A Morawetz type energy estimate for wave equation in $\mathbb{R}^2$ and application to elastic waves

TL;DR

This paper develops a modified Morawetz energy estimate for wave equations in two dimensions, enabling new proofs of global existence for certain quasilinear and elastic wave systems.

Contribution

It introduces a novel weighted Morawetz energy estimate for 2D wave equations, applicable to quasilinear and elastic wave systems, providing alternative proofs of global existence.

Findings

Established a non-negative weighted Morawetz energy estimate in 2D

Proved global existence for quasilinear wave equations with small data

Extended the estimate to certain elastic wave systems

Abstract

In this paper, we introduce a modified scaling Morawetz multiplier, which produces a weighted Morawetz type energy (non-negative) estimate for the inhomogeneous wave equation in $\mathbb{R}^2$. With this estimate in hand, an alternative proof of global existence for the Cauchy problem of quasilinear wave equation with small and compactly supported data is given. What is more, such weighted Morawetz type energy estimate also works for certain wave system with multiple speeds, which can be used to prove global existence of some admissible harmonic elastic wave system in $\mathbb{R}^2$.