STRICHARTZ ESTIMATES FOR THE WAVE EQUATION INSIDE CYLINDRICAL CONVEX DOMAINS

Abstract

We establish local-in-time Strichartz estimates for solutions of the model case Dirichlet wave equation inside cylindrical convex domains $\Omega \subset \mathbb {R}^ 3$ with smooth boundary $\partial \Omega \neq \emptyset $ . The key ingredients to prove Strichartz estimates are dispersive estimates, energy estimates, interpolation and $TT^*$ arguments. Strichartz estimates for waves inside an arbitrary domain $\Omega $ have been proved by Blair, Smith and Sogge [‘Strichartz estimates for the wave equation on manifolds with boundary’, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 1817–1829]. We provide a detailed proof of the usual Strichartz estimates from dispersive estimates inside cylindrical convex domains for a certain range of the wave admissibility.

1 Introduction

1.1 The cylindrical model problem

Let $\Omega =\{x\geq 0,(y,z)\in \mathbb {R}^2\}\subset \mathbb {R}^3$ with smooth boundary $\partial \Omega =\{x=0\}$ and let $\Delta =\partial _x^2+(1+x)\partial _y^2+\partial _z^2$ . We consider solutions of the linear Dirichlet wave equation inside $\Omega $ :

(1.1) $$ \begin{align} (\partial_t^2-\Delta)u=0, \quad u_{|_{t=0}}=u_0,\quad\partial_tu_{|_{t=0}}=u_1,\quad u_{|_{x=0}}=0. \end{align} $$

The Riemannian manifold $(\Omega ,\Delta )$ with $\Delta =\partial _x^2+(1+x)\partial _y^2+\partial _z^2$ can be locally seen as a cylindrical domain in $\mathbb {R}^3$ by taking cylindrical coordinates $(r,\theta ,z)$ , where we set $r=1-x/2,\theta =y$ and $z=z$ . The main goal of this work is to prove the Strichartz estimates inside cylindrical convex domains for the solution u to (1.1).

1.2 Some known results

Let us recall a few results about Strichartz estimates (see [10, Section 1]). Let $(\Omega ,g)$ be a Riemannian manifold without boundary of dimension $d\geq 2$ . Local-in-time Strichartz estimates state that

(1.2) $$ \begin{align} \|u\|_{L^q((-T,T);L^r(\Omega))}\leq C_T(\|u_0\|_{\dot{H}^\beta(\Omega)}+\|u_1\|_{\dot{H}^{\beta-1}(\Omega)}), \end{align} $$

where $\dot {H}^\beta $ denotes the homogeneous Sobolev space over $\Omega $ of order $\beta $ , $2\leq q,r\leq \infty $ and

$$ \begin{align*} \frac{1}{q}+\frac{d}{r}=\frac{d}{2}-\beta,\quad \frac{1}{q}\leq \frac{d-1}{2}\bigg(\frac{1}{2}-\frac{1}{r}\bigg). \end{align*} $$

Here $u=u(t,x)$ is a solution to the wave equation

$$ \begin{align*} (\partial_t^2-\Delta_g)u=0\,\,\text{in } (-T,T)\times \Omega, \quad u(0,x)=u_0(x),\quad \partial_tu(0,x)=u_1(x), \end{align*} $$

where $\Delta _g$ denotes the Laplace–Beltrami operator on $(\Omega ,g)$ . The estimates (1.2) hold on $\Omega =\mathbb {R}^d$ and $g_{ij}=\delta _{ij}.$

Blair et al. [4] proved the Strichartz estimates for the wave equation on a (compact or noncompact) Riemannian manifold with boundary. They proved that the Strichartz estimates (1.2) hold if $\Omega $ is a compact manifold with boundary and $ (q,r,\beta )$ is a triple satisfying

$$ \begin{align*} \frac{1}{q}+\frac{d}{r}=\frac{d}{2}-\beta \quad \text{for } \begin{cases} \dfrac{3}{q}+\dfrac{d-1}{r}\leq\dfrac{d-1}{2},\quad d\leq 4,\\[0.3 cm] \dfrac{1}{q}+\dfrac{1}{r}\leq \dfrac{1}{2},\quad d\geq 4. \end{cases} \end{align*} $$

