Pointwise convergence for the elastic wave equation

We consider the Cauchy problem for the elastic wave equation

where $\Delta^{\ast}$ denotes the Lamé operator defined by

The Laplacian $\Delta$ acts on each component of a vector field $u$. Moreover, the following standard condition on the Lamé constants $\lambda,\mu\in\mathbb{R}$ is imposed to guarantee the ellipticity of $\Delta^{\ast}$:

The equation in (1.1) has been widely used as a model of wave propagation in an elastic medium, where $u$ denotes the displacement field of the medium (see e.g., [7, 8]). In particular, it is the classical wave equation if $\lambda+\mu=0$.

In this paper, we are concerned with determining the optimal regularity $s>0$ such that

for all $f\in H^{s}(\mathbb{R}^{n})$. The elastic wave propagator $e^{it\sqrt{-\Delta^{\ast}}}$ provides the solution $u=\frac{1}{2}(e^{it\sqrt{-\Delta^{\ast}}}f+e^{-it\sqrt{-\Delta^{\ast}}}f)$ for the problem (1.1) (see [5]).

For the classical wave equation, the pointwise convergence has been studied by means of the maximal estimate

which implies the convergence. Cowling [4] proved (1.3) for $s>1/2$. On the other hand, it was shown by Walther [11] that (1.3) fails for $s\leq 1/2$. (See also [9] for maximal estimates in $L^{p}$.) While the convergence is rather well understood for the wave equation, to the best of the authors’ knowledge, there doesn’t seem to be any corresponding literature for the elastic wave equation. In this regard, it would be interesting to ask whether the results for the wave equation are still valid for the elastic case. We obtain the following.

By a standard argument, Theorem 1.1 implies that

for $f\in H^{s}(\mathbb{R}^{n})$ with $s>1/2$. If $s<1/2$, it follows from Stein’s maximal principle ([10]) that (1.5) fails. The convergence (1.5) says that the solution converges to initial data along the line $\{(x+vt\theta,t)\colon t\geq 0\}$ on the light cone with speed $v$; see Figure 1. The convergence along lines on the light cone is new even for the wave equation (the case $\lambda+\mu=0$).

Recently, the three of the authors and Lee [6] diagonalized the Lamé operator so that $\Delta^{\ast}=P\Delta P^{-1}$ with a certain invertible matrix $P$ to study the elastic wave equation. We utilize the diagonalization to prove Theorem 1.1 in the following sections.

Let us recall from [6, Section 2] the diagonalization process and notations to make this article self-contained (the readers are encouraged to refer to [6] for details; also see Figure 2). For the unit vector $e_{1}=(1,0,\ldots,0)^{t}\in\mathbb{R}^{n}$, let $S_{\pm}=\{\omega\in S^{n-1}\colon-1/\sqrt{2}\leq\omega\cdot(\pm e_{1})\leq 1\}$ and $\mathbb{R}_{\pm}^{n}=\{\xi\in\mathbb{R}^{n}\setminus\{0\}\colon\xi/|\xi|\in S_{\pm}\}$. For every $\omega\in S_{\pm}\setminus\{\pm e_{1}\}$, we define the arc $\mathcal{C}_{\pm}(\omega)=S_{\pm}\cap\,\mathrm{span}\{e_{1},\omega\}$, which is the intersection of $S_{\pm}$ and the great circle on $S^{n-1}$ passing through $e_{1}$ and $\omega$. Now we take $\rho_{\pm}(\omega)\in\mathrm{SO}(n)$ so that its transpose $\rho_{\pm}(\omega)^{t}$ is the unique rotation mapping $\omega$ to $\pm e_{1}$ along the arc $\mathcal{C}_{\pm}(\omega)$ and satisfying $\rho_{\pm}(\omega)^{t}y=y$ whenever $y\in\mathrm{span}\{e_{1},\omega\}^{\perp}$. When $\omega=\pm e_{1}$ we set $\rho_{\pm}(\omega)^{t}=I_{n}$. We define the maps $R_{\pm}:\mathbb{R}^{n}_{\pm}\rightarrow\mathrm{SO}(n)$ and the projections $\mathcal{P}_{\pm}$ by

where $\{\varphi_{+},\varphi_{-}\}$ is a smooth partition of unity on $S^{n-1}$ subordinate to the covering $\{\mathrm{int}S_{+},\mathrm{int}S_{-}\}$. We also set $D=-i\nabla$ and denote by $m(D)f=(m\widehat{f}\,)^{\vee}$ the Fourier multiplier defined by a bounded (matrix-valued) function $m$.

The positive square root of $\Lambda$ exists by the condition (1.2), so we have

with $\sqrt{\Lambda}(\xi)=\mathrm{diag}(\sqrt{\lambda+2\mu}|\xi|,\sqrt{\mu}|\xi|,\ldots,\sqrt{\mu}|\xi|)$. Furthermore, it follows that

where

In this section, we construct an example to show that the maximal estimate (1.4) implies $s\geq 1/2$. We need only to consider $\theta=e_{1}$. By (2.3), write

We represent the $n\times n$ orthogonal matrix $R_{\pm}=(r^{\pm}_{ij})_{1\leq i,j\leq n}$ as the block matrix

where $A_{\pm}=r^{\pm}_{11}$, $B_{\pm}=(r_{1j}^{\pm})_{2\leq j\leq n}$, $C_{\pm}=(r_{i1}^{\pm})_{2\leq i\leq n}$ and $D_{\pm}=(r_{ij}^{\pm})_{2\leq i,j\leq n}$. Then

where

Let us set $\alpha=\sqrt{\lambda+2\mu}+v$. For every $N\gg\alpha^{-1}$ we define

where $\angle(e_{1},x)$ denotes the angle between $e_{1}$ and $x$. It is easy to see that

If we take

then

Since $\cos u\geq 1-u^{2}/2$ we have for $(x,\xi)\in E\times F$

from which it follows that

since $v/\alpha<1$.

Let us define $f_{F}\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ by $\widehat{f_{F}}(\xi)=\chi_{F}(\xi)e_{1}$. By the definition of $F$ and $\mathcal{P}_{\pm}$ it is clear that $\mathcal{P}_{+}f_{F}=f_{F}$ and $\mathcal{P}_{-}f_{F}=0$. By (3.1) and (3.2), we have

We need to estimate the size of $A_{+}(\xi)$ and $B_{+}(\xi)$ for $\xi\in F$.

By the estimates (3.6), we have

Combining these with (3.4) we see that (3.5) is estimated by

for every $N$ large enough. Thus, we obtain

On the other hand,

By (3.8) and (3.9), the maximal estimate (1.4) implies

for all $N$ large enough. Therefore we conclude that the estimate (1.4) does not hold for $s<1/2$.

Let us fix a cutoff function $\phi\in C_{0}^{\infty}(\mathbb{R})$ such that $\phi(t)=1$ for $|t|\leq 1$ and $\phi(t)=0$ for $|t|\geq 2$, and denote

In order to prove (1.4) we need only to show the following:

Indeed, the estimates imply

Furthermore, by complex interpolation (see [2]) it follows from (4.1) and (4.3) that, for $0<s<1$,

By the Sobolev embedding $H^{s}(\mathbb{R})\hookrightarrow L^{\infty}(\mathbb{R})$ for $s>1/2$, we conclude

which completes the proof of (1.4).

Now let us show (4.1) and (4.2). Using the diagonalization (2.3) we write

Then, by the Plancherel theorem

On the other hand, $|\partial_{t}T_{v,\theta}f(x,t)|$ is estimated by

Then by the Plancherel theorem, we obtain (4.2).