A first order in time wave equation modeling nonlinear acoustics
Abstract
In this paper we focus on a small amplitude approximation of a NavierStokesFourier 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 wellposedness of the derived system is shown using the classical method of Galerkin approximation in combination with a fixed point argument. We first prove the wellposedness of a linearized equation using energy estimates and then the wellposedness of the nonlinear system using a NewtonKantorovich type argument. Based on this, we also obtain global in time wellposedness 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 eprints
 Pub Date:
 April 2024
 DOI:
 10.48550/arXiv.2404.11250
 arXiv:
 arXiv:2404.11250
 Bibcode:
 2024arXiv240411250K
 Keywords:

 Mathematics  Analysis of PDEs;
 35L60;
 35L50