Recently in [10], Ivanovici et al. deduced local-in-time Strichartz estimates (1.2) from the optimal dispersive estimates inside strictly convex domains of dimension $d\geq 2$ for a triple $(d,q,\beta )$ satisfying

$$ \begin{align*} \frac{1}{q}\leq\bigg(\frac{d-1}{2}-\frac{1}{4}\bigg)\bigg(\frac{1}{2}-\frac{1}{r}\bigg) \quad \text{and}\quad \beta=d\bigg(\frac{1}{2}-\frac{1}{r}\bigg)-\frac{1}{q}. \end{align*} $$

For $d\geq 3$ , this improves the range of indices for which sharp Strichartz estimates hold compared to the result by Blair et al. [4]. However, the results in [4] apply to any domains or manifolds with boundary. The latest results in [11] on Strichartz estimates inside the Friedlander model domain have been obtained for pairs $(q, r)$ such that

$$ \begin{align*}\frac{1}{q}\leq \bigg(\frac{1}{2}-\frac{1}{9}\bigg)\bigg(\frac{1}{2}-\frac{1}{r}\bigg).\end{align*} $$

This result improves on the known results for strictly convex domains for $d=2$ , while [10] only gives a loss of $\tfrac 14$ .

Let us also recall that dispersive estimates for the wave equation in $\mathbb {R}^ d$ follow from the representation of the solution as a sum of Fourier integral operators (see [1, 5, 8]). They read as follows:

(1.3) $$ \begin{align} \|\,\chi(hD_t)e^{\pm it\sqrt{-\Delta_{\mathbb{R}^d}}}\|_{L^1(\mathbb{R}^ d)\rightarrow L^\infty(\mathbb{R}^ d)}\leq Ch^{-d}\min\bigg\{1,\bigg(\frac{h}{|t|}\bigg)^{{(d-1)}/{2}}\bigg\}, \end{align} $$

where $\Delta _{\mathbb {R}^d}$ is the Laplace operator in $\mathbb {R}^d$ . Here and in the following, the function $\chi $ belongs to $C_0^\infty (]0,\infty [)$ and is equal to $1$ on $[1,2]$ and $D_t={(1}/{i})\partial _t$ . Inside strictly convex domains $\Omega _D$ of dimensions $d\geq 2$ , the optimal (local-in-time) dispersive estimates for the wave equation have been established by Ivanovici et al. [10]. More precisely, they have proved that

(1.4) $$ \begin{align} \|\,\chi(hD_t)e^{\pm it\sqrt{-\Delta_D}}\|_{L^1(\Omega_D)\rightarrow L^\infty(\Omega_D)}\leq Ch^{-d}\min\bigg\{1,\bigg(\frac{h}{|t|}\bigg)^{{(d-1)}/{2}-{1}/{4}}\bigg\}, \end{align} $$

where $\Delta _D$ is the Laplace operator on $\Omega _D$ . Due to the formation of caustics in arbitrarily small times, (1.4) induces a loss of $\tfrac 14$ powers of the $(h/|t|)$ factor compared to (1.3). The local-in-time dispersive estimates for the wave equation inside cylindrical convex domains in dimension $3$ have been derived in [13, 14] as follows:

$$ \begin{align*} \|\,\chi(hD_t)\mathcal{G}_a(t,x,y,z)\|_{L^1(\Omega)\rightarrow L^\infty(\Omega)}\leq Ch^{-3}\min\bigg\{1,\bigg(\frac{h}{|t|}\bigg)^{{3}/{4}}\bigg\}, \end{align*} $$

where $\mathcal {G}_a$ is the Green function for (1.1).

2 Main result

We now state our main result concerning the Strichartz estimates inside cylindrical convex domains in dimension $3$ .

