On the Formation of Shock for Quasilinear Wave Equations with Weak Intensity Pulse

Abstract

In this paper we continue to study the shock formation for the 3-dimensional quasilinear wave equation

with $G''(0)$ being a non-zero constant. Since ($\star$) admits global-in-time solution with small initial data, to present shock formation, we consider a class of large data. Moreover, no symmetric assumption is imposed on the data. Compared to our previous work (Miao and Yu in Invent Math 207(2):697–831, 2017), here we pose data on the hypersurface $\{(t,x)|t=-\,r_{0}\}$ instead of $\{(t,x)|t=-\,2\}$, with $r_{0}$ being arbitrarily large. We prove an a priori energy estimate independent of $r_{0}$. Therefore a complete description of the solution behavior as $r_{0}\rightarrow \infty$ is obtained. This allows us to relax the restriction on the profile of initial data which still guarantees shock formation. Since ($\star$) can be viewed as a model equation for describing the propagation of electromagnetic waves in nonlinear dielectric, the result in this paper reveals the possibility to use wave pulse with weak intensity to form electromagnetic shocks in laboratory. A main new feature in the proof is that all estimates in the present paper do not depend on the parameter $r_{0}$, which requires different methods to obtain energy estimates. As a byproduct, we prove the existence of semi-global-in-time solutions which lead to shock formation by showing that the limits of the initial energies exist as $r_{0}\rightarrow \infty$. The proof combines the ideas in Christodoulou (in: EMS monographs in mathematics, European Mathematical Society (EMS), Zurich, 2007) where the formation of shocks for 3-dimensional relativistic compressible Euler equations with small initial data is established, and the short pulse method introduced in Christodoulou (in: EMS monographs in mathematics, European Mathematical Society (EMS), Zurich, 2009) and generalized in Klainerman and Rodnianski (Acta Math 208(2):211–333, 2012), where the formation of black holes in general relativity is proved.

Appendices

Appendix A. Proof of (6.3)

Proof

In view of (2.44), we have

$$\begin{aligned} R_{i}=(R_{i})^{m}\partial _{m},\quad R_{i}=\Omega _{i}-\lambda _{i}\widehat{T}. \end{aligned}$$

Thus, we have

$$\begin{aligned}{}[R_{i},R_{j}]=&\left( R_{i}((R_{j})^{m})-R_{j}((R_{i})^{m})\right) \partial _{m}\nonumber \\ =&\left( R_{i}((\Omega _{j})^{m}-\lambda _{j}\widehat{T}^{m})-R_{j}((\Omega _{i})^{m}-\lambda _{i}\widehat{T}^{m})\right) \partial _{m}. \end{aligned}$$

(A.1)

On the other hand,

$$\begin{aligned} R_{i}(\Omega _{j})^{m}=&R_{i}(\varepsilon _{jkm}x^{k})=\varepsilon _{jkm}(R_{i})^{k}\\ =&-\varepsilon _{jmk}\left( (\Omega _{i})^{k}-\lambda _{i}\widehat{T}^{k}\right) =-\varepsilon _{jmk}\varepsilon _{ilk}x^{l}+\varepsilon _{jmk}\lambda _{i}\widehat{T}^{k}. \end{aligned}$$

Here $\varepsilon$ is the standard volume form in Euclidean space ${\mathbb {R}}^{3}$, and it satisfies

$$\begin{aligned} \varepsilon _{jmk}\varepsilon _{ilk}+\varepsilon _{mik}\varepsilon _{jlk}+\varepsilon _{ijk}\varepsilon _{mlk}=0. \end{aligned}$$

We then obtain

$$\begin{aligned} R_{i}(\Omega _{j})^{m}-R_{j}(\Omega _{i})^{m}=-\varepsilon _{ijk}\Omega _{k}^{m}+\left( \varepsilon _{ikm}\lambda _{j}-\varepsilon _{jkm}\lambda _{i}\right) \widehat{T}^{k}. \end{aligned}$$

(A.2)

Substituting (A.2) in (A.1) gives

$$\begin{aligned}{}[R_{i},R_{j}]=&-\varepsilon _{ijk}\Omega _{k} -\lambda _{i}\varepsilon _{jkm}\widehat{T}^{k}\partial _{m} +\lambda _{j}\varepsilon _{ikm}\widehat{T}^{k}\partial _{m}\nonumber \\&-\left( R_{i}(\lambda _{j})-R_{j}(\lambda _{i})\right) \widehat{T}-\left( \lambda _{j}R_{i}(\widehat{T}^{m})-\lambda _{i}R_{j}(\widehat{T}^{m})\right) \partial _{m}. \end{aligned}$$

