and have the required smallness 5.12. Then, under the assumptions 6.16 and for any $\tau \in [0, T]$, the solution $(F_{\lambda, \mathcal{A}}^+, G_{\lambda, \mathcal{A}}^+, \mathcal{E}_{\lambda}^{\text{evo}})$ to (98) given by proposition 6.19 obey the following estimates$^{45}$

$$ \sum_{k \le 1} \lambda^k (\| \mathcal{E}_{\lambda}^{\text{evo}}(\tau) \|_{H^{k+1}} + \| \partial_t \mathcal{E}_{\lambda}^{\text{evo}}(\tau) \|_{H^k}) + \lambda \| \partial_{tt}^2 \mathcal{E}_{\lambda}^{\text{evo}}(\tau) \|_{L^2} \le C_{0,1,2,\tau} \lambda^{>1/2} + \lambda^{1/2} C_0 \frac{c_1}{c_2} e^{\tau c_2} + \lambda^{1/2} C_0, $$

$$ \sum_{k \le 1} \lambda^k (\|(E_{\lambda}^{\text{evo}})^0(\tau)\|_{H_{\delta}^{k+1}} + \|\partial_t (E_{\lambda}^{\text{evo}})^0(\tau)\|_{H_{\delta+1}^k}) + \lambda \|\partial_{tt}^2 (E_{\lambda}^{\text{evo}})^0(\tau)\|_{L_{\delta+2}^2} \le C_{0,1,2,\tau} \lambda^{>1/2} + \lambda^{1/2} C_0 \frac{c_1}{c_2} e^{\tau c_2} + \lambda^{1/2} C_0, $$

$$ \max_{\mathcal{A} \in \mathcal{A}} \sum_{k \le 1} \lambda^k (\| F_{\lambda, \mathcal{A}}^+(\tau) \|_{H^{k+1}} + \| \partial_t F_{\lambda, \mathcal{A}}^+(\tau) \|_{H^k}) + \lambda \| \partial_{tt}^2 F_{\lambda, \mathcal{A}}^+(\tau) \|_{L^2} \le C_{0,1,2,\tau} \lambda^{>0} + C_0 \frac{c_1}{c_2} e^{\tau c_2} + C_0, $$

$$ \max_{\mathcal{A} \in \mathcal{A}} \sum_{k \le 1} (\lambda^k \| G_{\lambda, \mathcal{A}}^+(\tau) \|_{H^k}) + \lambda \| \partial_t G_{\lambda, \mathcal{A}}^+(\tau) \|_{L^2} \le C_{0,1,2,\tau} \lambda^{>0} + C_0 \frac{c_1}{c_2} e^{\tau c_2} + C_0, $$

for constants defined in 4.5 and 6.17.

*Proof.* Firstly, we add all the estimates from sections 6.3.2, 6.3.3 and 6.3.4 and we use propositions 8.4 and 8.1$^{46}$ with proposition 8.2 and the smallness of the initial data 5.12 to have control of the Sobolev norms in space. Then, we use propositions 8.3 and 8.5 to get the control on the time derivatives. The results on the smallness are immediate.

**Lemma 6.36.** For $\varepsilon > 0$, $c_0 > 0$, $\frac{1}{\eta_0} > 0$, $N \in \mathbb{N}$, $T > 0$ and any $C_0 = C_0(c_0, \frac{1}{\eta_0}, N, T)$ and $C_{0,1,2,\tau}(\cdot, \cdot, \cdot, \cdot) = C_{0,1,2,\tau}(c_0, \frac{1}{\eta_0}, N, T, \cdot, \cdot, \cdot)$ from 4.5 and 6.17 there exists $\lambda_0 > 0$, $c_1$ and $c_2$, such that for any $\lambda < \lambda_0$ and for any $0 \le \tau \le T$ we have

$$ C_0 \frac{c_1}{c_2} e^{\tau c_2} + C_0 + C_{0,1,2,\tau}(c_1, c_2, \tau) \lambda^{\varepsilon} \le \frac{1}{2} c_1 e^{\tau c_2}. $$

*Proof.* We pick any $c_2$ such that $\frac{C_0}{c_2} \le \frac{1}{6}$ then we pick $c_1$ such that $C_0 \le \frac{1}{6}c_1$ and $\lambda_0$ such that $C_{0,1,2,\tau}(c_1, c_2, T) \lambda^{\varepsilon} \le \frac{1}{6}c_1$. □

### 6.3.6 Exact solution to KGM in Lorenz gauge and gauge propagation

We prove the proposition 6.14. First, we rewrite it more precisely as follows:

**Proposition 6.37.** For a given admissible background initial data set 2.24 $(a_0^\alpha, \dot{a}_0^\alpha, \phi_0, \dot{\phi}_0, v_{\mathcal{A}}, \dot{v}_{\mathcal{A}}, \psi_{\mathcal{A}}, w_{\mathcal{A}}^\alpha)$, let $\lambda > 0$ and let $((e_\lambda^{\text{evo}})^\alpha, \epsilon_\lambda^{\text{evo}}, (\dot{e}_\lambda^{\text{evo}})^\alpha, \dot{\epsilon}_\lambda^{\text{evo}}, w_{\lambda, \mathcal{A}}^{+\alpha}, \check{w}_{\mathcal{A}\pm\mathcal{B}}^{+\alpha}, \psi_{\lambda, \mathcal{A}}^{+}, \check{\psi}_{\mathcal{A}\pm\mathcal{B}}^{+}, g_{W_{\lambda, \mathcal{A}}^{+}}^{+}, g_{\Psi_{\lambda, \mathcal{A}}^{+}}^{+}, (e_\lambda^{\text{ell}})^\alpha, \epsilon_\lambda^{\text{ell}}, (\dot{e}_\lambda^{\text{ell}})^\alpha, \dot{\epsilon}_\lambda^{\text{ell}})$ be **KGML** admissible and **KGM in Lorenz gauge admissible**. Let $(\Phi_\lambda, A_\lambda^\alpha)$ be the solution to **KGML** on $[0, t_\lambda]$ given by proposition 6.34 for these initial data assembled as in definition 5.5. Then, it is also a solution to **KGM in Lorenz gauge**. We have for all $t \in [0, t_\lambda]$