Theorem 2.1. Let $(\Omega ,\Delta )$ be defined as before. Let u be a solution of the wave equation on $\Omega $ :

$$ \begin{align*} (\partial_t^2-\Delta)u = 0\ \text{in } \Omega, \quad u_{|t=0}=u_0,\quad \partial_t u_{|t=0} =u_1,\quad u_{|x=0}=0. \end{align*} $$

Then for all T, there exists $C_T$ such that

$$ \begin{align*} \|u\|_{L^q((0,T);L^r(\Omega))}\leq C_T(\|u_0\|_{\dot{H}^{\beta}(\Omega)}+\|u_1\|_{\dot{H}^{\beta-1}(\Omega)}), \end{align*} $$

with

$$ \begin{align*} \frac{1}{q}\leq \frac{3}{4}\bigg(\frac{1}{2}-\frac{1}{r}\bigg) \quad \text{and}\quad \beta=3\bigg(\frac{1}{2}-\frac{1}{r}\bigg)-\frac{1}{q}. \end{align*} $$

To prove Theorem 2.1, we first prove the frequency-localised Strichartz estimates by utilising the frequency-localised dispersive estimates, interpolation and $TT^\ast $ arguments. We then apply the Littlewood–Paley square function estimates (see [2, 3, 12]) to get the Strichartz estimates (Theorem 2.1) in the context of cylindrical domains. For $d=3$ , Theorem 2.1 improves the range of indices for which the sharp Strichartz estimates hold. However, our result is restricted to cylindrical domains, while [4] applies to any domain.

3 Strichartz estimates for the model problem

Let us recall some notation. For any $I\subset \mathbb {R},\Omega \subset \mathbb {R}^d$ , we define the mixed space-time norms

$$ \begin{align*} \|u\|_{L^q(I;L^r(\Omega))}&:=\bigg(\int_I\|u(t,.)\|_{L^r(\Omega)}^q\,dt\bigg)^{1/q}\quad\text{if } 1\leq q<\infty,\\ \|u\|_{L^{\infty}(I;L^r(\Omega))}&:=\mathop{\mathrm{ess\, sup}}_{t\in I}\|u(t,.)\|_{L^r(\Omega)}. \end{align*} $$

3.1 Frequency-localised Strichartz estimates

In this section, we prove Theorem 3.1. The classical strategy is as follows. We begin by interpolating between the energy estimates and dispersive estimates. This yields a new estimate, which we further manipulate via a classical $L^p$ inequality to establish (3.8). This last step imposes conditions on the space-time exponent pair $(q,r)$ ; these are precisely the wave admissibility criteria. The classical inequalities used are the Young, Hölder and Hardy–Littlewood–Sobolev inequalities.

We first recall the Littlewood–Paley decomposition and some links with Sobolev spaces [1]. Let $\chi \in C_0^\infty (\mathbb {R}^*)$ and equal to $1$ on $[\tfrac 12,2]$ such that

$$ \begin{align*} \sum_{j\in\mathbb{Z}}\chi(2^{-j}\lambda)=1,\quad \lambda>0. \end{align*} $$

We define the associated Littlewood–Paley frequency cutoffs $\chi (2^{-j}\sqrt {-\Delta })$ using the spectral theorem for $\Delta $ and we have

$$ \begin{align*} \sum_{j\in\mathbb{Z}}\chi(2^{-j}\sqrt{-\Delta})=\mbox{Id}: L^2(\Omega)\longrightarrow L^2(\Omega). \end{align*} $$

This decomposition takes a single function and writes it as a superposition of a countably infinite family of functions $\chi $ each one having a frequency of magnitude $\sim 2^{j}$ for $j\geq 1$ . A norm of the homogeneous Sobolev space $\dot {H}^{\beta }$ is defined as follows: for all $\beta \geq 0$ ,

$$ \begin{align*} \|u\|_{\dot{H}^\beta}:=\bigg(\sum_{j\in\mathbb{Z}}2^{2j\beta}\|\chi(2^{-j}D_t)u\|_{L^2}^2\bigg)^{1/2}. \end{align*} $$

With this decomposition, the Littlewood–Paley square function estimate (see [2, 3, 12]) reads as follows: for $f\in L^r(\Omega )$ and for all $r\in [2,\infty [$ ,

(3.1) $$ \begin{align} \|f\|_{L^r(\Omega)}\leq C_r\bigg\|\bigg(\sum_{j\in\mathbb{Z}}|\chi(2^{-j}\sqrt{-\Delta})f|^2\bigg)^{1/2}\bigg\|_{L^r(\Omega)}. \end{align} $$

The proof follows from the classical Stein argument involving Rademacher functions and an appropriate Mikhlin–Hörmander multiplier theorem.

