Uniform stability and optimal time decay rates of the compressible pressureless Navier-Stokes system in the critical regularity framework
Fucai Li, Jinkai Ni, Zhipeng Zhang
TL;DR
This work analyzes the compressible pressureless Navier–Stokes system in $\mathbb{R}^d$ ($d\ge2$) and establishes global well-posedness in the critical Besov space $\dot B_{2,1}^{\frac{d}{2}} \times \dot B_{2,1}^{\frac{d}{2}-1}$, along with uniform stability under perturbations. It overcomes derivative losses caused by the nonlinear coupling, leveraging low-frequency regularity of the density fluctuation $a$ and refined commutator estimates to derive sharp decay rates for the velocity $u$ in $\dot B_{2,1}^\sigma$ for $\sigma$ up to $\tfrac{d}{2}+1$, while the density remains uniformly bounded in time. The paper distinguishes the long-time behavior from the isentropic NS system by showing the density does not decay, reflecting the absence of a dissipative pressure. It also explores upper and lower bounds for decay under boundedness or smallness of low-frequency data, including a more rapid decay in three dimensions under the smallness assumption. These results advance the understanding of critical-regularity analysis for pressureless fluids and provide new techniques for handling derivative loss in coupled nonlinear PDE systems.
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 $\dot{B}_{2,\infty}^{σ_0+1}$ and that the initial velocity is sufficiently small in $\dot{B}_{2,\infty}^{σ_0}$ with $σ_0 \in (-\frac{d}{2}, \frac{d}{2}-1]$, we overcome the challenge of derivative loss caused by nonlinearity and establish optimal decay estimates for $u$ in $\dot{B}_{2,1}^σ$ with $σ\in (σ_0, \frac{d}{2}+1]$. 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.