$$ \partial_{\alpha} A_{\lambda}^{\alpha}(t) = 0, \qquad (135) $$

$$ \partial_{\alpha} A_{0}^{\alpha}(t) = 0, \qquad (136) $$

$$ \forall \mathcal{A} \in \mathcal{A}, \partial_{\alpha} u_{\mathcal{A}} W_{\mathcal{A}}^{\alpha}, (t) = 0, \qquad (137) $$

$$ (138) $$

and if the parameters are initially polarized as

$$ \forall \mathcal{A} \in \mathcal{A}, \partial_{\alpha} u_{\mathcal{A}} w_{\lambda, \mathcal{A}}^{+\alpha} = 0, \qquad \forall (\mathcal{A}, \mathcal{B}) \in \mathcal{S}, \mathcal{A} < \mathcal{B}, (\partial_{\alpha} u_{\mathcal{A}} \pm \partial_{\alpha} u_{\mathcal{B}}) \check{w}_{\mathcal{A}\pm\mathcal{B}}^{+\alpha} = 0, \qquad (139) $$

$$ (140) $$

<footnote>$^{45}$We make a difference between $\mathcal{E}_{\lambda}^{\text{evo}}$ and $(E_{\lambda}^{\text{evo}})^0$ as in 6.15.</footnote>

<footnote>$^{46}$The linear term in the RHS of (146) and (153) is not a problem and is dealt with as the other linear terms (case c)).</footnote>
then we have

$$ \forall \mathcal{A} \in \mathcal{A}, \partial_{\alpha} u_{\mathcal{A}} W_{\lambda \mathcal{A}}^{+\alpha}(t) = 0, \quad \forall (\mathcal{A}, \mathcal{B}) \in \mathcal{S}, \mathcal{A} < \mathcal{B}, (\partial_{\alpha} u_{\mathcal{A}} \pm \partial_{\alpha} u_{\mathcal{B}}) \tilde{W}_{\mathcal{A} \pm \mathcal{B}}^{+\alpha}(t) = 0, \quad (141) $$

$$ (142) $$

$$ \sum_{\mathcal{A} \in \mathcal{A}} (\lambda^{1/2} \mathrm{Re}(e^{i \frac{u_{\mathcal{A}}}{\lambda} \overline{\partial_{\alpha} W_{\mathcal{A}}^{\alpha}}}) + \lambda^1 \mathrm{Re}(e^{i \frac{u_{\mathcal{A}}}{\lambda} \overline{\partial_{\alpha} W_{\lambda \mathcal{A}}^{+\alpha}}})) (t) + \sum_{\mathcal{A}, \mathcal{B} \in \mathcal{C}, \mathcal{A} < \mathcal{B}} \lambda^1 \mathrm{Re}(e^{i \frac{u_{\mathcal{A}} \pm u_{\mathcal{B}}}{\lambda} \overline{\partial_{\alpha} \tilde{W}_{\mathcal{A} \pm \mathcal{B}}^{+\alpha}}}) (t) + \partial_{\alpha} (E_{\lambda \mathcal{A}}^{\mathrm{ell}})^{\alpha}(t) + \partial_{\alpha} (E_{\lambda \mathcal{A}}^{\mathrm{evo}})^{\alpha}(t) = 0. \quad (143) $$

*Proof.* We use the argument of section 1.1.1, we have initial data satisfying the constraints so we take the divergence of the Maxwell equation to recover the propagation of gauge via a wave equation with zero initial data.

For the decomposition of the divergence of $A_\lambda$, we already have the equalities on the second line from (53) and (54).

For the propagation of polarization, we observed in remark 6.3 that the RHS of the evolution equation for $W_{\lambda \mathcal{A}}^{+\alpha}$ (resp. $\tilde{W}_{\mathcal{A} \pm \mathcal{B}}^{+\alpha}$) are orthogonal to $\partial_\alpha u_\mathcal{A}$ (resp. $\partial_\beta u_\mathcal{A} \pm \partial_\beta u_\mathcal{B}$). We contract the equations with their respective vector fields and use the geodesic equation for the characteristic phases to conclude.

Finally, the last line comes from all the remaining term. □

**Remark 6.38.** The set we exhibit in proposition 5.16 is polarized as $\tilde{w}_{\mathcal{A} \pm \mathcal{B}}^{+\alpha} = w_{\lambda \mathcal{A}}^{+\alpha} = 0$.

# 7 Conclusion

We put everything together and give some general remarks.

## 7.1 Proof of the theorems

For the first theorem 3.1, we only detail the proof of the point 1. Indeed, everything boils down to the construction of the MPHF solutions, the other points are direct consequences of our construction and the point 5 is proved using the Strichartz estimates for waves. The statement of point 1 is that for a given initial ansatz there exists $\lambda_0$ such that below it there is a family of multi phase high frequency solutions to KGM ($A_\lambda, \Phi_\lambda)_{\lambda<\lambda_0}$ on $[0, T]$. More precisely, for each $\lambda > 0$ we can build an almost approximate solution arising from the initial ansatz and an exact solution based on this almost approximate solution. We follow the steps of 1.5. The approximation (step (i)) is done in proposition 4.1. The rest of the theorem follows from the main propositions 6.13 (step (iii)), 6.14 (step (iv)) and 6.18 (step (v)). The construction of the initial data (step (ii)) is considered after.

