Lower bounds on the radius of spatial analyticity for the Kawahara equation

TL;DR

This paper establishes that solutions to the Kawahara equation with initially analytic data maintain a spatial analyticity radius that decays no faster than inversely proportional to time, providing lower bounds on analyticity over time.

Contribution

It provides the first lower bounds on the decay rate of the spatial analyticity radius for solutions to the Kawahara equation with analytic initial data.

Findings

The analyticity radius decays at most as 1/|t| over time.

Solutions preserve a minimal analyticity radius inversely proportional to time.

The results apply to solutions with initial data having a fixed radius of analyticity.

Abstract

In this paper we obtain lower bounds on the radius of spatial analyticity of solutions to the Kawahara equation $u_t + uu_x + \alpha u_{xxx} + \beta u_{xxxxx} = 0$, $\beta\neq0$, given initial data which is analytic with a fixed radius. It is shown that the uniform radius of spatial analyticity of solutions at later time $t$ can decay no faster than $1/|t|$ as $|t|\rightarrow\infty$.

Equations

$$\{\begin{array}{l}{u}_t+u{u}_x+\alpha u_{xxx}+\beta u_{xxxxx}=0,\\ u(0,x)=u_0(x),\end{array}$$

$$S_{{\sigma }_0}=\{x+iy:x,y\in \mathbb{R},|y|<{\sigma }_0\}.$$

$$\Vert f{\Vert }_{G^{\sigma ,s}}=\Vert e^{\sigma |D|}⟨D{⟩}^{s}f{\Vert }_{L^2},$$

$$u(t)\in G^{\sigma (t),s}(\mathbb{R})$$

$$G^{\sigma ,s}\subset G^{{\sigma }^{\prime },s^{\prime }}$$

$$\Vert f{\Vert }_{X^{s,b}}=\Vert ⟨\xi {⟩}^s⟨\tau -p(\xi ){⟩}^b\hat{f}(\tau ,\xi ){\Vert }_{L_{\tau ,\xi }^2},$$

$$\hat{f}(\tau ,\xi )={\int }_{{\mathbb{R}}^2}{e}^{-i(t\tau +x\xi )}f(t,x)dtdx.$$

$$\|f\|_{X^{s,b}_{\delta}}=\inf\big{\{}\|g\|_{X^{s,b}}:g=f\,\,\text{on}\,\,(0,\delta)\times\mathbb{R}\big{\}}.$$

$$\Vert f{\Vert }_{X^{\sigma ,s,b}}=\Vert e^{\sigma |{\partial }_x|}f{\Vert }_{X^{s,b}}.$$

$$\underset{t\in \mathbb{R}}{\sup }\Vert f(t){\Vert }_{G^{\sigma ,s}}\le C\Vert f{\Vert }_{X^{\sigma ,s,b}},$$

$$\Vert f{\Vert }_{X_{\delta }^{\sigma ,s,b}}\le C_{b,b^{\prime }}{\delta }^{b^{\prime }-b}\Vert f{\Vert }_{X_{\delta }^{\sigma ,s,b^{\prime }}},$$

$$\Vert {\chi }_{I}f{\Vert }_{X^{\sigma ,s,b}}\le C\Vert f{\Vert }_{X_{\delta }^{\sigma ,s,b}},$$

$$\Vert {\partial }_x(uv){\Vert }_{X^{\sigma ,s,b^{\prime }-1}}\le C_{s,b,b^{\prime }}\Vert u{\Vert }_{X^{\sigma ,s,b}}\Vert v{\Vert }_{X^{\sigma ,s,b}}$$

$$f(u)=\frac{1}{2}\partial_{x}\Big{(}(e^{\sigma|\partial_{x}|}u)^{2}-e^{\sigma|\partial_{x}|}u^{2}\Big{)}.$$

$$\Vert f(u){\Vert }_{X^{0,b-1}}\le C{\sigma }^{\rho }\Vert u{\Vert }_{X^{\sigma ,0,b}}^2$$

$$\widehat{v_1{v}_2}(\tau ,\xi )$$

$$={\int }_{{\mathbb{R}}^2}\widehat{v_1}({\tau }_1,{\xi }_1)\widehat{v_2}(\tau -{\tau }_1,\xi -{\xi }_1)d{\tau }_{1}d{\xi }_1$$

$$={\int }_{{\mathbb{R}}^2}\widehat{v_1}({\tau }_1,{\xi }_1)\widehat{v_2}({\tau }_2,{\xi }_2)d{\tau }_{1}d{\xi }_1$$

$$e^{\sigma|\xi_{1}|}e^{\sigma|\xi_{2}|}-e^{\sigma|\xi_{1}+\xi_{2}|}\leq\big{(}2\sigma\min\{|\xi_{1}|,|\xi_{2}|\}\big{)}^{\rho}e^{\sigma|\xi_{1}|}e^{\sigma|\xi_{2}|}$$

$$\Vert f(u){\Vert }_{X^{0,b-1}}$$

$$\displaystyle\sim\bigg{\|}\frac{\xi}{\langle\tau-p(\xi)\rangle^{1-b}}\int_{\mathbb{R}^{2}}e^{\sigma|\xi_{1}|}\hat{u}(\tau_{1},\xi_{1})e^{\sigma|\xi_{2}|}\hat{u}(\tau_{2},\xi_{2})-e^{\sigma|\xi|}\hat{u}(\tau_{1},\xi_{1})\hat{u}(\tau_{2},\xi_{2})\,d\tau_{1}d\xi_{1}\bigg{\|}_{L^{2}_{\tau,\xi}}$$