(A.3)

We note that

$$\begin{aligned} R_{i}(\widehat{T}^{m})\partial _{m}=\overline{\nabla }_{R_{i}}\widehat{T}=\theta ^{A}{}_{B}R^{B}_{i}X_{A}=-c^{-1}\underline{\chi }^{A}_{B}R^{B}_{i}X_{A}. \end{aligned}$$

(A.4)

Here $\overline{\nabla }$ is the covariant derivative with respect to the Euclidean metric, which is also the induced metric of g on each $\Sigma _{t}$. So if we apply $\Pi$ to both sides of (A.3), we obtain:

$$\begin{aligned}{}[R_{i},R_{j}]=&-\varepsilon _{ijk}R_{k} -\lambda _{i}\varepsilon _{jkm}\widehat{T}^{k}\Pi \partial _{m} +\lambda _{j}\varepsilon _{ikm}\widehat{T}^{k}\Pi \partial _{m}\nonumber \\&-\left( R_{i}(\lambda _{j})-R_{j}(\lambda _{i})\right) \widehat{T}-\left( \lambda _{j}R_{i}(\widehat{T}^{m})-\lambda _{i}R_{j}(\widehat{T}^{m})\right) \partial _{m}. \end{aligned}$$

(A.5)

On the other hand, by (2.46) we have

$$\begin{aligned}&-\lambda _{i}\varepsilon _{jkm}\widehat{T}^{k}\partial _{m} +\lambda _{j}\varepsilon _{ikm}\widehat{T}^{k}\partial _{m}\\ =&-\lambda _{i}\varepsilon _{jkm}y^{k}\partial _{m} +\lambda _{j}\varepsilon _{ikm}y^{k}\partial _{m}\\&-\lambda _{i}\varepsilon _{jkm}\frac{x^{k}}{\underline{u}-t}\partial _{m} +\lambda _{j}\varepsilon _{ikm}\frac{x^{k}}{\underline{u}-t}\partial _{m}\\ =&-\lambda _{i}\varepsilon _{jkm}y^{k}\partial _{m} +\lambda _{j}\varepsilon _{ikm}y^{k}\partial _{m}\\&-\lambda _{i}\frac{\Omega _{j}}{\underline{u}-t} +\lambda _{j}\frac{\Omega _{i}}{\underline{u}-t}. \end{aligned}$$

Applying $\Pi$-projection, we have

$$\begin{aligned}&-\lambda _{i}\varepsilon _{jkm}\widehat{T}^{k}\partial _{m} +\lambda _{j}\varepsilon _{ikm}\widehat{T}^{k}\partial _{m}\nonumber \\ =&-\lambda _{i}\varepsilon _{jkm}y^{k}\Pi \partial _{m} +\lambda _{j}\varepsilon _{ikm}y^{k}\Pi \partial _{m}\nonumber \\&-\lambda _{i}\frac{R_{j}}{\underline{u}-t}\partial _{m} +\lambda _{j}\frac{R_{i}}{\underline{u}-t}\partial _{m}. \end{aligned}$$

(A.6)

Substituting (A.4) and (A.6) into (A.5), we obtain the desired result. $\square$

Appendix B. Coordinate transformation is one-to-one

To prove that the transformation from $(t,\underline{u},\vartheta )$ to $(t,x^{1},x^{2},x^{3})$ is one-to-one, one needs to show

• The null geodesics on a level set of the optical function $\underline{u}$ do not intersect, namely, there is no conjugate point.

• The level set of $\underline{u}$ does not pinch. In other words, there is no cut loci along the null geodesics.

For the first statement, let $\gamma =\gamma (t)$ be an arbitrary generator of a null hypersurface $\underline{C}_{\underline{u}}$. It suffices to show that any Jacobi field $J\in TS_{t,\underline{u}}$ along $\gamma (t)$ does not vanish for all $t\in [-r_{0},t^{*})$. Note that according to the definition of $\underline{L}$, we have $\dot{\gamma }(t)=\underline{L}$. J satisfies the Jacobi equation:

$$\begin{aligned} \nabla _{\underline{L}}^{2}J=\mu ^{-1}(\underline{L}\mu )\nabla _{\underline{L}}J+R(\underline{L},J)\underline{L}. \end{aligned}$$

(B.1)