From proposition 6.13, we have for any $\lambda > 0$ a well structured multi phase high frequency local solution to KGML on $[0, t_\lambda]$. The part 1 of this proposition is proved in proposition 6.19 and its part 2 in proposition 6.34. This requires initial data admissible for KGML 5.9. Then, with the proposition 6.14, we recover a true solution to KGM in Lorenz gauge if the initial data are admissible for KGM in Lorenz gauge 5.11, this is proved in proposition 6.37. Finally, the good structure of the previously cited solutions is exploited to prove the proposition 6.18. This proposition states that for $\lambda < \lambda_0$ the family of solutions to KGM exists on $[0, T]$. For $\lambda > 0$, we have either $t_\lambda > T$ and the proof is done or, if $\lambda$ is small enough, that $t_\lambda$ can be improved up to $T$. To demonstrate the second case, we use the following method. We assume some estimates 6.16 from which we deduce the proposition 6.35. Then, using the lemma 6.36, we exhibit $\lambda_0$ and we show that the estimates of proposition 6.35 are in fact better than the assumptions 6.16 for $\lambda < \lambda_0$. This shows a contradiction, $t_\lambda$ cannot be the maximum time of existence, it can be pushed up to $T$. This requires a certain smallness for the error initial data 5.12.

We exhibit one generic set of initial data which fits the three criteria in proposition 5.16. This concludes the proof of the first theorem.

For the second theorem, the proof is direct with the structure of the solution and the structure of the error term in particular. We make some more remarks on that in the next section. The third point is detailed in section 4.2.

## 7.2 Convergence and smallness of the error term

It is interesting to compare our results to the ones of [10]. The author of [10] exhibits mono phase high frequency solutions to Yang Mills coupled with a scalar field and a spinor field. More particularly, the author shows how to
construct approximate solutions of any order with the WKB expansion

$$F_{M\lambda} = \sum_{0 \le i \le M-1} \lambda^{i/2} (\tilde{F}_i(x) + \lambda^{1/2} \tilde{F}_i^*(x, \frac{u}{\lambda}))$$

and how to construct exact solutions

$$F_{\lambda} = F_{M\lambda} + Z_{\lambda}$$

on a time interval uniform in $\lambda$ and for $M \ge n$ where $n$ is the dimension. The function $F_i(x, \cdot)$ is decomposed as a mean part $\tilde{F}_i(x)$ and a purely oscillatory part $\tilde{F}_i^*(x, \cdot)$. It has to be a more general profile than the $\cos(\frac{u}{\lambda})$ and $\sin(\frac{u}{\lambda})$ case of our paper to treat the harmonics. The construction of approximate solutions of [10] holds for a broader family of gauges than the only Lorenz gauge. Dealing with the non hyperbolicity requires to add a higher order parameter, see the remark 4.9. The following bounds for the Yang-Mills potential and the scalar field is obtained

$$\forall \lambda < \lambda_0, \lambda^{M/2} \|F_{\lambda}\|_{H^{1/2}} + \|Z_{\lambda}\|_{H^{1/2}} \le \lambda^{M/2} C,$$

for some constant $C$ which depends on the initial data.

The KGM system is a sub case of this more general case. On the other hand, the mono phase case of [10] is a sub case of the multi phase case treated here. We show in this paper how to construct an error term at order $M = 1$, that is below 3 (the dimension). Our first order approximation looks like

$$F_{1\lambda} = \tilde{F}_0(x) + \lambda^{1/2} \tilde{F}_0^*(x, \frac{u}{\lambda}) := F_0(x) + \lambda^{1/2} \tilde{F}(x) e^{i \frac{u}{\lambda}}.$$

In [10], it is not possible to give directly the error term at this order. This is due to the use of some product estimates (proposition 3.1.1 in [10] and 2.2.2 in [9]) which is consistent with the results of [19] for first order hyperbolic system. To get higher order approximate solutions one needs to build a hierarchical system of equations and to take care of harmonics which arise from the interactions between phases.

## 7.3 Regularity of the background

The background $B'_0 := (A_0, \Phi_0, w_{\mathcal{A}}, \psi_{\mathcal{A}}, du_{\mathcal{A}})$ defined in proposition 4.1 is more regular than the full solution (and in particular the error term). This extra regularity is mandatory and is used to have the well posedness of (99) and (87) and along the estimates in section 6.3. For example, the equation for $\mathcal{E}_{\lambda}^{\text{evo}}$ has $\square F_{\mathcal{A}}$ in its RHS so that $\partial \mathcal{E}_{\lambda}^{\text{evo}} \sim \square F_{\mathcal{A}}$ in terms of regularity. We see that $B_0 = (A_0, \Phi_0, w_{\mathcal{A}}, \psi_{\mathcal{A}})$ is one degree more regular than the error term. Moreover, the phases have the following regularity

$$\max_{\mathcal{A} \in \mathcal{H}} \sum_{j=0}^{5} \|u_{\mathcal{A}}\|_{C^j([0,T], H_{\delta_1}^{5-j})} \le c_0.$$

The phases need to be this regular to have the well-posedness of (51) with solutions in $H^3$ and to be able to use the commutator estimates of proposition 8.6 for (106). In fact, it is doable if we only have $\square u_{\mathcal{A}}$ in $H^3$ and $u_{\mathcal{A}}$ in $H_{\delta_1}^4$. Constructing the error term for a higher order approximation requires a more regular background.

