\vee\tilde{u}_{0}}^{\infty}\partial_{\xi}m_{+}^{0}\tilde{\nu}_{x}^{\tau_{0}}((-\infty, \xi])d\xi$$= \nu_{x}^{\tau_{0}}([\tilde{u}_{0}, \infty))\tilde{m}_{-}^{0}(\cdot,\tilde{u}_{0})+\int_{-\infty}^{\tilde{u}0}\tilde{m}_{-}^{0}d\nu_{x}^{\tau 0}-\int_{\tilde{u}0}^{u_{0}\vee\overline{\mathrm{u}}0}\nu_{x}^{\tau_{0}}([\xi, \infty))\tilde{\nu}_{x}^{\tau_{0}}((-\infty, \xi])d\xi$$+m_{+}^{0}( \cdot, u_{0}\vee\tilde{u}_{0})\tilde{\nu}_{x}^{\tau_{0}}((-\infty, u_{0}\mathrm{V}\tilde{u}_{0}])+\int_{u\mathrm{o}\vee\tilde{u}0}^{\infty}m_{+}^{0}d\tilde{\nu}_{+}^{\tau_{0}}$$\geq-\oint_{\overline{\mathrm{u}}_{0}}^{u\mathrm{o}\vee\overline{u}0}d\xi=-(u--\tilde{u}_{0})^{+}$

Here we used the fact that
. Thus we have.

$d\nu_{x}^{\tau_{0}}([\xi, \infty))/d\xi=-d\nu_{x}^{\mathcal{T}\mathrm{Q}}(\xi)$and$d\tilde{\nu}_{-}^{\tau 0}((-\infty, \xi])/d\xi=$$d\tilde{\nu}_{-}^{\tau_{0}}(\xi)$$\int_{\Omega_{\lambda}\mathrm{x}\mathbb{R}}f_{+}^{\tau_{0}}\tilde{f}_{-}^{\tau_{0}}\varphi^{\lambda}(0, \cdot)d\xi dx\geq-\int_{\Omega_{\lambda}}(u_{0}-\tilde{u}_{0})^{+}\varphi^{\lambda}(0, \cdot)dx$.

(2.12)
Finally, we calculate analogously the boundary term by using Lemma
2.4:$\int_{\mathbb{R}}(\mathrm{n}_{1}.

a)f_{+}^{\tau}\tilde{f}_{-}^{\tau}d\xi$$=- \int_{-\infty}^{\tilde{u}_{b}}\partial_{\xi}\tilde{m}_{-}^{b}\nu_{t,y}^{\tau}([\xi, \infty))d\xi-\oint_{\tilde{u}_{b}}^{u_{b}\vee\tilde{u}_{b}}(\mathrm{n}_{1}\cdot a)\nu_{t,y}^{\tau}([\xi, \infty))\tilde{\nu}_{t,y}^{\tau}((-\infty,\xi])d\xi$$+ \int_{u_{b}\vee\tilde{u}_{b}}^{\infty}\partial_{\xi}m_{+}^{b}\nu_{t,y}^{\tau}((-\infty, \xi])d\xi$$\leq M\oint_{\tilde{u}_{b}}^{u_{b}\vee\tilde{u}_{b}}d\xi=M(u_{b}-\tilde{u}_{b})^{+}$

where

$y$

$-d\nu_{t,y}^{\tau}(\xi)$

stands forand$\overline{x}_{0}$,and we used the fact that
as well as the fact that$d\tilde{\nu}_{t,y}^{\tau}((-\infty, \xi])/d\xi=d\tilde{\nu}_{t,y}^{\tau}(\xi)$$d\nu_{t,y}^{\tau}([\xi, \infty))/d\xi=$$m_{+}^{b}\geq$185
K. Kobayasi

for $\xi\leq ub$ by virtue of (2.21) and thecorresponding inequality associated wiyh , respectively. This implies

0 for$\xi\geq u_{b}$and$\tilde{m}_{-}^{b}\geq 0$$\tilde{u}$$\int_{\Sigma_{\lambda}\cross \mathbb{R}}(\mathrm{n}_{1}.

.

a)f_{+}^{\tau}\tilde{f}_{-}^{\tau}\overline{\varphi}^{\lambda}d\xi dtd\overline{x}\underline{<}M\int_{\Sigma_{\lambda}}(u_{b}-\tilde{u}_{b})^{+}\overline{\varphi}^{\lambda}dtd\overline{x}$(2.13)

Moreover$\int_{\Sigma_{\lambda}\mathrm{x}\mathbb{R}}\beta’(\xi)\nabla_{\overline{x}}h(\overline{x})$.

(2.14)$\nabla_{\overline{x}}(f_{+}^{b}\tilde{f}_{-}^{b})\overline{\varphi}^{\lambda}d\xi dtd\overline{x}$$=- \int_{\Sigma_{\lambda}\mathrm{x}\mathbb{R}}\beta’(\xi)\mathrm{d}\mathrm{i}\mathrm{v}_{\overline{x}}(\overline{\varphi}^{\lambda}\nabla_{\overline{x}}h(\overline{x}))f_{+}^{b}\tilde{f}_{-}^{b}d\xi dtd\overline{x}$$=- \int_{\Sigma_{\lambda}}(\beta(u_{b})-\beta(\tilde{u}_{b}))^{+}(\triangle_{\tilde{x}}h(\overline{x})\overline{\varphi}^{\lambda}+\nabla_{\tilde{x}}h(\overline{x})\cdot\nabla_{\overline{x}}\overline{\varphi}^{\lambda})dtd\overline{x}$$\geq\int_{\Sigma_{\lambda}}(\beta(u_{b})-\beta(\tilde{u}_{b}))^{+}(-L\overline{\varphi}^{\lambda}+\nabla_{\tilde{x}}h(\overline{x})\cdot\nabla_{\overline{x}}\overline{\varphi}^{\lambda})dtdi$.

Combining (2.10) with (2.11) through (2.14) and choosing appropriate, we arrive at the estimate
test functions$\varphi’ \mathrm{s}$$\frac{1}{T}\int_{Q_{\lambda}}(u-\tilde{u})^{+}dtdx$$\leq\int_{\Omega_{\lambda}}(u_{0}-\tilde{u}_{0})^{+}dx+M\int_{\Sigma_{\lambda}}(u_{b}-\tilde{u}_{b})^{+}dtd\overline{x}+\frac{L}{2}\int_{\Sigma_{\lambda}}(\beta(u_{b})-\beta(\tilde{u}_{b}))^{+}dtd\overline{x}$, $N$ , we obtain the desired estimate (1.9),
By summing overand the proof of Theorem is complete.
$\mathrm{i}=0,1$$\cdots$

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Comparison theorem for degenerate PDEs

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