On the sharp local well-posedness for the modified Ostrovsky, Stepanyams and Tsimring equation

Abstract

In this paper, we consider the modified Ostrovsky, Stepanyams and Tsimring equation ut+uxxx−η(Hux+Huxxx)+u2ux=0. We prove that the associated initial value problem is locally well-posed in Sobolev spaces Hs(R) for s>−1∕2. We also prove that our result is sharp in the sense that the flow map of this equation fails to be C3 in Hs(R) for s<−1∕2. Moreover, we prove that for any s>1∕2 and T>0, its solution converges in C([0,T];Hs(R)) to that of the mKdV equation if η tends to 0.