Optimality of the Decay Estimate of Solutions to the Linearised Curl-Free Compressible Navier-Stokes Equations

TL;DR

This paper establishes the optimal decay estimates in Besov norms for solutions to the linearised curl-free compressible Navier-Stokes equations, demonstrating the bounds are tight in the $L^$-norm.

Contribution

It proves the optimality of decay estimates for the curl-free part of solutions to linearised compressible Navier-Stokes equations in Besov norms.

Findings

Decay estimates are optimal in the $L^$-norm.

Lower bounds match the upper decay rates.

Results confirm the sharpness of previous estimates.

Abstract

We discuss optimal estimates of solutions to the compressible Navier-Stokes equations in Besov norms. In particular, we consider the estimate of the curl-free part of the solution to the linearised equations, in the homogeneous case. We prove that our estimate is optimal in the $L^\infty$-norm by showing that the norm is bounded from below by the same decay rate.