Uniform stability and optimal time decay rates of the compressible pressureless Navier-Stokes system in the critical regularity framework

TL;DR

This paper proves global stability and decay rates for solutions to the pressureless Navier-Stokes system in critical Besov spaces, revealing new uniform density bounds and asymptotic behaviors distinct from classical models.

Contribution

The authors establish the first global well-posedness and optimal decay estimates for the pressureless Navier-Stokes system in critical regularity frameworks, overcoming derivative loss challenges.

Findings

Global well-posedness in critical Besov spaces

Optimal decay rates for velocity in specific Besov spaces

Density remains uniformly bounded over time

Abstract

This paper investigates the Cauchy problem for the compressible pressureless Navier-Stokes system in $\mathbb{R}^d$ with $d \geq 2$. Unlike the standard isentropic compressible Navier-Stokes system, the density in the pressureless model lacks a dissipative mechanism, leading to significant coupling effects from nonlinear terms in the momentum equations. We first prove the global well-posedness and uniform stability of strong solutions to the compressible pressureless Navier-Stokes system in the critical Besov space $\dot{B}_{2,1}^{\frac{d}{2}} \times \dot{B}_{2,1}^{\frac{d}{2}-1}$. Then, under the additional assumption that the low-frequency component of the initial density belongs to $\dot{B}_{2,\infty}^{\sigma_0+1}$ and that the initial velocity is sufficiently small in $\dot{B}_{2,\infty}^{\sigma_0}$ with $\sigma_0 \in (-\frac{d}{2}, \frac{d}{2}-1]$, we overcome the challenge of derivative loss caused by nonlinearity and establish optimal decay estimates for $u$ in $\dot{B}_{2,1}^{\sigma}$ with $\sigma \in (\sigma_0, \frac{d}{2}+1]$. In particular, it is shown that the density remains uniformly bounded in time which reveals a new asymptotic behavior in contrast to the isentropic compressible Navier-Stokes system where the density exhibits a dissipative structure and decays over time.