We define the frequency localisation $v_j$ of u by $v_j=\chi (2^{-j}\sqrt {-\Delta })u$ . Hence, $u=\sum _{j\geq 0}v_j$ . Let $h=2^{-j}$ . We deduce from the dispersive estimates inside cylindrical convex domains established in [13, 14] the frequency-localised dispersive estimates for the solution $v_j=\chi (hD_t)u$ of the (frequency-localised) wave equation

(3.2) $$ \begin{align} (\partial_t^2-\Delta)v_j=0 \text{ in } \Omega,\quad {v_j}_{|t=0}=\chi(hD_t)u_0,\quad \partial_t{v_j}_{|t=0}=\chi(hD_t)u_1,\quad {v_j}_{|\partial\Omega}=0, \end{align} $$

which read as follows:

(3.3) $$ \begin{align} \|\dot{\mathcal U}(t)\chi(hD_t)u_0\|_{L^\infty}\lesssim h^{-3}\min\bigg\{1,\bigg(\frac{h}{t}\bigg)^{{3}/{4}}\bigg\}\|\chi(hD_t)\,u_0\|_{L^1},\\ \|\mathcal U(t)\chi(hD_t)u_1\|_{L^\infty}\lesssim h^{-2}\min\bigg\{1,\bigg(\frac{h}{t}\bigg)^{{3}/{4}}\bigg\}\|\chi(hD_t)\,u_1\|_{L^1},\nonumber \end{align} $$

where we use the notation

$$ \begin{align*} \mathcal U(t):=\frac{\sin(t\sqrt{-\Delta})}{\sqrt{-\Delta}}& \quad\text{and}\quad \dot{\mathcal U}(t):=\cos(t\sqrt{-\Delta}). \end{align*} $$

These estimates yield the following Strichartz estimates.

Theorem 3.1 (Frequency-localised Strichartz estimates).

Let $(\Omega ,\Delta )$ be defined as before. Let $v_j$ be a solution of the (frequency-localised) wave equation (3.2). Then for all T, there exists $C_T$ such that

(3.4) $$ \begin{align} h^\beta\|\dot{\mathcal U}(t)\chi(hD_t)u_0\|_{L_{t}^q(L_x^r)}\lesssim \|\chi(hD_t)u_0\|_{L^2}, \end{align} $$

(3.5) $$ \begin{align} h^{\beta-1}\|\mathcal U(t)\chi(hD_t)u_1\|_{L_{t}^q(L_x^r)}\lesssim \|\chi(hD_t)u_1\|_{L^2}, \end{align} $$

with

$$ \begin{align*}q\in ]2,\infty],\quad r\in[2,\infty],\quad \frac{1}{q}\leq\alpha_3\bigg(\frac{1}{2}-\frac{1}{r}\bigg),\quad \alpha_3=\frac{3}{4},\quad \beta=3\bigg(\frac{1}{2}-\frac{1}{r}\bigg)-\frac{1}{q}.\end{align*} $$

Remark 3.2. If ${1}/{q}=\alpha _3({1}/{2}-{1}/{r})$ , then $\beta =(3-\alpha _3)({1}/{2}-{1}/{r})$ .

Proof of Theorem 3.1.

We prove only (3.4) since (3.5) follows analogously. We have the frequency-localised dispersive estimates in $\Omega $ in (3.3) for $|t|\geq h$ ,

(3.6) $$ \begin{align} \|\dot{\mathcal U}(t)\chi(hD_t)u_0\|_ {L^\infty}\lesssim h^{-3}\bigg(\frac{h}{t}\bigg)^{\alpha_3}\|\chi(hD_t)u_0\|_{L^1}, \end{align} $$

and the energy estimates,

(3.7) $$ \begin{align} \|\dot{\mathcal U}(t)\chi(hD_t)u_0\|_ {L^2}\lesssim\|\chi(hD_t)u_0\|_{L^2}. \end{align} $$