$$\displaystyle\sim\bigg{\|}\frac{\xi}{\langle\tau-p(\xi)\rangle^{1-b}}\int_{\mathbb{R}^{2}}\big{(}e^{\sigma|\xi_{1}|}e^{\sigma|\xi_{2}|}-e^{\sigma|\xi|}\big{)}\hat{u}(\tau_{1},\xi_{1})\hat{u}(\tau_{2},\xi_{2})\,d\tau_{1}d\xi_{1}\bigg{\|}_{L^{2}_{\tau,\xi}}$$

$$\displaystyle\lesssim\bigg{\|}\frac{\xi}{\langle\tau-p(\xi)\rangle^{1-b}}\int_{\mathbb{R}^{2}}\big{[}2\sigma\min(|\xi_{1}|,|\xi_{2}|)\big{]}^{\rho}e^{\sigma|\xi_{1}|}e^{\sigma|\xi_{2}|}\hat{u}(\tau_{1},\xi_{1})\hat{u}(\tau_{2},\xi_{2})\,d\tau_{1}d\xi_{1}\bigg{\|}_{L^{2}_{\tau,\xi}},$$

$$\min (|{\xi }_1|,|{\xi }_2|)\le 2\frac{(1+|{\xi }_1|)(1+|{\xi }_2|)}{(1+|{\xi }_1+{\xi }_2|)}=2\frac{⟨{\xi }_1⟩⟨{\xi }_2⟩}{⟨\xi ⟩},$$

$$\Vert f(u){\Vert }_{X^{0,b-1}}$$

$$\displaystyle=\sigma^{\rho}\Big{\|}\partial_{x}\Big{(}e^{\sigma|\partial_{x}|}\langle\partial_{x}\rangle^{\rho}u\cdot e^{\sigma|\partial_{x}|}\langle\partial_{x}\rangle^{\rho}u\Big{)}\Big{\|}_{X^{0,-\rho,b-1}}$$

$$\displaystyle\lesssim\sigma^{\rho}\big{\|}e^{\sigma|\partial_{x}|}\langle\partial_{x}\rangle^{\rho}u\big{\|}^{2}_{X^{0,-\rho,b}}$$

$$={\sigma }^{\rho }\Vert u{\Vert }_{X^{\sigma ,0,b}}^2$$

$$\delta =c_0(1+\Vert u_0{\Vert }_{G^{\sigma .s}}{)}^{-a}$$

$$\Vert u{\Vert }_{X_{\delta }^{\sigma ,s,b}}\le C\Vert u_0{\Vert }_{G^{\sigma ,s}}$$

$$\{\begin{array}{l}{u}_t+\alpha u_{xxx}+\beta u_{xxxxx}=F(t,x),\\ u(0,x)=f(x).\end{array}$$

$$u(t,x)=e^{itp(-i{\partial }_x)}f(x)+{\int }_0^t{e}^{i(t-t^{\prime })p(-i{\partial }_x)}F(t^{\prime },\cdot )d{t}^{\prime },$$

$$e^{itp(-i{\partial }_x)}f(x)=\frac{1}{(2\pi )}{\int }_{\mathbb{R}}{e}^{ix\xi }{e}^{itp(\xi )}\hat{f}(\xi )d\xi .$$

$$\Vert e^{itp(-i{\partial }_x)}f{\Vert }_{X_{\delta }^{\sigma ,s,b}}\le C_b\Vert f{\Vert }_{G^{\sigma ,s}}$$

Full text

This research was supported by NRF-2019R1F1A1061316.

1. Introduction

Consider the Cauchy problem for the Kawahara equation

[TABLE]

where $u:\mathbb{R}^{1+1}\rightarrow\mathbb{R}$ and $\alpha$, $\beta$ are real constants with $\beta\neq 0$. This fifth-order KdV type equation has been derived to model gravity-capillary waves on a shallow layer and magneto-sound propagation in plasmas ([13, 15]).

The well-posedness of the above Cauchy problem with initial data in Sobolev spaces $H^{s}$ has been studied by several authors (see e.g. [9, 25, 7, 6]). In particular, it was shown in [9] that (1.1) has a global solution when $s=0$. This was improved by Wang, Cui and Deng [25] to $s>-1/2$ and then by Chen, Li, Miao and Wu [7] to $s>-7/4$. More recently, Chen and Guo [6] obtained the global well-posedness for $s\geq-7/4$.

In this paper, we are concerned with the persistence of spatial analyticity for the solutions of (1.1), given initial data in a class of analyticity functions. While the well-posedness theory in Sobolev spaces is well developed, nothing is known about the spatial analyticity for the Kawahara equation. From now on, we focus on the situation where we consider a real-analytic initial data with uniform radius of analyticity $\sigma_{0}>0$, so there is a holomorphic extension to a complex strip

[TABLE]

Now, it is natural to ask whether this property may be continued analytically to a complex strip $S_{\sigma(t)}$ for all later times $t$, but with a possibly smaller and shrinking radius of analyticity $\sigma(t)>0$.

