Leray weak solutions of the Incompressible Navier Stokes system on exterior domains via the artificial compressibility method

TL;DR

This paper investigates Leray weak solutions of the incompressible Navier-Stokes equations in exterior domains, employing an hyperbolic artificial compressibility method and dispersive wave analysis to establish convergence.

Contribution

It introduces a hyperbolic artificial compressibility approach combined with dispersive wave estimates to analyze weak solutions in exterior domains, addressing convergence challenges.

Findings

Demonstrates convergence of the hyperbolic artificial compressibility method

Decomposes pressure into acoustic components for analysis

Uses Strichartz estimates to handle dispersive waves

Abstract

In this paper we study the Leray weak solutions of the incompressible Navier Stokes equation in an exterior domain.We describe, in particular, an hyperbolic version of the so called artificial compressibility method investigated by J.L.Lions and Temam. The convergence of these type of approximation show in general a lack of strong convergence due to the presence of acoustic waves. In this paper we face this difficulty by taking care of the dispersive nature of these waves by means of the Strichartz estimates or waves equations satisfied by the pressure. We actually decompose the pressure in different acoustic components, each one of them satisfies a specific initial boundary value problem. The strong convergence analysis of the velocity field will be achieved by using the associated Leray-Hodge decomposition.