Convergence problem of the Kawahara equation on the real line

In this paper, we consider the convergence problem of the Kawahara equationut+α∂x5u+β∂x3u+∂x(u2)=0 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 Lemma 3.1, Lemma 3.2 in this paper; we also present the proof of Lemma 3.3 which shows that s>−12 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 Hs(R)(s≥14); more precisely, we provelimt→0⁡u(x,t)=u(x,0),a.e.x∈R, where u(x,t) is the solution to the Kawahara equation with initial data u(x,0). Lastly, we showlimt→0⁡supx∈R⁡|u(x,t)−U(t)u0|=0 with rough data in Hs(R)(s>−12).