On Morawetz estimates for the elastic wave equation

TL;DR

The paper advances Morawetz-type estimates for the elastic wave equation by decoupling the system into two scalar wave equations via Fourier projections, reducing the problem to the classical wave propagator. It proves a known spatial-weight Morawetz bound and introduces a new space-time weighted Morawetz bound, the latter achieved through Littlewood–Paley theory with $A_2$ weights and a detailed frequency-localized—followed by bilinear—analysis. The core contributions are explicit solution formulas in Fourier space, frequency-localized estimates for the wave propagator, and bilinear interpolation arguments that yield robust weighted $L^2$ bounds under precise regularity and weight conditions. Collectively, these results extend Morawetz-type control to elastic waves with stronger singular weights in space-time, while requiring weaker initial data regularity than previous purely spatial-weight approaches, with potential implications for dispersive and scattering analyses in elastic media.

Abstract

We establish Morawetz-type estimates for solutions to the elastic wave equation with singular weights of the form $|x|^{-α}$ or $|(x,t)|^{-α}$. In particular, we show that space-time weights $|(x,t)|^{-α}$ admit stronger singularities and require weaker regularity assumptions on the initial data compared to purely spatial weights $|x|^{-α}$.

Figures (1)

• Figure 1: The region of $(\alpha,s)$ for \ref{['mor']} and \ref{['mor2']}.

Theorems & Definitions (12)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Proposition 3.1
• proof : Proof of \ref{['wmor2']}
• Lemma 3.2
• Lemma 4.1
• Lemma 4.2: BL, Section 3.13, Exercise 5(a)
• Lemma 4.3
• Lemma 5.1