Optimal regularity and exponential stability for the Blackstock-Crighton equation in $L_p$-spaces with Dirichlet and Neumann boundary conditions

TL;DR

This paper establishes optimal regularity, exponential stability, and well-posedness for the Blackstock-Crighton equation in bounded domains with Dirichlet and Neumann boundary conditions, using an $L_p$-framework and nonlinear analysis techniques.

Contribution

It introduces an optimal $L_p$-setting for the Blackstock-Crighton equation and proves long-time well-posedness and exponential stability for small data.

Findings

Solutions decay exponentially under Dirichlet conditions.

Solutions decay to zero with zero-mean Neumann data.

The linearized model has maximal $L_p$-regularity.

Abstract

The Blackstock-Crighton equation models nonlinear acoustic wave propagation in thermo-viscous fluids. In the present work we investigate the associated inhomogeneous Dirichlet and Neumann boundary value problems in a bounded domain and prove long-time well-posedness and exponential stability for sufficiently small data. The solution depends analytically on the data. In the Dirichlet case, the solution decays to zero and the same holds for Neumann conditions if the data have zero mean. We choose an optimal $L_p$-setting, where the regularity of the initial and boundary data are necessary and sufficient for existence, uniqueness and regularity of the solution. The linearized model with homogeneous boundary conditions is represented as an abstract evolution equation for which we show maximal $L_p$-regularity. In order to eliminate inhomogeneous boundary conditions, we establish a general higher regularity result for the heat equation. We conclude that the linearized model induces a topological linear isomorphism and then solve the nonlinear problem by means of the implicit function theorem.