The elastic wave equation of limited smoothness

TL;DR

This paper proves a decoupling result for P and S waves in linear isotropic elasticity with limited smoothness, showing that the wave components can be effectively separated and regularized under certain conditions.

Contribution

It introduces a novel diagonalization technique for the elastic system with twice-differentiable Lamé parameters, enabling wave component decoupling with limited smoothness.

Findings

P and S wave components are regularized of order one on L^2 data.

Diagonalization of the elastic system is achieved modulo a bounded operator.

The method depends on the specific structure of the conjugation operator's symbol.

Abstract

We establish a decoupling result for the $P$ and $S$ waves of linear, isotropic elasticity, in the setting of twice-differentiable Lam\'e parameters. Precisely, we show that the $P\leftrightarrow S$ components of the wave propagation operator are regularizing of order one on $L^2$ data, by establishing the diagonalization of the elastic system modulo a $L^2$-bounded operator. Effecting the diagonalization in the setting of twice-differentiable coefficients depends upon the symbol of the conjugation operator having a particular structure.