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.
Categories: math.AP