We apply the Riesz–Thorin interpolation theorem [9] to the operator $\dot {\mathcal {U}}(t)\chi (hD_t)$ for fixed time $t\in \mathbb {R}$ . Interpolating between (3.6) and (3.7) with $\theta =1-{2}/{r}$ yields

(3.8) $$ \begin{align} \|\dot{\mathcal U}(t)\chi(hD_t)u_0\|_ {L^r}\lesssim h^{(-3+\alpha_3)(1-{2}/{r})}t^{-\alpha_3(1-{2}/{r})}\|\chi(hD_t)u_0\|_{L^{r'}}, \end{align} $$

for $2\leq r\leq \infty $ , where $r'$ denotes the exponent conjugate to r (that is, ${1}/{r}+{1}/{r'}=1$ ). Let T be the operator solution defined by

$$ \begin{align*} T: \phi_0\in L^2 \longmapsto T\phi_0=\dot{\mathcal U}(t)\chi(hD_t)\phi_0\in L_t^qL_x^r. \end{align*} $$

Its adjoint is given by

$$ \begin{align*} T^*: \psi\in L_t^{q'}L_x^{r'}\longmapsto T^*\psi=\int \dot{\mathcal U}(t)\chi^*(hD_t)\psi(t) \,dt\in L^ 2. \end{align*} $$

Moreover,

$$ \begin{align*} T^*T: \psi\in L_t^{q'}L_x^{r'}\longmapsto T^*T\psi=\int \dot{\mathcal U}(t-s)\chi^*(hD_t)\chi(hD_t)\psi(s) \,ds\in L_t^qL_x^r. \end{align*} $$

By the $TT^*$ argument in [7], it is sufficient to prove

$$ \begin{align*} \| T^*T\psi\|_{L_t^qL_x^r}&\lesssim h^{-2\beta}\| \psi\|_{L_t^{q'}L_x^{r'}}. \end{align*} $$

We have

(3.9) $$ \begin{align} \| T^*T\psi\|_{L_t^qL_x^r}&=\bigg\|\int \dot{\mathcal U}(t-s)\chi^*(hD_t)\chi(hD_t)\psi(s) \,ds\bigg\|_{L_t^qL_x^r},\nonumber\\ &\lesssim h^{-2(3-\alpha_3)({1}/{2}-{1}/{r})}\bigg\|\int|t-s|^{-2\alpha_3({1}/{2}-{1}/{r})}\|\psi\|_ {L_x^{r'}}\,ds\bigg\|_{L_t^q}. \end{align} $$

When ${1}/{q}< \alpha _3({1}/{2}-{1}/{r})$ , we use Young’s inequality which states that

(3.10) $$ \begin{align} \|K\ast u\|_{L^q}\leq \|K\|_{L^{\tilde r}}\|u\|_ {L^p} \quad\mbox{for } 1\leq p,q\leq\infty, \end{align} $$

where $1+{1}/{q}={1}/{\tilde r}+{1}/{p}$ . We apply (3.10) with $\tilde r=q/2, p=q'$ and ${1}/{q}+{1}/{q'}=1$ to get the estimate

$$ \begin{align*} \bigg\|\int_ h^\infty|t-s|^{-2\alpha_3({1}/{2}-{1}/{r})}\|\psi\|_ {L_x^{r'}}\,ds\bigg\|_{L_t^q} & \leq \|\psi\|_{L_t^{q'}L_x^{r'}}\|t^{-2\alpha_3({1}/{2}-{1}/{r})}\|_{L_{|t|\geq h}^{q/2}} \\ & \leq h^{-2\alpha_3({1}{/2}-{1}/{r})+{2}/{q}}\| \psi\|_{L_t^{q'}L_x^{r'}}. \end{align*} $$

Since ${1}{/q}< \alpha _3({1}/{2}-{1}/{r})$ ,

$$ \begin{align*} \|t^{-2\alpha_3({1}/{2}-{2}/{r})}\|_{L_{|t|\geq h}^{q/2}}=\bigg(\int_h^\infty t^{-2\alpha_3({1}/{2}-{2}/{r})q/2} \,dt\bigg)^{2/q}\simeq h^{-2\alpha_3({1}/{2}-{1}/{r})+{2}/{q}}. \end{align*} $$