This type of problem was first introduced by Kato and Masuda [11], and has recently received a lot of attention for the Korteweg-de Vries (KdV) equation. It was shown by Bona and Grujić [2] that the radius $\sigma(t)$ for the KdV equation can decay no faster than $e^{-t^{2}}$ as $|t|\rightarrow\infty$. This was improved greatly by Bona, Grujić and Kalisch [3] to a polynomial decay rate of $|t|^{-12}$. Later, Selberg and da Silva [18] obtained a further refinement, $\sigma(t)\geq c|t|^{-4/3+\varepsilon}$, where the $\varepsilon$ exponent was also removed by Tesfahun [24]. The rate was again improved by Huang and Wang [10] up to $|t|^{-1/4}$. See also a recent related result for the quartic generalised KdV equation by Selberg and Tesfahun [20].

In spite of these many works for KdV equations, there have been no results on this issue for the Kawahara equation (1.1) which is a fifth-order KdV type equation. Motivated by this, we aim here to obtain the spatial analyticity for the Kawahara equation.

The Gevrey space, denoted $G^{\sigma,s}(\mathbb{R})$, $\sigma\geq 0$, $s\in\mathbb{R}$, is a suitable function space to study analyticity of solution. In our case, it will be used with the norm

[TABLE]

where $\langle D\rangle=1+|D|$ and $D$ denotes the derivative. According to the Paley-Wiener theorem111The proof given for $s=0$ in [12] applies also for $s\in\mathbb{R}$ with some obvious modifications. (see e.g. [12], p. 209), a function $f$ belongs to $G^{\sigma,s}$ with $\sigma>0$ if and only if it is the restriction to the real line of a function $F$ which is holomorphic in the strip $S_{\sigma}=\{x+iy:x,y\in\mathbb{R},\,|y|<\sigma\}$ and satisfies $\sup_{|y|<\sigma}\|F(x+iy)\|_{H_{x}^{s}}<\infty$. Therefore every function in $G^{\sigma,s}$ with $\sigma>0$ has an analytic extension to the strip $S_{\sigma}$. Based on this property of the Gevrey space, which is the key to studying spatial analyticity of solution, our result below gives a lower bound $|t|^{-1}$ on the radius of analyticity $\sigma(t)$ of the solution to (1.1) as the time $t$ tends to infinity.

Theorem 1.1**.**

Let $u$ be the global $C^{\infty}$ solution of (1.1) with $u_{0}\in G^{\sigma_{0},s}(\mathbb{R})$ for some $\sigma_{0}>0$ and $s\in\mathbb{R}$. Then, for all $t\in\mathbb{R}$

[TABLE]

with $\sigma(t)\geq c|t|^{-1}$ as $|t|\rightarrow\infty$. Here, $c>0$ is a constant depending on $\|u_{0}\|_{G^{\sigma_{0},s}(\mathbb{R})}$.

It should be noted that the existence of the global $C^{\infty}$ solution in the theorem is always guaranteed. Indeed, observe that ${G^{0,s}}$ coincides with the Sobolev space $H^{s}$ and the embeddings

[TABLE]

hold for all $0\leq\sigma^{\prime}<\sigma$ and $s,s^{\prime}\in\mathbb{R}$. As a consequence of this embedding with $\sigma^{\prime}=0$ and the existing global well-posedness theory in $H^{s^{\prime}}(=G^{0,s^{\prime}})$ for $s^{\prime}\geq-7/4$, the Cauchy problem (1.1) has a unique smooth solution for all time, given initial data $u_{0}\in G^{\sigma_{0},s}$ for any $\sigma_{0}>0$ and $s\in\mathbb{R}$.

We close this section with further references on the spatial analyticity for other dispersive equations such as Schrödinger equations [4, 23, 1], Klein-Gordon equations [16] and Dirac-Klein-Gordon equations [19, 17].

The outline of this paper is as follows: In Section 2 we introduce some function spaces such as Bourgain and Gevrey-Bourgain spaces, and their basic properties which will be used in later sections. In Section 3 we present a bilinear estimate (Lemma 3.1) in Gevrey-Bourgain spaces. By making use of a contraction argument involving this estimate, we prove that in a short time interval $0\leq t\leq\delta$ with $\delta>0$ depending on the norm of the initial data, the radius of analyticity remains strictly positive. Next, we prove an approximate conservation law, although the conservation of $G^{\sigma_{0},s}$-norm of the solution does not hold exactly, in order to control the growth of the solution in the time interval $[0,\delta]$, measured in the data norm $G^{\sigma_{0},s}$. Section 4 is concerned with the proofs of such a local result and the almost conservation law. In Section 5, we finish the proof of Theorem 1.1 by iterating the local result based on the conservation law. The final section, Section 6, is devoted to the proof of Lemma 3.1.

Throughout this paper, the letter $C$ stands for a positive constant which may be different at each occurrence. We denote $A\lesssim B$ and $A\sim B$ to mean $A\leq CB$ and $B\lesssim A\lesssim B$, respectively. We also use $A\ll B$ to mean $A\leq cB$ for some small constant $c>0$.

2. Function spaces

In this section we introduce some function spaces and their basic properties which will be used in later sections for the proof of Theorem 1.1.

For $s,b\in\mathbb{R}$, we use $X^{s,b}=X^{s,b}(\mathbb{R}^{2})$ to denote the Bourgain space defined by the norm

[TABLE]

where $p(\xi)=\alpha\xi^{3}-\beta\xi^{5}$, $\langle\cdot\rangle=1+|\cdot|$ and $\widehat{f}$ denotes the space-time Fourier transform given by

[TABLE]

The restriction of the Bourgain space, denoted $X^{s,b}_{\delta}$, to a time slab $(0,\delta)\times\mathbb{R}$ is a Banach space when equipped with the norm

[TABLE]

We also introduce the Gevrey-Bourgain space $X^{\sigma,s,b}=X^{\sigma,s,b}(\mathbb{R}^{2})$ defined by the norm

[TABLE]

Its restriction $X^{\sigma,s,b}_{\delta}$ to a time slab $(0,\delta)\times\mathbb{R}$ is defined in a similar way as above, and when $\sigma=0$ it coincides with the Bourgain space $X^{s,b}$.

