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On a higher dimensional version of the Benjamin-Ono equation. (English) Zbl 1435.35300

Summary: We consider a higher dimensional version of the Benjamin-Ono equation, \( \partial_t u -\mathcal{R}_1\Delta u+u\partial_{x_1} u=0\), where \(\mathcal{R}_1\) denotes the Riesz transform with respect to the first coordinate. We first establish sharp space-time estimates for the associated linear equation. These estimates enable us to show that the initial value problem for the nonlinear equation is locally well-posed in \(L^2\)-Sobolev spaces \(H^s(\mathbb{R}^d)\), with \(s>5/3\) if \(d=2\) and \(s>d/2+1/2\) if \(d\ge 3\). We also provide ill-posedness results.

MSC:

35Q35 PDEs in connection with fluid mechanics
35B65 Smoothness and regularity of solutions to PDEs
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
35R25 Ill-posed problems for PDEs
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
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