On a Higher Dimensional Version of the Benjamin--Ono Equation

Abstract

We consider a higher dimensional version of the Benjamin-Ono equation, ∂tu−R 1 ∆u+u∂x 1 u = 0, where 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 (R d ), with s > 5/3 if d = 2 and s > d/2 + 1/2 if d 3. We also provide ill-posedness results.With d = 1, the available local well-posedness theory has been based on compactness methods. Indeed, Molinet, Saut and Tzvetkov [19] proved that the problem cannot be solved in L 2 -Sobolev spaces H s by Picard iteration. We will show that

Dispersive Estimates for Full Dispersion KP Equations

“…This is not a simple matter since the symbol p and p defined in (1.3) and (1.4) are non-homogeneous and also non-polynomial. Similar difficulties occur for other non-standard dispersive equation such as the Novikov-Veselov equation [10] or a higher dimensional version of the Benjamin-Ono equation [8].…”

“…x at the regularity level s > 7 4 . Note that similar estimates have already been used for nonlinear dispersive equations (see for instance to [2,4,8,12,13,[16][17][18]28]).…”

“…As in [4,8,12,13,[16][17][18] the main ingredient in our analysis is a refined Strichartz estimates for solutions of the non-homogeneous linear equation…”

“…Finally, the proof that u belongs to the class (1.21) and of the continuous dependence of the flow follows from the Bona-Smith argument [3]. Since it is a classical argument, we skip the proof and refer the readers to [8,18] for more details in this setting.…”

“…when β > 0, and 8) when β = 0. The FDKP equation may be therefore seen as a natural (weakly transverse) two-dimensional version of the Whitham equation, with and without surface tension.…”