# 8 Appendix

## 8.1 Estimates for evolution equations

**Proposition 8.1.** For $f$ a function defined on $[0, T'] \times \mathbb{R}^3$ solution to

$$\begin{cases} \square f = h, \\ f(t=0) = f_0 \in H_{\delta'}^{m+1} \quad \partial_t f(t=0) = f_1 \in H_{\delta'}^m, \end{cases} \qquad (144)$$

we have the energy estimate

$$\|\partial f(t)\|_{H_{\delta'}^m} \le C(\|\partial f(0)\|_{H_{\delta'}^m} + \int_0^t \|h(s)\|_{H_{\delta'}^m} + \|\partial f(s)\|_{H_{\delta'}^m} ds), \qquad (145)$$
and more precisely for $\delta' = 0$

$$ \|\partial f(t)\|_{H^m} \le C(\|\partial f(0)\|_{H^m} + \int_0^t \|h(s)\|_{H^m} ds). \quad (146) $$

*Proof.* We detail the energy estimate for $f$ for $m = 0$ and for any $\delta'$

$$ \Box f(1 + |x|^2)^{\delta'} \partial_t f = h(1 + |x|^2)^{\delta'} \partial_t f, $$

$$ \int_{\mathbb{R}^3} \Box f(1 + |x|^2)^{\delta'} \partial_t f dx = \int_{\mathbb{R}^3} h(1 + |x|^2)^{\delta'} \partial_t f dx \\ = -\frac{d}{dt} \int_{\mathbb{R}^3} (1 + |x|^2)^{\delta'} \frac{\partial_t f^2 + |\nabla f|^2}{2} dx - 2\delta' \int_{\mathbb{R}^3} (1 + |x|^2)^{\delta'-1} x_i \partial_i f \partial_t f dx, $$

where $|(1 + |x|^2)^{\delta'-1} x_i| \le (1 + |x|^2)^{\delta'}$. We can control this term in $L^2$ norm with the weight $\delta'$. We finally get

$$ \frac{d}{dt} \frac{1}{2} \|\partial f(t)\|_{L_{\delta'}^2}^2 \le C(\|\partial f(t)\|_{L_{\delta'}^2}^2 \|h(s)\|_{L_{\delta'}^2} + \|\partial f(t)\|_{L_{\delta'}^2}^2), \\ \|\partial f(t)\|_{L_{\delta'}^2} \le C(\|\partial f(0)\|_{L_{\delta'}^2} + \int_0^t \|h(s)\|_{L_{\delta'}^2} + \|\partial f(s)\|_{L_{\delta'}^2} ds). $$

Then, we commute the equation with $\partial_{\vec{m}}$ where $\vec{m}$ is a multi-index with $|\vec{m}| = m$. We multiply by $\partial_t \vec{m} f$ instead of $\partial_t f$ and the adjusted weight to get the Sobolev norms. □

Moreover, having bounds on $\partial f = (\partial_t f, \nabla f)$ and on $f_0$ gives us bounds on $f(t)$ for finite t.

**Proposition 8.2.** With $f$ solution to (144) we have $\forall t \in [0, T']$

$$ \frac{d}{dt} \|f(t)\|_{L^2(t)} \le \|\partial_t f(t)\|_{L^2}, \quad (147) $$

and so for $m \ge 0$

$$ \|f\|_{H_{\delta'-1}^{m+1}}(t) \le (\|f_0\|_{H_{\delta'-1}^{m+1}} + (t+1)\sup_{s \in [0, T']} \|\partial f(s)\|_{H_{\delta'}^m}). \quad (148) $$

*Proof.* By direct calculation. □

**Proposition 8.3.** The time derivatives of solutions to (144) obey the following inequalities

$$ \|\partial_t f(t)\|_{H_{\delta'}^m} \le C(\|\partial f(0)\|_{H_{\delta'}^m} + \int_0^t \|h(s)\|_{H_{\delta'}^m} + \|\partial f(s)\|_{H_{\delta'}^m} ds), \quad (149) $$

$$ \|\partial_{tt}^2 f(t)\|_{H_{\delta'+1}^{m-1}} \le (\|f(t)\|_{H_{\delta'-1}^{m+1}} + \|h(t)\|_{H_{\delta'+1}^{m-1}}). \quad (150) $$

*Proof.* The first inequality is a specific case of (146) and the second comes directly using the equation. □

**Proposition 8.4.** For $f$ a function defined on $[0, T'] \times \mathbb{R}^3$ solution to

$$ \begin{cases} 2\partial^\alpha u \partial_\alpha f + \Box u f = j, \\ f(t=0) = f_0 \in H^m, \end{cases} \quad (151) $$

with $u$ a smooth characteristic phase satisfying

$$ \min_{(t,x) \in [0, T'] \times \text{Supp}(f)} |\partial_t u(t, x)| > \eta > 0 \quad \|\partial u\|_{H^{m'}} + \|\Box u\|_{H^{m'}} + \|\partial_t \partial u\|_{H^{m'-1}} + \|\partial_t \Box u\|_{H^{m'-1}} \le C_u, \quad (152) $$

for $m' = \max(m, 2)$. We have the energy estimate

$$ \|f\|_{L^\infty([0, T'], H^m)} \le C_u (\|f(0)\|_{H^m} + \int_0^{T'} \|j(s)\|_{H^m} + \|f(s)\|_{H^m} ds). \quad (153) $$
Proof. In the case $m = 0$, we multiply by f

$$ \partial_{\alpha}(\partial^{\alpha} u_{\mathcal{A}} f^2) = jf, $$

so we have a conservative transport equation, by Stokes theorem$^{47}$ we get

$$ \int_{\mathbb{R}^3} (\partial^0 u_{\mathcal{A}} f^2)(t)dx = \int_{\mathbb{R}^3} (\partial^0 u_{\mathcal{A}} f^2)(0)dx + \int_0^t \int_{\mathbb{R}^3} (jf)(s)dxds, $$

and so

$$ \eta \|f(t)\|_{L^2}^2 \leq C_u \|f(0)\|_{L^2}^2 + \|f\|_{L^{\infty}([0,T'],L^2)} \int_0^t \|j\|_{L^2} ds, $$

which implies

$$ \|f(t)\|_{L^{\infty}([0,T'],L^2)} \leq C_{u,\eta} (\|f(0)\|_{L^2} + \int_0^{T'} \|j\|_{L^2} ds). $$

For $m > 1$ we commute the equation with $\partial_{\vec{m}}$, where $\vec{m}$ is a multi-index with $|\vec{m}| = m$, and we get

$$ (2\partial^{\alpha}u\partial_{\alpha} + \Box u)\partial_{\vec{m}}f = [2\partial^{\alpha}u\partial_{\alpha} + \Box u, \partial_{\vec{m}}]f + \partial_{\vec{m}}j, $$

where

$$ \|[2\partial^{\alpha}u\partial_{\alpha} + \Box u, \partial_{\vec{m}}]f(t)\|_{L^2} \leq C(\|\partial u\|_{H^{m'}} + \|\Box u\|_{H^{m'}})\|f(t)\|_{H^m}, $$

for $m' = \max(m, 2)$.