The Gevrey-modification of the Bourgain spaces was used already by Bourgain [5] to study persistence of analyticity of solutions of the Kadomtsev-Petviashvili equation. He proved that the radius of analyticity remains positive as long as the solution exists. His argument is quite general and applies also to the KdV and Kawahara equations, but it does not give any lower bound on the radius $\sigma(t)$ as $|t|\rightarrow\infty$.

We now list some basic properties of those spaces. When $\sigma=0$, the proofs of the first two lemmas below can be found in Section 2.6 of [22], and the third lemma follows by the argument used for Lemma 3.1 of [8]. But, by the substitution $f\rightarrow e^{\sigma|\partial_{x}|}f$, the properties of $X^{s,b}$ and its restrictions carry over to $X^{\sigma,s,b}$.

Lemma 2.1**.**

Let $\sigma\geq 0$, $s\in\mathbb{R}$ and $b>1/2$. Then, $X^{\sigma,s,b}\subset C(\mathbb{R},G^{\sigma,s})$ and

[TABLE]

where $C>0$ is a constant depending only on $b$.

Lemma 2.2**.**

Let $\sigma\geq 0$, $s\in\mathbb{R}$, $-1/2<b<b^{\prime}<1/2$ and $\delta>0$. Then

[TABLE]

where $C_{b,b^{\prime}}>0$ is a constant depending only on $b$ and $b^{\prime}$.

Lemma 2.3**.**

Let $\sigma\geq 0$, $s\in\mathbb{R}$, $-1/2<b<1/2$ and $\delta>0$. Then, for any time interval $I\subset[0,\delta]$,

[TABLE]

where $\chi_{I}(t)$ is the characteristic function of $I$, and $C>0$ is a constant depending only on $b$.

3. Bilinear estimates in Gevrey-Bourgain spaces

In this section we present a bilinear estimate in Gevrey-Bourgain spaces, Lemma 3.1, which plays a key role in obtaining the local well-posedness and almost conservation law in the next section. With the aid of it, we shall also deduce an estimate, Lemma 3.2, which is another useful tool particularly in obtaining the almost conservation law.

Lemma 3.1**.**

For all $\sigma\geq 0$ and $s>-7/4$, there exist $1/2<b<1$ and $\varepsilon>0$ such that

[TABLE]

for any $b^{\prime}$ satisfying $b\leq b^{\prime}<b+\varepsilon$. Here, $C_{s,b,b^{\prime}}>0$ is a constant depending only on $s$, $b$ and $b^{\prime}$.

It is worth comparing this lemma with the analogous result by Chen, Li, Miao and Wu [7] (cf. Proposition 2.2), where the proof for (3.1) is given only in the case $b^{\prime}=b$ to obtain local well-posedness in Sobolev spaces. On the contrary, it is crucial for the present issue to have $b^{\prime}$ that can range over a small interval for which (3.1) holds. Similarly as in [7], we apply Tao’s $[k;Z]$-multiplier norm method [21] to our case where $b\leq b^{\prime}<b+\varepsilon$.

We shall postpone the detailed proof of Lemma 3.1 until the last section, Section 6. Instead here we derive the following lemma from Lemma 3.1, which, along with the function $f$ defined here, plays a crucial role in obtaining the almost conservation law in the next section.

Lemma 3.2**.**

Let

[TABLE]

Given $0\leq\rho\leq 1$, there exist $1/2<b<1$ and $C>0$ such that

[TABLE]

for all $\sigma>0$ and $v\in X^{\sigma,0,b}$.

Proof.

Notice first that

[TABLE]

with $\tau_{2}=\tau-\tau_{1}$ and $\xi_{2}=\xi-\xi_{1}$. With this as well as the estimate222This estimate can be found in Lemma 12 of [18].

[TABLE]

where $\sigma>0$, $0\leq\rho\leq 1$ and $\xi_{1},\xi_{2}\in\mathbb{R}$, one can see that

[TABLE]

where $\tau_{2}=\tau-\tau_{1}$ and $\xi_{2}=\xi-\xi_{1}$. Here, from the triangle inequality,

[TABLE]

and therefore

[TABLE]

as desired. Here we used Lemma 3.1 with $\sigma=0$, $s=-\rho$ and $b^{\prime}=b$ for the second inequality. ∎

4. Local well-posedness and almost conservation law

In this section we first establish the local well-posedness and then the almost conservation law, by making use of the bilinear estimate in the previous section. They lie at the core of the proof of Theorem 1.1 in the next section.

4.1. Local well-posedness

Based on Picard’s iteration in the $X_{\delta}^{\sigma,s,b}$-space and Lemma 2.1, we establish the following local well-posedness in $G^{\sigma,s}$, with a lifespan $\delta>0$. Equally the radius of analyticity remains strictly positive in a short time interval $0\leq t\leq\delta$, where $\delta>0$ depends on the norm of the initial data.

