Well-posedness and weak rotation limit for the Ostrovsky equation

Abstract

We consider the Cauchy problem of the Ostrovsky equation. We first prove the time local well-posedness in the anisotropic Sobolev space H s , a with s > − a / 2 − 3 / 4 and 0 ⩽ a ⩽ − 1 by the Fourier restriction norm method. This result include the time local well-posedness in H s with s > − 3 / 4 for both positive and negative dissipation, namely for both β γ > 0 and β γ < 0 . We next consider the weak rotation limit. We prove that the solution of the Ostrovsky equation converges to the solution of the KdV equation when the rotation parameter γ goes to 0 and the initial data of the KdV equation is in L 2 . To show this result, we prove a bilinear estimate which is uniform with respect to γ.