Fourier analysis methods for the compressible Navier-Stokes equations

TL;DR

This paper reviews Fourier analysis techniques, especially Littlewood-Paley and paradifferential calculus, applied to the compressible Navier-Stokes equations, highlighting new decay estimates for small solutions in critical regularity settings.

Contribution

It provides a comprehensive overview of Fourier analysis methods for compressible Navier-Stokes equations and introduces new time decay estimates for global small solutions.

Findings

New decay estimates for small solutions in critical regularity.

Effective use of Littlewood-Paley and paradifferential calculus techniques.

Results applicable to whole space and torus domains in multiple dimensions.

Abstract

In the last three decades, Fourier analysis methods have known a growing importance in the study of linear and nonlinear PDE's. In particular, techniques based on Littlewood-Paley decomposition and paradifferential calculus have proved to be very efficient for investigating evolutionary fluid mechanics equations in the whole space or in the torus. We here give an overview of results that we can get by Fourier analysis and paradifferential calculus, for the compressible Navier-Stokes equations. We focus on the Initial Value Problem in the case where the fluid domain is the whole space or the torus in dimension at least two, and also establish some asymptotic properties of global small solutions. The time decay estimates in the critical regularity framework that are stated at the end of the survey are new, to the best of our knowledge.