Theorem 4.1**.**

Let $\sigma>0$ and $s>-7/4$. Then, for any $u_{0}\in G^{\sigma,s}$, there exist $\delta>0$ and a unique solution u of the Cauchy problem (1.1) on the time interval $[0,\delta]$ such that $u\in C([0,\delta],G^{\sigma,s})$ and the solution depends continuously on the data $u_{0}$. Here we have

[TABLE]

for some constants $c_{0}>0$ and $a>2$ depending only on $s$. Furthermore, if $1/2<b<1$, the solution $u$ satisfies

[TABLE]

with a constant $C>0$ depending only on $b$.

Proof.

Fix $\sigma>0$, $s>-7/4$ and $u_{0}\in G^{\sigma,s}$. By Lemma 2.1 we shall employ an iteration argument in the space $X^{\sigma,s,b}_{\delta}$ instead of $G^{\sigma,s}$.

Consider first the Cauchy problem for the linearised Kawahara equation

[TABLE]

By Duhamel’s principle, the solution can be then written as

[TABLE]

where the Fourier multiplier $e^{itp(-i\partial_{x})}$ with symbol $e^{itp(\xi)}$ is given by

[TABLE]

(Recall from Section 2 that $p(\xi)=\alpha\xi^{3}-\beta\xi^{5}$.) Then the following $X_{\delta}^{\sigma,s,b}$-energy estimate follows directly from Proposition 2.1 in [7] (see also [14]).

Lemma 4.2**.**

Let $\sigma\geq 0$, $s\in\mathbb{R}$, $1/2<b\leq 1$ and $0<\delta\leq 1$. Then we have

[TABLE]

and

[TABLE]

with a constant $C_{b}>0$ depending only on $b$.

Now, let $\left\{u^{(n)}\right\}^{\infty}_{n=0}$ be the sequence defined by

[TABLE]

and for $n\in\mathbb{Z}^{+}$

[TABLE]

Applying (4.3), we first write

[TABLE]

and

[TABLE]

By Lemma 4.2 we have

[TABLE]

and Lemmas 4.2, 2.2 and 3.1 combined imply

[TABLE]

with $1/2<b<b^{\prime}<1$. By induction together with (4.4) and (4.1), it follows that for all $n\geq 0$

[TABLE]

if we choose $\delta$ sufficiently small so that

[TABLE]

Using Lemmas 4.2, 2.2 and 3.1 together with (4.6) and (4.7) in that order, we therefore get

[TABLE]

which guarantees the convergence of the sequence $\left\{u^{(n)}\right\}^{\infty}_{n=0}$ to a solution $u$ with the bound (4.6). Furthermore, (4.1) follows easily from (4.7) and $0<b^{\prime}-b<1/2$.

Now assume that $u$ and $v$ are solutions to the Cauchy problem $\eqref{Kawahara}$ for initial data $u_{0}$ and $v_{0}$, respectively. Then similarly as above, again with the same choice of $\delta$ and for any $\delta^{\prime}$ such that $0<\delta^{\prime}<\delta$, we have

[TABLE]

provided $\|u_{0}-v_{0}\|_{G^{\sigma,s}}$ is sufficiently small, which proves the continuous dependence of the solution on the initial data.

Finally, it remains to show the uniqueness of solutions. Assume $u,v\in C_{t}G^{\sigma,s}$ are solutions to $\eqref{Kawahara}$ for the same initial data $u_{0}$ and let $w=u-v$. Then $w$ satisfies $w_{t}+\alpha w_{xxx}+\beta w_{xxxxx}+wu_{x}+vw_{x}=0$. Multiplying both sides by $w$ and integrating in space yields

[TABLE]

Using $2wvw_{x}=(vw^{2})_{x}-v_{x}w^{2}$ and integrating by parts, we then have

[TABLE]

We may here assume that $w$ and its all spatial derivatives decay to zero as $|x|\rightarrow\infty$.333This property can be shown by approximation using the monotone convergence theorem and the Riemann-Lebesgue lemma whenever $u\in X_{\delta}^{\sigma,1,b}$. See the argument in [18], p. 1018. It follows then that

[TABLE]

By Hölder’s inequality, this implies

[TABLE]

Here we used the fact that

[TABLE]

for all $2\leq q\leq\infty$ and $\sigma>0$. By Grönwall’s inequality, we now conclude that $w=0$. ∎

4.2. Almost conservation law

We have established the existence of local solutions; we would like to apply the local result repeatedly to cover time intervals of arbitrary length. This, of course, requires some sort of control on the growth of the norm on which the local existence time depends. The following approximate conservation will allow us (see Section 5) to repeat the local result on successive short-time intervals to reach any target time $T>0$, by adjusting the strip width parameter $\sigma$ according to the size of $T$.

Theorem 4.3**.**

Let $0\leq\rho\leq 1$, $\frac{1}{2}<b<1$ and $\delta$ be as in Theorem 4.1. Then there exists $C>0$ such that for any $\sigma>0$ and any solution $u\in X^{\sigma,0,b}_{\delta}$ to the Cauchy problem (1.1) on the time interval $[0,\delta]$, we have the estimate

[TABLE]

Proof.

