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

TL;DR

This paper proves the well-posedness of a nonlinear acoustic model with nonhomogeneous boundary conditions using spectral analysis, Galerkin's method, and interpolation theory, establishing regularity in fractional Sobolev spaces.

Contribution

It introduces a novel approach combining spectral decomposition and interpolation to analyze well-posedness of nonlinear acoustics models with complex boundary conditions.

Findings

Well-posedness established for models with Dirichlet and Neumann boundary conditions.

Solutions exhibit fractional Sobolev regularity due to spectral analysis.

Classical Sobolev regularity achieved for Hodge/Lions 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.