Global existence for a damped wave equation and convergence towards a solution of the Navier-Stokes problem

TL;DR

This paper proves global existence for a damped wave equation in 2D and 3D, showing its convergence towards Navier-Stokes solutions with improved methods and less restrictive initial data assumptions.

Contribution

It improves previous results by relaxing initial data regularity and simplifying proofs, avoiding Strichartz estimates, and achieving near-optimal regularity for convergence.

Findings

Established global existence for damped wave equations in 2D and 3D.

Proved convergence of damped wave solutions to Navier-Stokes solutions.

Simplified proof techniques with less regularity requirements.

Abstract

In two and three space dimensions, and under suitable assumptions on the initial data, we show global existence for a damped wave equation which approaches, in some sense, the Navier-Stokes problem. The proofs are based on a refined energy method. In this paper, we improve the results in two papers by Y. Brenier, R. Natalini and M. Puel and by M. Paicu and G. Raugel. We relax the regularity of the initial data of the former, even though we still use energy methods as a principal tool. Regarding the second paper, the improvement consists in the simplicity of the proofs since we do not use any Strichartz estimate and in requiring less regularity for the convergence to the Navier-Stokes problem. Indeed, the convergence result we obtain is near-optimal regularity.