Let $0\leq\delta^{\prime}\leq\delta$. Setting $v(t,x)=e^{\sigma|\partial_{x}|}u(t,x)$ and applying $e^{\sigma|\partial_{x}|}$ to (1.1), we obtain

[TABLE]

where $f(u)$ is as in (3.2):

[TABLE]

Multiplying both sides by $v$ and integrating in space yield

[TABLE]

As before, we may here assume that $v$ and its all spatial derivatives decay to zero as $|x|\rightarrow\infty$. Using this fact and integration by parts, we have

[TABLE]

and furthermore, the second, third and fourth terms on the left side vanish. Subsequently integrating in time over the interval $[0,\delta^{\prime}]$, we obtain

[TABLE]

Now by Hölder’s inequailty, Lemma 2.3 and Lemma 3.2,

[TABLE]

Since $1-b<b$, we therefore get

[TABLE]

as desired. ∎

5. Proof of Theorem 1.1

By invariance of the Kawahara equation under the reflection $(t.x)\rightarrow(-t,-x)$, we may restrict to positive times. By the embedding (1.2), the general case $s\in\mathbb{R}$ will reduce to $s=0$ as shown in the end of this section.

5.1. The case $s=0$

Combining (4.2) and (4.8), we first note that

[TABLE]

Let $u_{0}=u(0)\in G^{\sigma_{0},0}$ for some $\sigma_{0}>0$ and $\delta$ be as in Theorem 4.1. For arbitrarily large $T$, we want to show that the soution $u$ to (1.1) satisfies

[TABLE]

where

[TABLE]

with a constant $c>0$ depending on $\|u_{0}\|_{G^{\sigma_{0},0}}$ and $\sigma_{0}$.

Now fix $T$ arbitrarily large. It suffices to show

[TABLE]

for $\sigma$ satisfying (5.2), which in turn implies $u(t)\in G^{\sigma(t),0}$ as desired.

To prove (5.3), we first choose $n\in\mathbb{Z}^{+}$ so that $n\delta\leq T\leq(n+1)\delta$. Using induction we shall show for any $k\in\left\{1,2,\cdots,n+1\right\}$ that

[TABLE]

and

[TABLE]

provided $\sigma$ satisfies

[TABLE]

Indeed, for $k=1$, we have from (5.1) that

[TABLE]

where we used the fact that $\|u_{0}\|_{G^{\sigma,0}}\leq\|u_{0}\|_{G^{\sigma_{0},0}}$ and $C\sigma^{\rho}\|u_{0}\|_{G^{\sigma_{0},0}}\leq 1$ which are a direct consequence of (5.6). Now assume (5.4) and (5.5) hold for some $k\in\left\{1,2,\cdots,n\right\}$. Applying (5.1), (5.5) and (5.4), we then have

[TABLE]

Combining this with the induction hypothesis (5.4) for $k$, we get

[TABLE]

which proves (5.4) for $k+1$. Since $k+1\leq n+1\leq T/\delta+1\leq 2T/\delta$, from (5.6) we also get

[TABLE]

which, along with (5.7), proves (5.5) for $k+1$.

Finally, the condition (5.6) is satisfied for

[TABLE]

Particularly when $\rho=1$, the constant $c$ in (5.2) may be given as

[TABLE]

which depends only on $\|u_{0}\|_{G^{\sigma_{0},0}}$.

5.2. The general case $s\in\mathbb{R}$.

Recall that (1.2) states

[TABLE]

For any $s\in\mathbb{R}$ we use this embedding to get

[TABLE]

From the local well-posedness result, there is a $\delta=\delta(\|u_{0}\|_{G^{\sigma_{0}/2,0}})$ such that

[TABLE]

Similarly as in the case $s=0$, for $T$ fixed greater than $\delta$, we have $u(t)\in G^{\sigma^{\prime},0}$ for $t\in[0,T]$ and $\sigma^{\prime}\geq c/T$ with $c>0$ depending on $\|u_{0}\|_{G^{\sigma_{0}/2,0}}$ and $\sigma_{0}$. Applying the embedding again, we conclude

[TABLE]

where $\sigma=\sigma^{\prime}/2$.

6. Proof of Lemma 3.1

This last section is devoted to the proof of Lemma 3.1.

6.1. Preliminaries

Before we begin the proof, we shall introduce some notations in Tao’s $[k;Z]$-multiplier norm method [21] with $k=3$ and $Z=\mathbb{R}^{2}$. Let $\Gamma_{3}(\mathbb{R}^{2})$ denote the hyperplane

[TABLE]

where each $\rho_{i}=(\tau_{i},\xi_{i})$ is an ordered pair of real numbers $\tau_{i},\xi_{i}$. Note here that $\tau_{1}+\tau_{2}+\tau_{3}=\xi_{1}+\xi_{2}+\xi_{3}=0$. We will also denote $\rho$, $\xi$ and $\tau$ the triplets $(\rho_{1},\rho_{2},\rho_{3})$, $(\tau_{1},\tau_{2},\tau_{3})$ and $(\xi_{1},\xi_{2},\xi_{3})$, respectively. We endow $\Gamma_{3}(\mathbb{R}^{2})$ with the obvious measure

[TABLE]

where $d\rho_{i}=d\tau_{i}d\xi_{i}$ for $i=1,2$.

For any function $m:\Gamma_{3}(\mathbb{R}^{2})\rightarrow\mathbb{C}$, we define the $[3,\mathbb{R}^{2}]$-multiplier norm $\|m\|_{[3,\mathbb{R}^{2}]}$ to be the best constant $c$ such that the inequality

[TABLE]

holds for all test functions $f_{j}$ on $\mathbb{R}^{2}$.