**Proposition 8.5.** The time derivatives of (151) obey the following inequalities

$$ \|\partial_t f\|_{L^{\infty}([0,T'],H^m)} \leq C_{u,\eta}(\|f\|_{L^{\infty}([0,T'],H^{m+1})} + \|j\|_{L^{\infty}([0,T'],H^m)}), \quad (154) $$

$$ \|\partial_{tt}^2 f\|_{L^{\infty}([0,T'],H^{m-1})} \leq C_{u,\eta}(\|f\|_{L^{\infty}([0,T'],H^{m+1})} + \|j\|_{L^{\infty}([0,T'],H^m)}). \quad (155) $$

Proof. For the first inequality we use directly the transport equation. In the second case we apply one derivative on the transport equation and use the previous one to control the appearing first order in time derivatives. □

## 8.2 Commutator

**Proposition 8.6.** Let $g$ be a smooth Lorentzian metric, we define $\Box := D_{\alpha}D^{\alpha}$ for $D$ the covariant derivative and we contract the indices with respect to this metric.

Let $f$ defined on $[0, T'] \times \mathbb{R}^3$ be compactly supported and solution to

$$ \begin{cases} 2\partial^{\alpha}u\partial_{\alpha}f + \Box uf = j, \\ f(t=0) = f_0 \in H^2, \end{cases} \quad (156) $$

for $u$ a smooth characteristic phase such that there exists $C_u$ such that for every compact set $\Omega \subset \mathbb{R}^3$

$$ \|\Box \partial u\|_{L^{\infty}([0,T'], L^{\infty}(\Omega))} + \|\Box \Box u\|_{L^{\infty}([0,T'], H^{1/2}(\Omega))} + \|\partial \partial u\|_{L^{\infty}([0,T'], L^{\infty}(\Omega))} \leq C_u. \quad (157) $$

We suppose that

$$ (\Box f)|_{t=0} = G_0, \quad G_0 \in H^1. \quad (158) $$

<footnote>$^{47}$We ignore the border terms as they will be useless in our application.</footnote>
Let $G$ defined on $[0, T'] \times \mathbb{R}^3$ be solution to

$$ \begin{cases} 2\partial^\alpha u \partial_\alpha G + \Box u G = \Box j + [(2\partial^\alpha u \partial_\alpha + \Box u), \Box] f, \\ G(t=0) = G_0. \end{cases} \qquad (159) $$

Then, $G$ stays in $H^1$ for all $t \in [0, T']$ and

$$ \|G(t)\|_{H^1} \le C_{g,u} \int_0^t (\|\Box j(s)\|_{H^1} + \|\partial j(s)\|_{H^1} + \|f(s)\|_{H^2}) ds + \|G_0\|_{H^1}, \qquad (160) $$

where we use the pointwise in time estimate

$$ \|[(2\partial^\alpha u \partial_\alpha + \Box u), \Box] f(t)\|_{L^2} \le C_{g,u} (\|G(t)\|_{L^2} + \|\partial j(t)\|_{L^2} + \|f(t)\|_{H^1}). \qquad (161) $$

The result is true independently of the metric $g$.

**Proof.** We see that if $u = k^\alpha x_\alpha$ for $k$ a constant vector, that is if $u$ is the phase of a plane wave, we have $\partial_\alpha u = k_\alpha$ so that

$$ [(2\partial^\alpha u \partial_\alpha + \Box u), \Box] = [(2k^\alpha \partial_\alpha), \Box] = 0, \qquad (162) $$

so the result is direct.

Now we do the general case, to simplify the calculations we put ourselves in wave coordinates $g^{\mu\nu} \Gamma_{\mu\nu}^{\gamma} = 0$.

$$ \begin{aligned} [(2\partial^\alpha u \partial_\alpha + \Box u), \Box] f &= 2\partial^\alpha u \partial_\alpha (g^{\mu\nu}) \partial_\mu \partial_\nu f - 2\partial^\alpha u \partial_\alpha (g^{\mu\nu} \Gamma_{\mu\nu}^\gamma) \partial_\gamma f - 4\Box \partial^\alpha u \partial_\alpha f - 4g^{\mu\nu} \partial_\mu \partial^\alpha u \partial_\nu \partial_\alpha f - \Box \Box u f \\ &= -4(\Gamma_{\rho\alpha}^\mu g^{\rho\nu} \partial^\alpha u + g^{\mu\alpha} \partial_\alpha \partial^\nu u) \partial_\mu \partial_\nu f - 4(\Box \partial^\alpha u) \partial_\alpha f - \Box \Box u f = -4(\nabla^\mu \partial^\nu u) \partial_\mu \partial_\nu f - 4(\Box \partial^\alpha u) \partial_\alpha f - \Box \Box u f. \end{aligned} $$

We can write the metric as

$$ g^{\mu\nu} = -\frac{1}{2}(\partial^\mu u \partial^\nu \bar{u} + \partial^\mu \bar{u} \partial^\nu u) + \delta^{MN} e_M^\mu e_N^\nu, $$

where $e_{M,N}$ are spacelike vector fields orthonormal to $\partial u$. We set $\partial \bar{u}$ as the only null vector field such that $g(\partial u, \partial \bar{u}) = -2$ and $g(E_M, \partial \bar{u}) = 0$.

