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Well-posedness of a first-order-in-time model for nonlinear acoustics with nonhomogeneous boundary conditions

Pascal Lehner

Abstract

We study well-posedness of a first-order-in-time model for nonlinear acoustics with nonhomogeneous boundary conditions in fractional Sobolev spaces. The analysis proceeds by first establishing well-posedness of an abstract parabolic-type semilinear evolution equation. These results are then applied to concrete operators and function spaces that capture the boundary conditions relevant for realistic modeling. Our approach is based on the spectral decomposition of a positive definite self-adjoint operator, with solution regularity characterized via the domains of its fractional powers. Employing Galerkin's method and the Newton-Kantorovich theorem, we prove well-posedness for the abstract nonlinear system with possibly nonhomogeneous boundary data. The connection between (spectral) fractional powers of the Laplacian and fractional Sobolev spaces due to interpolation theory allows us to transfer these results to the nonlinear acoustic model under nonhomogeneous Dirichlet and Neumann boundary conditions, yielding fractional Sobolev regularity. For Hodge/Lions boundary conditions, we establish well-posedness with solutions in classical Sobolev spaces of integer order.

Well-posedness of a first-order-in-time model for nonlinear acoustics with nonhomogeneous boundary conditions

Abstract

We study well-posedness of a first-order-in-time model for nonlinear acoustics with nonhomogeneous boundary conditions in fractional Sobolev spaces. The analysis proceeds by first establishing well-posedness of an abstract parabolic-type semilinear evolution equation. These results are then applied to concrete operators and function spaces that capture the boundary conditions relevant for realistic modeling. Our approach is based on the spectral decomposition of a positive definite self-adjoint operator, with solution regularity characterized via the domains of its fractional powers. Employing Galerkin's method and the Newton-Kantorovich theorem, we prove well-posedness for the abstract nonlinear system with possibly nonhomogeneous boundary data. The connection between (spectral) fractional powers of the Laplacian and fractional Sobolev spaces due to interpolation theory allows us to transfer these results to the nonlinear acoustic model under nonhomogeneous Dirichlet and Neumann boundary conditions, yielding fractional Sobolev regularity. For Hodge/Lions boundary conditions, we establish well-posedness with solutions in classical Sobolev spaces of integer order.
Paper Structure (21 sections, 10 theorems, 91 equations)

This paper contains 21 sections, 10 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Contributions
  4. Choice of boundary conditions
  5. Abstract system
  6. Auxiliary linear problem
  7. Galerkin Approximation
  8. Energy estimates
  9. Existence and uniqueness
  10. Contraction argument for the bilinear problem
  11. Nonhomogeneous boundary conditions
  12. Application to nonlinear acoustics
  13. Preliminaries
  14. Dirichlet Laplacian
  15. Neumann Laplacian
  16. ...and 6 more sections

Key Result

Theorem 2.1

Let $T>0$ and Assumption se:assumption hold with $\sigma \in [0,1]$. Then, for small enough $r_G>0$ and all ${u_*} \in B_{r_G}^{X}(0)$ as well as any there exists a unique solution of equation eq:full_system_abstract. Moreover, the a priori estimate holds for some $C_G>0$.

Theorems & Definitions (21)

  • Remark 2.1
  • Theorem 2.1
  • proof
  • Lemma 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Lemma 3.1
  • proof
  • ...and 11 more