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

In this paper, we consider the modified Ostrovsky, Stepanyams and Tsimring equation u t + u x x x − η ( H u x + H u x x x ) + u 2 u x = 0 . We prove that the associated initial value problem is locally well-posed in Sobolev spaces H s ( 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 C 3 in H s ( R ) for s − 1 ∕ 2 . Moreover, we prove that for any s > 1 ∕ 2 and T > 0 , its solution converges in C ( [ 0 , T ] ; H s ( R ) ) to that of the mKdV equation if η tends to 0.