We know that $\partial_\nu u D^\mu \partial^\nu u = 0$ (35) and $\partial_\nu u D^\nu \partial^\mu u = 0$ (38), we deduce that locally we can write the tensor $\nabla^\mu \partial^\nu u$ as

$$ a \partial u \otimes \partial u + b(\partial u \otimes e_{M,N} + e_{M,N} \otimes \partial u) + c(e_{M,N} \otimes e_{M,N}), $$

$a, b$ and $c$ depend on $\partial \partial u$ and $\partial g$. All the other components are zero.

We conclude that schematically

$$ [(2\partial^\alpha u \partial_\alpha + \Box u), \Box] f = f_1(\partial \partial u, \partial g) \partial(\partial^\alpha u \partial_\alpha f) + f_2(\partial \partial u, \partial g) \partial f + f_3(\partial \partial u, \partial g) \partial_M \partial_N f + \Box \partial u \partial f + \Box \Box u f, $$

for some functions $f_1, f_2$ and $f_3$.

We know that the hessian $\partial_M \partial_N f$ can be estimated in $L^2$ on surfaces of $u$ and $\bar{u}$ constant via $\Delta_{M,N} F$ and $\partial F$, see [21].

We use the identity

$$ \Delta_{M,N} f = \Box f + \frac{1}{2}(\partial^\mu u \partial^\nu \bar{u} + \partial^\mu \bar{u} \partial^\nu u) \partial_\mu \partial_\nu f, $$

to get

$$ \|\partial_M \partial_N f\|_{L^2} \le C_{u,g} (\|\Box f\|_{L^2} + \|\partial(\partial^\alpha u \partial_\alpha f)\|_{L^2} + \|\partial f\|_{L^2}). $$

Putting everything together, we deduce

$$ \|[(2\partial^\alpha u \partial_\alpha + \Box u), \Box] f(t)\|_{L^2} \le C_{u,g} (\|\Box f\|_{L^2} + \|\partial(\partial^\alpha u \partial_\alpha f)\|_{L^2} + \|f\|_{H^1}) = C_{u,g} (\|G\|_{L^2} + \|\partial j\|_{L^2} + \|f\|_{H^1}). $$

This is the desired inequality.

<figure><img src="image_3.png" /></figure>
## 8.3 Sobolev inequalities

Here we recall the classical Sobolev embedding and interpolation results.

**Proposition 8.7.** Let $n \in \mathbb{N}^*$ be the dimension $k > 0$, $1 \le p < \infty$, $1 \le q \le \infty$, $k > l$ and $\frac{1}{p} \ge \frac{1}{q} \ge \frac{k-l}{n}$

$$W^{k,p} \hookrightarrow W^{l,q}.$$

*Proof.* From standard results.

**Proposition 8.8.** Let $n \in \mathbb{N}^*$ be the dimension $k > 0$, $1 \le p < \infty$, $1 \le q < \infty$,

$$\frac{1}{p} - \frac{k}{n} \ge 0 \Rightarrow W^{k,p} \xrightarrow{c} L^q,$$

$$\frac{1}{p} - \frac{k}{n} < 0 \Rightarrow W^{k,p} \xrightarrow{c} L^{\infty}.$$

*Proof.* From standard results.

**Proposition 8.9.** We have the embedding of weighted Sobolev spaces into the weighted $C_{\delta'}^m$ spaces for $m \in \mathbb{N}$, $1 \le p \le \infty$, $n/p < s < \infty$ and $\delta' \le \delta + \frac{n}{p}$

$$W_{p,\delta}^{s+m} \hookrightarrow C_{\delta'}^m.$$

*Proof.* From [3].

**Proposition 8.10.** For $s_\theta, s_0, s_1 \in \mathbb{R}$, $1 \le p_\theta, p_0, p_1 \le \infty$ and $0 < \theta < 1$ with $\frac{1}{p_\theta} = \frac{1-\theta}{p_0} + \frac{\theta}{p_1}$ and $s_\theta = (1-\theta)s_0 + (\theta)s_1$ we have

$$\|f\|_{W^{s_\theta, p_\theta}} \le (\|f\|_{W^{s_0, p_0}})^{1-\theta} (\|f\|_{W^{s_1, p_1}})^{\theta}. \qquad (163)$$

*Proof.* From standard results.

**Proposition 8.11.** We have the Gagliardo-Nirenberg inequality: Let $1 \le r, q \le \infty$, $j, m \in \mathbb{N}$ such that $j < m$, $p \ge 1$ then for

$$\frac{1}{p} = \frac{j}{n} + \theta \left( \frac{1}{r} - \frac{m}{n} \right) + (1-\theta) \frac{1}{q}, \quad \frac{j}{m} \le \theta \le 1,$$

we have

$$\|\partial_x^j f\|_{L^p(\mathbb{R}^n)} \le C (\|\partial_x^m f\|_{L^r(\mathbb{R}^n)})^\theta (\|f\|_{L^q(\mathbb{R}^n)})^{1-\theta}.$$

*Proof.* From standard results.

## 8.4 Laplacian inversion

**Definition 8.12.** We define the classical Laplacian as $\Delta = \partial_i \partial^i$ where

$$\Delta : W_{\delta,p}^{2+m} \to W_{\delta+2,p}^m.$$

**Theorem 8.1.** For $n \ge 3$, $\delta \ne -2 + \frac{n}{p} \mod(\mathbb{N})$ and $-\delta \ne \frac{n}{p} \mod(\mathbb{N})$ with

$$-n/p < \delta < -2 + n/p,$$
the Laplacian operator is an isomorphism from $W_{\delta,p}^{2+m}$ to $W_{\delta+2,p}^m$ and is in particular invertible.

