On a higher dimensional version of the Benjamin--Ono equation

2019/01/15 by Linares, Felipe, Riaño, Oscar G., Rogers, Keith M. +2 · 2 citations
#35B65 #35Q35 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1901.04817

Abstract

We consider a higher dimensional version of the Benjamin--Ono equation, ∂_t u -R₁Δu+u∂_x1 u=0, where R₁ 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 L²-Sobolev spaces H^s(ℝ^d), with s>5/3 if d=2 and s>d/2+1/2 if d≥ 3. We also provide ill-posedness results.

Cited by

• A numerical method for the fractional Zakharov-Kuznetsov equation
• A monotonicity formula for the fractional Laplacian and instability results for the Shrira equation

Related

• The IVP for a higher dimensional version of the Benjamin-Ono equation in\n weighted Sobolev spaces
• Sharp well-posedness results for the generalized Benjamin-Ono equation with high nonlinearity
• Well-posedness for the generalized Benjamin-Ono equations with arbitrary large initial data in the critical space
• Sharp well-posedness results of the Benjamin-Ono equation in H^s(\mathbbT,ℝ) and qualitative properties of its solution
• Global well-posedness and limit behavior for a higher-order Benjamin-Ono equation