Capitalised variables such as $N_{j}$, $L_{j}$, $H$ are presumed to be dyadic, i.e., these variables range over numbers of the form $2^{k}$ for $k\in\mathbb{Z}$. Let $N_{1}$, $N_{2}$, $N_{3}>0$. It will be convenient to define the quantities $N_{max}\geq N_{med}\geq N_{min}$ to be the maximum, median and minimum of $N_{1}$, $N_{2}$, $N_{3}$, respectively. Similarly we define $L_{max}\geq L_{med}\geq L_{min}$ whenever $L_{1}$, $L_{2}$, $L_{3}>0$. We also adopt the following summation conventions. Any summation of the form $L_{max}\sim\cdots$ is a sum over the three dyadic variables $L_{1}$, $L_{2}$, $L_{3}\gtrsim 1$, thus for instance

[TABLE]

Similarly, any summation of the form $N_{max}\sim\cdots$ is a sum over the three dyadic variables $N_{1}$, $N_{2}$, $N_{3}>0$, thus for instance

[TABLE]

If $\tau_{j}$ and $\xi_{j}$ are given for $j=1,2,3$, we define

[TABLE]

where $p(\xi_{j})=\alpha\xi_{j}^{3}-\beta\xi_{j}^{5}$ is as in Section 2. Here we will let

[TABLE]

By dyadic decomposition of $\xi_{j}$, $\lambda_{j}$ and $|h(\xi)|$, one is led to consider

[TABLE]

where $X_{N_{1},N_{2},N_{3};H;L_{1},L_{2},L_{3}}$ is the multiplier

[TABLE]

Since the following identities

[TABLE]

and

[TABLE]

hold, from the support of the multiplier, one can see that $X_{N_{1},N_{2},N_{3};H;L_{1},L_{2},L_{3}}$ vanishes unless both

[TABLE]

are satisfied. We then recall the following fundamental estimates on dyadic blocks for the Kawahara equation.

Lemma 6.1** ([7]).**

Let $H,N_{1},N_{2},N_{3},L_{1},L_{2},L_{3}>0$ obey (6.2), (6.3) and $N_{max}\sim N_{med}\gtrsim 1$. Then

[TABLE]

and

• •

If $N_{max}\sim N_{min}$ and $L_{max}\sim H$,

[TABLE]

• •

If $N_{\sigma(2)}\sim N_{\sigma(3)}\gg N_{\sigma(1)}$ and $H\sim L_{\sigma(1)}\gtrsim L_{\sigma(2)},\ L_{\sigma(3)}$,

[TABLE]

where $\sigma$ is a permutation in $\{1,2,3\}$.

• •

In all other cases,

[TABLE]

6.2. Proof

Now we turn to the proof of Lemma 3.1. By the definition of $X^{s,b}$-norms and the dual characterisation of $L^{2}$ space, we may show that

[TABLE]

To show this, we first decompose dyadically $|\xi_{j}|,|\lambda_{j}|,|h(\xi)|$ as $|\xi_{j}|\sim N_{j}$, $|\lambda_{j}|\sim L_{j}$, $|h(\xi)|\sim H$. By some properties of the $[k;Z]$-multiplier norm, one may restrict the multiplier in (6.9) to the region $L_{j}\gtrsim 1$ $(j=1,2,3)$ and $\max\{N_{1},N_{2},N_{3}\}\gtrsim 1$. Consequently, it suffices to show that

[TABLE]

By Lemma 3.11 (Schur’s test) in [21], this estimate is reduced to showing that

[TABLE]

and

[TABLE]

for all $N\gtrsim 1$.

6.2.1. Proof of (6.2)

Since $N_{max}\sim N_{med}\sim N\gtrsim 1$, by (6.5) and (6.8), it suffices to show

[TABLE]

We only need to consider two cases: $N_{1}\sim N_{2}\sim N$, $N_{3}=N_{min}$ and $N_{1}\sim N_{3}\sim N$, $N_{2}=N_{min}$. (The other case $N_{2}\sim N_{3}\sim N$, $N_{1}=N_{min}$ then follows by symmetry.)

In the former case, the estimate (6.12) can be further reduced to

[TABLE]

since $1-b^{\prime}<1/2<b$. Then performing the $L$ summations, we reduce to

[TABLE]

which holds if $4(1+b-b^{\prime})+2s>0$. So we require $0\leq b^{\prime}-b<1+s/2$, which is possible if $s>-2$. Therefore, (6.12) follows.

In the latter case, the estimate (6.12) can be reduced to

[TABLE]

Before performing the $L$ summations, we need to divide cases into two parts, $N_{min}\leq 1$ and $N_{min}\geq 1$, as follows:

[TABLE]

This holds clearly if $s>-7/2$, and therefore (6.12) holds for $s>-7/2$.

6.2.2. Proof of (6.6)

The conditions $N_{max}\sim N_{med}\sim N\gtrsim 1$ and $L_{max}\sim H$ in (6.2) are divided into three cases corresponding to each of (6.6), (6.7) and (6.8). By (6.5) we also see that $L_{max}\sim N^{4}_{max}N_{min}$, which is used repeatedly below.

The case for (6.6)

Now we shall consider the first case, which reduces to

[TABLE]

Performing the $L$ summations, we reduce to

[TABLE]

which holds if $1+s+5(1-b^{\prime})\geq 0$. So we require $1/2<b^{\prime}\leq(6+s)/5$, which is possible if $s>-7/2$. Therefore, (6.13) follows.

