Three evolution problems modelling the interaction between acoustic waves and non-locally reacting surfaces

TL;DR

This paper rigorously derives and analyzes three evolution models describing acoustic wave interactions with non-locally reacting surfaces, providing new well-posedness and regularity results especially for the Lagrangian and Eulerian formulations.

Contribution

It offers a rigorous derivation of the wave equation with acoustic boundary conditions from physically transparent models and establishes new well-posedness and regularity results for these models.

Findings

Rigorous derivation of the wave equation from Lagrangian and Eulerian models.

New well-posedness results for all three models.

Optimal regularity results for the Eulerian and Lagrangian models.

Abstract

The paper deals with three evolution problems arising in the physical modelling of acoustic phenomena of small amplitude in a fluid, bounded by a surface of extended reaction. The first one is the widely studied wave equation with acoustic boundary conditions, which derivation from the physical model is not fully mathematically satisfactory. The other two models studied in the paper, in the Lagrangian and Eulerian settings, are physically transparent. In the paper the first model is derived from the other two in a rigorous way, also for solutions merely belonging to the natural energy spaces. The paper also gives several well-posedness and optimal regularity results for the three problems considered, which are new for the Eulerian and Lagrangian models.