Convolutions of singular measures and applications to the Zakharov system

Abstract

Uniform L 2 -estimates for the convolution of singular measures with respect to transversal submanifolds are proved in arbitrary space dimension. The results of Bennett-Bez are used to extend previous work of Bejenaru-Herr-Tataru. As an application, it is shown that the 3D Zakharov system is locally well-posed in the full subcritical regime.

Scattering for the Zakharov System in 3 Dimensions

“…From the mathematical side, there has been considerable work on local and global wellposedness of solutions with rough data through the works of Kenig, Ponce and Vega [15], Bourgain and Colliander [1], Ginibre, Tsutsumi and Velo [7], Bejenaru, Herr, Holmer and Tataru [2] and Bejenaru and Herr [3] (cf. the references in the cited works for previous well-posedness results).…”

Convolution estimates for singular measures and some global nonlinear Brascamp—Lieb inequalities

“…The work of Bennett & Bez was then used by Bejenaru & Herr [1] to extend [2] to arbitrary dimensions under the natural scaling condition of the codimensions adding up to the space dimension. All three papers treat the nonlinearity in a perturbative fashion.…”

Global well-posedness of the fractional Klein-Gordon-Schrödinger system with rough initial data

“…In three-dimensional case, global well-posedness was obtained for s 1 0 and s 1 − 1/2 s 2 s 1 + 1. These results are based on the sharp local well-posedness result in two-dimensional case by Bejenaru et al [2] as well as the three-dimensional estimates by Bejenaru and Herr [1] for the Zakharov system.…”