Convergence problem of the Kawahara equation on the real line

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 $H^{s}(\R)(s>-\frac{1}{2})$.