Weighted Strichartz Estimates with Angular Regularity and their Applications

TL;DR

This paper develops weighted Strichartz estimates with angular regularity, leading to new results on wave and Schrödinger equations, including proving Strauss' conjecture for certain dimensions and establishing global well-posedness for some nonlinear Schrödinger equations.

Contribution

It introduces an optimal dual trace estimate involving angular regularity and applies it to derive generalized Morawetz and weighted Strichartz estimates for evolution equations.

Findings

Proved Strauss' conjecture for 2≤n≤4 with mild rough data.

Established global well-posedness for small data in certain nonlinear Schrödinger equations.

Derived new weighted Strichartz estimates with angular regularity.

Abstract

In this paper, we establish an optimal dual version of trace estimate involving angular regularity. Based on this estimate, we get the generalized Morawetz estimates and weighted Strichartz estimates for the solutions to a large class of evolution equations, including the wave and Schr\"{o}dinger equation. As applications, we prove the Strauss' conjecture with a kind of mild rough data for $2\le n\le 4$, and a result of global well-posedness with small data for some nonlinear Schr\"{o}dinger equation with $L^2$-subcritical nonlinearity.