Moreover, for $f$ solution to

$$\Delta f = h,$$

we have

$$\|f\|_{H_{\delta}^{2}} \lesssim \|h\|_{L_{\delta+2}^{2}}. \qquad (164)$$

*Proof.* From [18]. □

**Remark 8.13.** In the case $n=3$, $p=2$ and $p'=2$ we have $-3/2 < \delta < -1/2$. With $m=0$, we see that the source term $h$ must be in $L_{\delta+2}^2 \subset L^2(\delta+2>0)$ but if $\text{Supp}(h) \subset K$ for $K$ compact then

$$\|h\|_{L_{\delta+2}^2} \le C \|h\|_{L^2(K)}.$$

We can deduce the following useful lemma.

**Lemma 8.14.** Let

$$L = -\Delta + g, \qquad (165)$$

for $g \ge 0$ a compactly supported function in $H^2$ with $\|g\|_{H^2} \le C_g$. Then, for $\delta \in (-3/2, -1/2)$ the operator $L$ is an isomorphism from $H_\delta^2$ to $L_{\delta+2}^2$ and is in particular invertible.

Moreover, for $f$ solution to

$$Lf = h, \qquad (166)$$

We have

$$\|f\|_{H_{\delta}^{2}} \lesssim \|h\|_{L_{\delta+2}^{2}}. \qquad (167)$$

*Proof.* First, we show that the multiplication by $g$ is a compact operator from $H_\delta^2$ to $L_{\delta+2}^2$. Let $B$ be a ball of finite radius such that $\text{Supp}(g) \subset B$. For $f \in H_\delta^2$ with $\|f\|_{H_\delta^2} \le C$ we have

$$\|gf\|_{H^1(B)} \le \|g\|_{H^2(B)} \|f\|_{H^2(B)} \le c(B) \|f\|_{H_\delta^2} \|g\|_{H^2} \le c(C, B, C_g).$$

Using the Rellich-Kondrachov theorem we deduce that $\{gf|f \in H_\delta^2, \|f\|_{H_\delta^2} \le C\}$ is a compact subset of $L^2(B)$ and so a compact subset of $L_{\delta+2}^2$. Now we show the invertibility.

The classical Laplacian is already an isomorphism from $H_\delta^2$ to $L_{\delta+2}^2$ for $\delta \in (-3/2, -1/2)$ so that it is Fredholm of index 0. Adding a compact perturbation do not modify the index of the operator.

We deduce that the operator $L$ is Fredholm of index 0. For such an operator, the injectivity will imply the invertibility.

To prove the injectivity we use the maximum principle for Sobolev regularity.

Let $f$ be in $H_\delta^2$ such that

$$Lf = 0,$$

then because $g \ge 0$ we can use the maximum principle (see chapter 8 of [4] for example) for any set $\Omega$ of finite radius $R$ with smooth boundary we have

$$\sup_{\Omega} f \le \sup_{\partial\Omega} f|_{\partial\Omega},$$

$$\inf_{\Omega} f \ge \inf_{\partial\Omega} f|_{\partial\Omega},$$

where the trace of $f$ is its restriction to the border as $f$ is continuous (by Sobolev embedding in Hölder spaces from proposition 8.9). In fact, we have even more than continuity, $f$ also obeys the following inequality

$$|f(x)| \le \frac{C \|f\|_{H_{\delta}^{2}}}{|x|^{\beta}},$$
for some $C > 0$ and for $0 \le \beta < \delta + 3/2$ (here $\delta \in (-3/2, -1/2)$) due to the fact that the Sobolev spaces are weighted.

We have

$$|\sup_{\Omega} f| \le \frac{C ||f||_{H_{\delta}^2}}{R^{\beta}},$$

$$|\inf_{\Omega} f| \le \frac{C ||f||_{H_{\delta}^2}}{R^{\beta}}.$$

Letting the $R$ go to $+\infty$ we deduce that $f$ is identically 0. This proves the injectivity of $L$ and thus its invertibility.

To prove the inequality (167), we work by contradiction.

First, because $L$ is asymptotic to a Laplacian in the senses of [1] (for $q = 4$) we have the following inequality for every $f$ in $H_{\delta}^2$

$$||f||_{H_{\delta}^2} \le C(||Lf||_{L_{\delta+2}^2} + ||f||_{L^2(B_R)}), \quad (168)$$

which is from theorem 1.10 of [1]. The constant $C$ and $R$ depend on $g$ and $\delta$.

If the inequality (167) is false, then there exists a sequence $f_n$ such that

$$||f_n||_{H_{\delta}^2} = 1,$$

$$\lim_{n \to \infty} ||L f_n||_{L_{\delta+2}^2} = 0.$$

Because the sequence is uniformly bounded there exists a subsequence that converges weakly to $f$ in $H_{\delta}^2$ and that converges strongly in $L^2(B_R)$ by Rellich-Kondrachov.

We have $\forall \varphi \in C_c^{\infty}$

$$\lim_{n \to \infty} \int L f_n \varphi dx = 0$$

by assumptions and

$$\int (L f_n - L f) \varphi dx = \int (-\Delta f_n + \Delta f) \varphi dx + \int g(f_n - f) \varphi dx \le \int -\Delta (f_n - f) \varphi dx + ||g u||_{L^2(\Omega)} ||f_n - f||_{L^2(\Omega)},$$

which implies that

$$\lim_{n \to \infty} \int L f_n \varphi dx = \int L f \varphi dx.$$

We deduce that $L f = 0$ and thus $f = 0$ by injectivity.

