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

TL;DR

The article develops a modified scaling Morawetz approach to obtain a non-negative, weighted Morawetz energy in two dimensions, enabling an alternative global existence proof for small-data quasilinear wave equations and extending to wave systems with multiple speeds such as elastic waves. Central to the method is the Morawetz-type energy $\tilde{M}_\kappa$ and its equivalence to the standard energy, together with null-condition improvements that yield time-decay in nonlinear terms. The work then applies these ideas to admissible harmonic elastodynamics, using a Helmholtz decomposition and a two-speed framework to establish weighted $L^2$ estimates and energy bounds, leading to almost global or global small-data well-posedness under specific algebraic null-condition constraints. Overall, the paper broadens the applicability of Morawetz-type energies in 2D, delivering robust tools for managing multi-speed quasilinear systems and elastic wave models.

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 $\mathbb{R}^2$.

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Remark 3.1
• Lemma 3.1
• Lemma 3.2
• Lemma 4.1