Time-weighted estimates for the Blackstock equation in nonlinear ultrasonics

TL;DR

This paper analyzes the well-posedness of the Blackstock equation in nonlinear ultrasonics, demonstrating local and global solvability with less restrictive initial conditions and exponential decay of solutions.

Contribution

It introduces a time-weighted energy framework exploiting the equation's parabolic-like nature, improving well-posedness results for the Blackstock equation.

Findings

Local well-posedness under weaker regularity assumptions

Global solvability with exponential decay

Enhanced understanding of nonlinear ultrasonics dynamics

Abstract

High frequencies at which ultrasonic waves travel give rise to nonlinear phenomena. In thermoviscous fluids, these are captured by Blackstock's acoustic wave equation with strong damping. We revisit in this work its well-posedness analysis. By exploiting the parabolic-like character of this equation due to strong dissipation, we construct a time-weighted energy framework for investigating its local solvability. In this manner, we obtain the small-data well-posedness on bounded domains under less restrictive regularity assumptions on the initial conditions compared to the known results. Furthermore, we prove that such initial boundary-value problems for the Blackstock equation are globally solvable and that their solution decays exponentially fast to the steady state.