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

Abstract

This paper investigates the Cauchy problem for the compressible pressureless Navier-Stokes system in ${\mathbb{R}}^d$ with $d\ge 2$. 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.

31 pages