An improved Strichartz estimate for systems with divergence free data

TL;DR

This paper improves Strichartz estimates for wave and Schrödinger systems with divergence-free inhomogeneities, allowing the use of L^1_x norms, which broadens the applicability of such estimates in PDE analysis.

Contribution

It introduces an enhanced Strichartz estimate leveraging div-curl inequalities, accommodating L^1_x inhomogeneity norms, a novel extension in the analysis of PDE systems.

Findings

Improved Strichartz estimate for divergence-free systems

Allows L^1_x norms of inhomogeneity

Utilizes div-curl inequalities of Bourgain-Brezis and van Schaftingen

Abstract

Using the div-curl inequalities of Bourgain-Brezis [?MR2057026] and van Schaftingen [?MR2078071], we prove an improved Strichartz estimate for systems of inhomogeneous wave and Schrodinger equations, for which the inhomogeneity is a divergence-free vector field at each given time. The novelty of the result is that one can allow $L^1_x$ norms of the inhomogeneity in the right hand side of the estimate.