Then (3.9) becomes

$$ \begin{align*} \| T^*T\psi\|_{L_t^qL_x^r} &\lesssim h^{-2(3-\alpha_3)({1}/{2}-{1}/{r})}\bigg\|\int|t-s|^{-2\alpha_3({1}/{2}-{1}/{r})}\|\psi\|_ {L_x^{r'}}\,ds\bigg\|_{L_t^q},\\ &\lesssim h^{-2[3({1}/{2}-{1}/{r})-\frac{1}{q}]}\| \psi\|_{L_t^{q'}L_x^{r'}} \lesssim h^{-2\beta}\| \psi\|_{L_t^{q'}L_x^{r'}}. \end{align*} $$

When ${1}/{q}= \alpha _3({1}/{2}-{1}/{r})$ , we instead use the Hardy–Littlewood–Sobolev inequality (see [9, Theorem 4.5.3]) which says that for $K(t)=|t|^{-1/\gamma }$ and $1<\gamma <\infty $ ,

(3.11) $$ \begin{align} \|K\ast u\|_{L^{\tilde r}(\mathbb{R})}\lesssim \|u\|_ {L^{p'}(\mathbb{R})} \quad\mbox{for } \frac{1}{\gamma}=\frac{1}{p}+\frac{1}{\tilde r}. \end{align} $$

We apply (3.11) with $\tilde r=q, p=q$ and ${1}/{\gamma }={2}/{q}=2\alpha _3({1}/{2}-{1}/{r})$ to show that $t^{-2/q} *: L^{q'}\rightarrow L^{q}$ is bounded for $q>2$ . Hence, from (3.9),

$$ \begin{align*} \| T^*T\psi\|_{L_t^qL_x^r}&\lesssim h^{-2(3-\alpha_3)({1}{/2}-{1}/{r})}\| \psi\|_{L_t^{q'}L_x^{r'}} \lesssim h^{-2\beta}\| \psi\|_{L_t^{q'}L_x^{r'}}.\\[-2.5pc] \end{align*} $$

3.2 Homogeneous Strichartz estimates

We can restate Theorem 2.1 as follows.

Theorem 3.3 (Theorem 2.1).

Let $(\Omega ,\Delta )$ be defined as before. Let u be a solution of the wave equation on $\Omega $ :

(3.12) $$ \begin{align} &(\partial_t^2-\Delta)u=0\ \text{in } \Omega,\quad u_{|t=0}=u_0,\quad \partial_t u_{|t=0}=u_1,\quad u_{|x=0}=0. \end{align} $$

Then for all T, there exists $C_T$ such that

$$ \begin{align*} \|u\|_{L^q((0,T);L^r(\Omega))}\leq C_T(\|u_0\|_{\dot{H}^{\beta}(\Omega)}+\|u_1\|_{\dot{H}^{\beta-1}(\Omega)}), \end{align*} $$

with

$$ \begin{align*} \frac{1}{q}\leq\frac{3}{4}\bigg(\frac{1}{2}-\frac{1}{r}\bigg) \quad\text{and}\quad \beta=3 \bigg(\frac{1}{2}-\frac{1}{r}\bigg)-\frac{1}{q}. \end{align*} $$

Proof. Using the square function estimates (3.1),

$$ \begin{align*} \|u\|_{L_t^qL_x^r}\lesssim \bigg(\sum_{j}\|v_j\|_{L_t^qL_x^r}^2\bigg)^{1/2}. \end{align*} $$

Indeed,

$$ \begin{align*} \|u\|_{L^r(\Omega)} & \lesssim\bigg\|\bigg(\sum_{j\geq 0}|v_j|^2\bigg)^{1/2}\bigg\|_{L^r(\Omega)}= \bigg\|\sum_{j\geq 0}|v_j|^2\bigg\|_{L^{r/2}(\Omega)}^{1/2} \\ & \lesssim\bigg\{\sum_{j\geq 0}\|v_j^2\|_{L^{r/2}(\Omega)}\bigg\}^{1/2}=\bigg\{\sum_{j\geq 0}\|v_j\|_{L^{r}(\Omega)}^2\bigg\}^{1/2}. \end{align*} $$