Let $(E_{A}=E_{A}(t), A=1,2)$ be an orthonormal basis of $TS_{t,\underline{u}}$. The Jacobi field J has the expansion with respect to this basis:

$$\begin{aligned} J(t)=\sum _{A=1}^{2}J^{A}(t)E_{A}(t). \end{aligned}$$

The components $J^{A}(t)$ satisfies the following ODE system:

$$\begin{aligned} \frac{d^{2}J^{A}(t)}{dt^{2}}=\mu ^{-1}\frac{d\mu (t)}{dt}\frac{dJ^{A}(t)}{dt}-\sum _{B=1}^{2}\underline{\alpha }_{AB}(t)J^{B}(t). \end{aligned}$$

(B.2)

Here we used the fact

$$\begin{aligned} R(\underline{L},J)\underline{L}=-\sum _{A,B=1}^{2}\underline{\alpha }_{AB}J^{B}E_{A}, \end{aligned}$$

which can be shown by a direct computation. On the other hand,

$$\begin{aligned} \frac{dJ^{A}}{dt}=&g(\nabla _{\underline{L}}J,E_{A})=g(\nabla _{J}\underline{L},E_{A})+g([\underline{L},J],E_{A}) =\underline{\chi }(J,E_{A})=\sum _{B=1}^{2}\underline{\chi }_{AB}J^{B}. \end{aligned}$$

(B.3)

Here we used the fact

$$\begin{aligned}{}[\underline{L},J]=\mu \nabla _{{\widehat{\underline{L}}}}J-\nabla _{J}(\mu {\widehat{\underline{L}}})=-J(\mu ){\widehat{\underline{L}}},\quad \Rightarrow \quad g([\underline{L},J],E_{A})=0. \end{aligned}$$

In particular, (B.3) holds at the initial time $t=-\,r_{0}$. Therefore the solution to (B.2) is given by

$$\begin{aligned} J^{A}(t)=\sum _{B=1}^{2}M^{A}_{B}(t)J^{B}(-r_{0}) \end{aligned}$$

(B.4)

for some matrix M. This implies

$$\begin{aligned}&\sum _{B=1}^{2}\frac{dM^{A}_{B}}{dt}J^{B}(-r_{0})=\frac{dJ^{A}}{dt}=\sum _{B=1}^{2} \underline{\chi }_{AB}(t)J^{B}(t)=\sum _{B,C=1}^{2}\underline{\chi }_{AC}(t)M^{C}_{B}J^{B}(-r_{0})\nonumber \\ \Rightarrow \quad&\frac{dM^{A}_{B}}{dt}=\sum _{C=1}^{2}\underline{\chi }_{AC}M^{C}_{B}. \end{aligned}$$

(B.5)

This together with a direct computation shows that for any $t_{1}, t_{2}\in [-r_{0},t^{*})$, we have

$$\begin{aligned} \det M(t_{2})=\det M(t_{1})\exp \left( \int _{t_{1}}^{t_{2}}{\text {tr}}\underline{\chi }(t')dt'\right) \end{aligned}$$

(B.6)

If J has a conjugate point, namely, there is a $t_{c}\in [-r_{0},t^{*})$ such that $J(t_{c})=0$, then (B.4) implies that $\det M(t_{c})=0$. Without loss of generality, we can assume that for any $t\in [-r_{0},t_{c})$, . But this contradicts with (B.6) and estimate on $\underline{\chi }$ (see (3.20) and (3.21)). Therefore J(t) does not admit a conjugate point.

Now we turn to the second statement. It suffices to prove that the surface $S_{t,\underline{u}}$ does not pinch. In fact, we will show that $S_{t,\underline{u}}$ is convex, which can be seen as follows: The second fundamental form of $S_{t,\underline{u}}$ as a submanifold in $\Sigma _{t}$ is given by (see (2.30)):

$$\begin{aligned} \theta =-c^{-1}\underline{\chi }. \end{aligned}$$

(B.7)

In view of the estimate (3.14) on c and the estimate (3.20), (3.21) on $\underline{\chi }$, $\theta$ is close to the second fundamental form of a round sphere in Euclidean space. Therefore $\theta$ is positive definite. This implies that $S_{t,\underline{u}}$ is convex, which means that $S_{t,\underline{u}}$ does not pinch. Alternatively, this second statement can be seen in a more intuitive way: The estimate (3.39) implies that the angle between the outward unit normal $\widehat{T}$ to $S_{t,\underline{u}}$ and the outward unit normal $\partial _{r}$ to a round sphere is small. If $S_{t,\underline{u}}$ pinches at some point $p\in S_{t,\underline{u}}$, then $\widehat{T}$ must deviate largely from $\partial _{r}$ near p, which contradicts with the fact that $\widehat{T}$ and $\partial _{r}$ have a small angle. This completes the proof for the fact that the transfromation between the optical coordinate and rectangular coordinate is one-to-one.

Notations

For the reader’s convenience we give the definitions of some of the symbols used commonly in this work.