Optimal regularity and exponential stability for the Blackstock–Crighton equation in L p -spaces with Dirichlet and Neumann boundary conditions

Abstract

Abstract. The Blackstock-Crighton equation models nonlinear acoustic wave propagation in monatomic gases. 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 is 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 solves the nonlinear problem by means of the implicit function theorem.

Asymptotic behaviors for Blackstock's model of thermoviscous flow

“…for example, [6,7,9,10,29]. Actually, a natural idea is reversing the procedure to replace ψ tt by ∆ψ, and this procedure has been employed in Kuznetsov's equation [17] already.…”

“…Concerning some studies on linear or nonlinear Blackstock's model under Becker's assumption (cf. next subsection), we refer the interested readers to [9,6,7,10,29,11,18] for Dirichlet or Neumann boundary value problems.…”

“…Our first purpose in the present paper is to understand the linear Blackstock's model (10) by employing the phase space analysis, which helps us to understand the underlying physical phenomena in a certain condition and plays the crucial role for investigating the corresponding nonlinear problem (8). In Section 2, by using asymptotic expansions of characteristic roots and WKB analysis, we derive optimal decay estimates with weighted L 1 data for high dimensions n 5 as well as three asymptotic profiles of solution including heat equations and diffusion-wave equations.…”

“…Let us formally take into account the limit case with vanishing thermal diffusivity in the Cauchy problem (10), which turns out to be the viscoelastic damped wave equation (see [35,28,15,24,25,26,3,2] and references therein) whose initial data are inherited from (10) as follows:…”

“…It means that no heat is exchanged during the compression and decompression phases of sound propagation. Formally, (11) seems to be the limit equation for (10) with vanishing thermal diffusivity κ = 0. In Section 3, under the consistency assumption ψ 2 = bν∆ψ 1 + ∆ψ 0 , we rigorously justify singular limits (limiting processes) from the linear Blackstock's model (10) to the viscoselastic damped wave equation (11) as the thermal diffusivity tends to zero, for instance, we conclude in Subsection 3.…”