A first order in time wave equation modeling nonlinear acoustics
Abstract
In this paper we focus on a small amplitude approximation of a Navier-Stokes-Fourier system modeling nonlinear acoustics. Omitting all third and higher order terms with respect to certain small parameters, we obtain a first order in time system containing linear and quadratic pressure and velocity terms. Subsequently, the well-posedness of the derived system is shown using the classical method of Galerkin approximation in combination with a fixed point argument. We first prove the well-posedness of a linearized equation using energy estimates and then the well-posedness of the nonlinear system using a Newton-Kantorovich type argument. Based on this, we also obtain global in time well-posedness for small enough data and exponential decay. This is in line with semigroup results for a linear part of the system that we provide as well.
- Publication:
-
arXiv e-prints
- Pub Date:
- April 2024
- DOI:
- 10.48550/arXiv.2404.11250
- arXiv:
- arXiv:2404.11250
- Bibcode:
- 2024arXiv240411250K
- Keywords:
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- Mathematics - Analysis of PDEs;
- 35L60;
- 35L50