Optimal regularity and long-time behavior of solutions for the Westervelt equation

TL;DR

This paper proves the existence, uniqueness, and exponential decay of solutions for the Westervelt equation under small initial data, using maximal Lp-regularity techniques for quasilinear parabolic equations.

Contribution

It establishes optimal regularity and long-time decay results for the Westervelt equation with a novel application of maximal Lp-regularity methods.

Findings

Existence of a unique global solution under small initial data

Solution converges exponentially to zero over time

Achieves optimal Lp-regularity for solutions

Abstract

We investigate an initial-boundary value problem for the quasilinear Westervelt equation which models the propagation of sound in fluidic media. We prove that, if the initial data are sufficiently small and regular, then there exists a unique global solution with optimal $L_p$-regularity. We show furthermore that the solution converges to zero at an exponential rate as time tends to infinity. Our techniques are based on maximal $L_p$-regularity for abstract quasilinear parabolic equations.