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.
Highlights
An acoustic wave propagates through a medium as a local pressure change
The present work provides an analysis of the Blackstock–Crighton–Kuznetsov equation
Results on well-posedness for the Kuznetsov and the Westervelt equation with homogeneous Dirichlet [19] and inhomogeneous Dirichlet [22], [21] and Neumann [20] boundary conditions have been shown in an L2( )-setting on spatial domains ⊂ Rn of dimension n ∈ {1, 2, 3}
Summary
An acoustic wave propagates through a medium as a local pressure change. Nonlinear effects typically occur in case of acoustic waves of high amplitude which are used for several medical and industrial purposes such as lithotripsy, thermotherapy, ultrasound cleaning or welding and sonochemistry. Results on well-posedness for the Kuznetsov and the Westervelt equation with homogeneous Dirichlet [19] and inhomogeneous Dirichlet [22], [21] and Neumann [20] boundary conditions have been shown in an L2( )-setting on spatial domains ⊂ Rn of dimension n ∈ {1, 2, 3}. There are results on optimal regularity and long-time behavior of solutions for the Westervelt equation with homogeneous Dirichlet [26] and for the Kuznetsov equation with inhomogeneous Dirichlet [27] boundary conditions in L p( )-spaces, where the spatial domain ⊂ Rn is of arbitrary dimension. In Appendix B, we prove some higher regularity results for the heat equation with inhomogeneous Dirichlet or Neumann boundary conditions and inhomogeneous initial conditions in a far more general framework than needed in the main text. We explicitly state all necessary compatibility conditions between initial and boundary data and show how these are used to construct a solution with high regularity
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