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Local well-posedness and a priori bounds for the modified Benjamin-Ono equation. (English) Zbl 1236.35121
Summary: We prove that the complex-valued modified Benjamin-Ono (mBO) equation is analytically locally well posed if the initial data \(\phi\) belongs to \(H^s\) for \(s\geq 1/2\) with \(\|\phi\|_{L^2}\) sufficiently small, without performing a gauge transformation. The key ingredient is that the logarithmic divergence in the high-low frequency interaction can be overcome by a combination of \(X^{s,b}\) structure and smoothing effect structure. We also prove that the real-valued \(H^\infty\) solutions to the mBO equation satisfy a priori local-in-time \(H^s\) bounds in terms of the \(H^s\) size of the initial data for \(s>1/4\).

35Q35 PDEs in connection with fluid mechanics
35Q53 KdV equations (Korteweg-de Vries equations)
35B45 A priori estimates in context of PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence