Convergence problem of the Kawahara equation on the real line

TL;DR

This paper investigates the convergence behavior of solutions to the Kawahara equation on the real line with rough initial data, establishing bilinear estimates and proving pointwise convergence in certain Sobolev spaces.

Contribution

It introduces new bilinear estimates and demonstrates pointwise convergence of solutions with rough data, extending understanding of the Kawahara equation's behavior.

Findings

Established bilinear estimates using Strichartz and frequency techniques.

Proved pointwise convergence of solutions with initial data in H^s for s ≥ 1/4.

Showed that s > -1/2 is necessary for certain bilinear estimates.

Abstract

In this paper, we consider the convergence problem of the Kawahara equation \begin{eqnarray*} &&u_{t}+\alpha\partial_{x}^{5}u+\beta\partial_{x}^{3}u+\partial_{x}(u^{2})=0 \end{eqnarray*} on the real line with rough data. Firstly, by using Strichartz estimates as well as high-low frequency idea, we establish two crucial bilinear estimates, which are just Lemmas 3.1-3.2 in this paper; we also present the proof of Lemma 3.3 which shows that $s>-\frac{1}{2}$ is necessary for Lemma 3.2. Secondly, by using frequency truncated technique and high-low frequency technique, we show the pointwise convergence of the Kawahara equation with rough data in $H^{s}(\R)(s\geq\frac{1}{4})$; more precisely, we prove \begin{eqnarray*} &&\lim\limits_{t\rightarrow0}u(x,t)=u(x,0), \qquad a.e. x\in\R, \end{eqnarray*} where $u(x,t)$ is the solution to the Kawahara equation with initial data $u(x,0).$ Lastly, we show \begin{eqnarray*} &&\lim\limits_{t\rightarrow0}\sup\limits_{x\in\SR}|u(x,t)-U(t)u_{0}|=0 \end{eqnarray*} with rough data in $H^{s}(\R)(s>-\frac{1}{2})$.