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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 $$s>1/4$$.

##### MSC:
 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