The case for (6.7)

Next we consider the second case which is divided into $N_{1}\sim N_{2}\gg N_{3},H\sim L_{3}\gtrsim L_{1},L_{2}$ and $N_{2}\sim N_{3}\gg N_{1},H\sim L_{1}\gtrsim L_{2},L_{3}$. (The other case $N_{1}\sim N_{3}\gg N_{2},H\sim L_{2}\gtrsim L_{1},L_{3}$ then follows by symmetry.) In the former case the estimate (6.2) reduces to

[TABLE]

We then decompose the left-hand side of (6.14) into two parts $N_{3}\leq 1$ and $N_{3}>1$:

[TABLE]

We first consider the part $I_{1}$. When $N^{4}N_{3}\geq\frac{N}{N_{3}}L_{med}$, which is equivalent to $N_{3}\geq(L_{med}/N^{3})^{1/2}$, we see that

[TABLE]

Performing the $N_{3}$ summation, we have the desired bound

[TABLE]

if $2s+4(1-b^{\prime})+3/2>0$. So we require $1/2<b\leq b^{\prime}<(4s+11)/8$, which is possible if $s>-7/4$. On the other hand, when $N_{3}<(L_{med}/N^{3})^{1/2}$,

[TABLE]

Performing the $N_{3}$ summation, we have

[TABLE]

if $2s+4(1-b^{\prime})+\frac{3}{2}(\frac{1}{2}+b^{\prime})>0$ when $(L_{med}/N^{3})^{\frac{1}{2}(\frac{1}{2}+b^{\prime})}\leq 1$, and if $2s+4+3b-4b^{\prime}>0$ when $(L_{med}/N^{3})^{\frac{1}{2}(\frac{1}{2}+b^{\prime})}\geq 1$. So we require $1/2<b\leq b^{\prime}<(8s+19)/10$ and $1/2<b^{\prime}+3(b^{\prime}-b)<4+2s$, but this is possible if $s>-7/4$. Now we consider the part $I_{2}$ that has the following trivial bound

[TABLE]

Performing the $N_{3}$ summation, we conclude that

[TABLE]

if $2s+4(1-b^{\prime})+3/2>0$ when $s-1/2+b^{\prime}<0$, and if $2s+4(1-b^{\prime})+3/2>s-1/2+b^{\prime}$ when $s-1/2+b^{\prime}\geq 0$. So we require $1/2<b\leq b^{\prime}<\min\{(4s+11)/8,(s+6)/5\}$. This is possible if $s>-7/4$. Consequently, we get the desired estimate (6.14) if $s>-7/4$.

Next we deal with the remaining case where $N_{2}\sim N_{3}\gg N_{1},H\sim L_{1}\gtrsim L_{2},L_{3}$. In this case, applying (6.7) to (6.2), we may show

[TABLE]

We first decompose the left-hand side of this inequality into two parts $N_{1}\leq 1$ and $N_{1}>1$:

[TABLE]

Here we used the fact that $H\sim N^{4}N_{min}$ and $\min\{a,b\}\leq\sqrt{ab}$. Note first that the $L$ summations in $J_{1}$ vanish unless $N^{4}N_{1}\gtrsim 1$. Using this, we get

[TABLE]

since $b>1/2$ and $N\gtrsim 1$. Performing the $L$ summations in $J_{2}$, we see that

[TABLE]

if we take $1/2<b\leq b^{\prime}<3/4$. (Otherwise, the sum may diverge.) When $-s-b>0$, we conclude

[TABLE]

if $4b+s+b-1/4>0$. So we require $\max\{1/2,(1/4-s)/5\}<b\leq b^{\prime}<3/4$ and $b<-s$. This is possible if $-7/2<s<-1/2$. Furthermore, we can have the range $1/2<b\leq b^{\prime}<3/4$ if $-9/4\leq s<-1/2$. When $-s-b\leq 0$,

[TABLE]

which holds when $1/2<b\leq b^{\prime}<3/4$. Consequently, we get the desired estimate.

The case for (6.8)

Lastly, using (6.8), the desired estimate (6.2) reduces to

[TABLE]

To show this, we need to divide the case into $N_{1}\sim N_{2}\sim N,$ $N_{3}=N_{min}$ and $N_{1}\sim N_{3}\sim N,$ $N_{2}=N_{min}$. (The other case $N_{2}\sim N_{3}\sim N,$ $N_{1}=N_{min}$ then follows by symmetry.) In the former case, the above estimate further reduces to

[TABLE]

Then performing the $L$ summations, we reduce to

[TABLE]

which holds if $2+2s+4(1-b^{\prime})>0$. So we require $1/2<b\leq b^{\prime}<(s+3)/2$. This is possible if $s>-2$. In the latter case, we reduce to

[TABLE]

Using $\min\{a,b\}\leq\sqrt{ab}$ and then performing the $L$ summations, we further reduce to

[TABLE]

We then decompose the left-hand side of (6.15) into two parts $N_{2}\leq 1$ and $N_{2}>1$:

[TABLE]

It is easy to see that $K_{1}\lesssim N^{-4(1-b^{\prime})}\lesssim 1$ if $b^{\prime}<3/4$. On the other hand,

[TABLE]

if $b^{\prime}\leq 1$ when $b^{\prime}-s-3/4\geq 0$, and if $4(1-b^{\prime})-(b^{\prime}-s-3/4)>0$ when $b^{\prime}-s-3/4<0$. So we get the desired bound for all $s$ with $1/2<b\leq b^{\prime}<\min\{(4s+19)/20,s+3/4\}$. This completes the proof.