Now, using the inequality (168), we find out that for any $\varepsilon > 0$ there exists $m \in \mathbb{N}$ such that for any $n \ge m$

$$1 = ||f_n||_{H_{\delta}^2} \le C(\varepsilon + ||f_n||_{L^2(B_R)}),$$

and we have $\lim_{n \to \infty} ||f_n||_{L^2(B_R)} = ||f||_{L^2(B_R)}$ as the convergence is strong. We deduce that $f \ne 0$ which is a contradiction. Thus, we obtain the inequality (167). □

## 8.5 Strichartz

**Definition 8.15.** We say that $(q, r)$ are Strichartz admissible if:

$$\begin{array}{l} q \ge 2, \\ (q, r, n) \ne (2, \infty, 3), \end{array} \qquad \begin{array}{l} \frac{2}{q} \le (n-1)(\frac{1}{2} - \frac{1}{r}), \\ r < \infty. \end{array}$$

**Theorem 8.2.** For $(q_1, r_1)$ et $(q_2, r_2)$ Strichartz admissible with

$$\frac{1}{q_1} + \frac{n}{r_1} = \frac{1}{q_2'} + \frac{n}{r_2'} - 2 = \frac{n}{2} - \gamma, \quad (169)$$

and $f$ solution to the Cauchy problem

$$\begin{cases} \Box f = j, \\ f(0) = f_0, \partial_t f(0) = \dot{f}_0, \end{cases} \quad (170)$$

then

$$||u||_{L^{q_1}([0,T], L^{r_1})} \le (||S||_{L^{q_2'}([0,T], L^{r_2'})} + ||u_0||_{\dot{H}^\gamma} + ||\dot{u}_0||_{\dot{H}^{\gamma-1}}). \quad (171)$$
## 8.6 Product estimates

We recall the following Sobolev product estimates result.

**Proposition 8.16.** For $1 < p < \infty$, $r, s \in \mathbb{R}$, $r, s < \frac{n}{p}$, $r + s > \max(\frac{n}{p} - \frac{n}{p'}, 0)$ with $\frac{1}{p} + \frac{1}{p'} = 1$ and, if $p > 2$, $||f||_{W^{p',-r}}$ or $||g||_{W^{p',-s}}$ we have

$$ ||fg||_{W^{p,t}} \le ||f||_{W^{p,r}} ||g||_{W^{p,s}}, \quad (172) $$

for $t = r + s - \frac{n}{p}$.

Proof. From [22].

We give two lemmas on product estimates based the Strichartz estimates.

**Lemma 8.17.** Let $f$ and $g$ be two functions defined on $[0, T'] \times \mathbb{R}^3$ for $T' > 0$. For $\nu \in (0, 1/2)$ we have the inequality

$$ ||fg||_{L^1([0,T'],L^2)} \le C ||f||_{L^p([0,T'],H^{1/2-\nu})} (||\Box g||_{L^1([0,T'],L^2)} + ||g(0)||_{\dot{H}^1} + ||\partial_t g(0)||_{L^2}), \quad (173) $$

with $p$ given by $\nu > 0$.

Proof. For $\frac{1}{p} + \frac{1}{p'} = 1$, $2 < \alpha = \frac{3}{1+\nu} < 3$ and $\frac{1}{2} = \frac{1}{\alpha} + \frac{1}{\beta}$ we have

$$ \begin{aligned} ||fg||_{L^1([0,T'],L^2)} &\le ||f||_{L^p([0,T'],L^\alpha)} ||g||_{L^{p'}([0,T'],L^\beta)} \\ &\le ||f||_{L^p([0,T'],H^{1/2-\nu})} ||g||_{L^{p'}([0,T'],L^\beta)}. \end{aligned} $$

We set ($r_2 = 2, q_2 = \infty$), they are Strichartz admissible, we want $p'$ such that $(\beta, p')$ are Strichartz admissible and such that

$$ \frac{1}{p'} + \frac{3}{\beta} = \frac{1}{q_2'} + \frac{3}{r_2'} - 2 = \frac{1}{2}, $$

this implies that $p' = \frac{\alpha}{3-\alpha}$ and so $p = \frac{\alpha}{2\alpha-3}$. Then, we use the Strichartz estimates on $||g||_{L^{p'}([0,T'],L^\beta)}$ to get the desired inequality. □

**Lemma 8.18.** Let $f$ and $g$ be two functions defined on $[0, T'] \times \mathbb{R}^3$ for $T' > 0$, for $\theta \in (0, 1/2)$ we have the inequality

$$ ||fg||_{L^1([0,T'],L^2)} \le ||f||_{L^p([0,T'],L^2)} (||g||_{L^q([0,T'],H^2)})^\theta (||\Box g||_{L^1([0,T'],L^2)} + ||g(0)||_{\dot{H}^1} + ||\partial_t g(0)||_{L^2})^{1-\theta}, $$

where $p$ and $q$ depend on $\theta$.

Proof. We have

$$ ||fg||_{L^1([0,T'],L^2)} \le ||f||_{L^p([0,T'],L^2)} ||g||_{L^{p'}([0,T'],L^\infty)}, \quad (174) $$

with Gaglirado-Nirenberg 8.11 we get

$$ ||fg||_{L^1([0,T'],L^2)} \le ||f||_{L^p([0,T'],L^2)} (||g||_{L^{p'}([0,T'],H^2)})^\theta (||g||_{L^{p'}([0,T'],L^{r_1})})^{1-\theta}, \quad (175) $$

where $r_1 = 6(\frac{1-\theta}{\theta}) > 6$ with $\theta \in (0, 1/2)$. We set ($r_2 = 2, q_2 = \infty$) and $p_1 = p'(1 - \theta)$ and search for ($r_1, p_1$) Strichartz admissible with

$$ \frac{1}{p_1} + \frac{3}{r_1} = \frac{1}{q_2'} + \frac{3}{r_2'} - 2 = \frac{1}{2}. $$

We find $p' = \frac{2}{1-\theta}$ so that $p = \frac{2}{1+\theta}$. Finally, we use the Strichartz estimates on $||g||_{L^{p'}([0,T'],L^{r_1})}$ to recover the desired inequality. □
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