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

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.

Weak Solutions of Navier–stokes Equations Constructed by Artificial Compressibility Method Are Suitable

“…Temam showed the convergence of this approximation on bounded domain. Recently in [6] and [7] the result was extended in the case of the whole space and exterior domain, respectively. In [6] the convergence towards Leray weak solutions of the Navier-Stokes equations is achieved by using the dispersive structure of the system.…”

Singular limit in Hopf bifurcation for doubly diffusive convection equations II: bifurcation and stability

“…Convergence result was also established in the framework of weak solutions on three-dimensional bounded domains. Donatelli [5,6] and Donatelli and Marcati [7,8] proved similar convergence results in the case of unbounded domains on any finite time interval by using the wave equation structure of the pressure and the dispersive estimates.…”

Singular limit in Hopf bifurcation for doubly diffusive convection equations I: linearized analysis at criticality

“…Convergence result was also established in the framework of weak solutions on three-dimensional bounded domains. In this direction, Donatelli [6,7] and Donatelli and Marcati [8,9] proved convergence results which reflect a dispersive aspect of the system in the case of unbounded domains by using the wave equation structure of the pressure and the dispersive estimates.…”