Hence,

$$ \begin{align*} \|u\|_{L_t^qL_x^r}&\lesssim \bigg\|\bigg\{\sum_{j\geq 0}\|v_j\|_{L_x^{r}}^2\bigg\}^{1/2}\bigg\|_{L_t^q}=\bigg\{\bigg\|\sum_{j\geq 0}\|v_j\|_{L_x^{r}}^2\bigg\|_{L_t^{q/2}}\bigg\}^{1/2},\\ &\lesssim \bigg\{\sum_{j\geq 0}\|\|v_j\|_{L_x^r}^2\|_{L_t^{q/2}}\bigg\}^{1/2}=\bigg\{\sum_{j\geq 0}\|v_j\|_{L_t^qL_x^r}^2\bigg\}^{1/2}. \end{align*} $$

The solution u to the wave equation (3.12) with localised initial data in frequency $1/h=2^j$ is given by

$$ \begin{align*} u=\sum_j v_j \quad\text{where}\ v_j=\dot{\mathcal U}(t)\chi(2^{-j}D_t)u_0+ \mathcal U(t)\chi(2^{-j}D_t)u_1. \end{align*} $$

Therefore,

$$ \begin{align*} \begin{aligned} \|u\|_{L_t^qL_x^r}&\lesssim \bigg( \sum_{j}\|\dot{\mathcal U}(t)\chi(2^{-j}D_t)u_0\|_{L_t^qL_x^r}^2+\|\mathcal U(t)\chi(2^{-j}D_t)u_1\|_{L_t^qL_x^r}^2\bigg)^{1/2},\\ &\lesssim \bigg( \sum_j2^{2j\beta}\|\chi(2^{-j}D_t)u_0\|_ {L^2}^2+2^{2j(\beta-1)}\|\chi(2^{-j}D_t)u_1\|_ {L^2}^2\bigg)^{1/2},\\ &\lesssim \bigg(\sum_j2^{2j\beta}\|\chi(2^{-j}D_t)u_0\|_{L^2}^2\bigg)^{1/2}+\bigg(\sum_j2^{2j(\beta-1)}\|\chi(2^{-j}D_t)u_1\|_{L^2}^2\bigg)^{1/2},\\ &\lesssim \|u_0\|_{\dot{H}^\beta(\Omega)}+\|u_1\|_{\dot{H}^{\beta-1}(\Omega)}, \end{aligned} \end{align*} $$

where we used Minkowski’s inequality in the third line.

4 Application

We can use the Strichartz estimates (Theorem 2.1) to obtain the well posedness of the following energy critical nonlinear wave equation in $(\Omega , \Delta )$ :

(4.1) $$ \begin{align} \begin{aligned} &\quad (\partial_t^2-\Delta)u+u^5=0\quad \text{in } \mathbb{R}_t\times\Omega,\\ &u_{|t=0}=u_0,\quad \partial_t u_{|t=0}=u_1,\quad u_{|x=0}=0. \end{aligned} \end{align} $$

The solutions to (4.1) satisfy an energy conservation law:

$$ \begin{align*}E(u(t),\partial_t u(t))=\int_\Omega\bigg(\frac{1}{2}|\nabla u(t,x)|^2+\frac{1}{2}|\partial_t u(t,x)|^2+\frac{1}{6}|u(t,x)|^6\bigg)\,dx=E(u_0, u_1).\end{align*} $$

For initial data $(u_0, u_1)\in H_0^1(\Omega )\times L^2(\Omega )$ , Theorem 2.1 allows the Strichartz triplet $q=5, r=10,\,\beta =1$ and we get

$$ \begin{align*}\|u\|_{L^5((0,T); L^{10}(\Omega))}\leq C_T (\|u_0\|_{H^1(\Omega)}+\|u_1\|_{L^2(\Omega)}).\end{align*} $$

As a consequence, the critical nonlinear wave equation (4.1) is locally well posed in

$$ \begin{align*}X_T=C^0([0,T]; H_0^1(\Omega))\cap L^5((0,T); L^{10}(\Omega))\times C^0([0,T]; L^2(\Omega)).\end{align*} $$

Moreover, with the arguments in [6], we can extend local to global existence for arbitrary (finite energy) data.