Well-posedness and weak rotation limit for the Ostrovsky equation. (English) Zbl 1181.35253
Summary: 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 \(0\leqslant a\leqslant - 1\) 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 \(\beta \gamma >0\) and \(\beta \gamma <0\). 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 \(\gamma \) 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 \(\gamma \).
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
Keywords:
Ostrovsky equation; KdV equation; well-posedness; Cauchy problem; Fourier restriction norm; low regularity; weak rotation limitReferences:
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