Well-posedness and numerical treatment of the Blackstock equation in nonlinear acoustics

TL;DR

This paper establishes the well-posedness and decay properties of the Blackstock equation in nonlinear acoustics, and introduces a finite element method for its numerical solution, supported by numerical experiments.

Contribution

It proves global well-posedness for small initial data and develops a finite element scheme for the Blackstock equation, with numerical validation.

Findings

Global well-posedness for small initial data

Exponential decay of solution energy

Finite element method effectively simulates nonlinear sound waves

Abstract

We study the Blackstock equation which models the propagation of nonlinear sound waves through dissipative fluids. Global well-posedness of the model with homogeneous Dirichlet boundary conditions is shown for small initial data. To this end, we employ a fixed-point technique coupled with well-posedness results for a linearized model and appropriate energy estimates. Furthermore, we obtain exponential decay for the energy of the solution. We present additionally a finite element-based method for solving the Blackstock equation and illustrate the behavior of solutions through several numerical experiments.