Sharp weighted Strichartz estimates and critical inhomogeneous Hartree equations

TL;DR

This paper establishes local existence results for the inhomogeneous Hartree equation at Sobolev critical indices by deriving weighted Strichartz estimates, addressing a gap in the well-posedness theory for this model.

Contribution

It introduces sharp weighted Strichartz estimates and proves local well-posedness for the critical inhomogeneous Hartree equation, a problem previously unresolved.

Findings

Derived all weighted $L^p$ Strichartz estimates with singular weights.

Proved local existence of solutions at Sobolev critical indices.

Extended well-posedness theory to inhomogeneous Hartree models.

Abstract

We study the Cauchy problem for the inhomogeneous Hartree equation in this paper. Although its well-posedness theory has been extensively studied in recent years, much less is known compared to the classical Hartree model of homogeneous type. In particular, the problem of Sobolev initial data with the Sobolev critical index remains unsolved. The main contribution of this paper is to establish the local existence of solutions to the inhomogeneous equation in the critical cases. To do so, we obtain all possible $L^p$ Strichartz estimates with singular weights.