The Pressureless Euler-Navier-Stokes System
Valentin Lemarié
TL;DR
This work analyzes the coupled pressureless Euler–Navier–Stokes system in $\mathbb{R}^d$ ($d\ge2$) with a density $\rho$ and velocity $w$ for the compressible, pressureless part and an incompressible velocity $u$ with pressure $P$ for the Navier–Stokes part, under the state $ (\rho,w,u)=(0,0,0)$. Employing a critical Besov space framework and Littlewood–Paley theory, the authors establish global existence and uniqueness for small data in the scale $\dot B^{d/2}_{2,1}\times(\dot B^{d/2-1}_{2,1}\cap\dot B^{d/2+1}_{2,1})\times\dot B^{d/2-1}_{2,1}$, with precise a priori bounds and persistence of density positivity via characteristics. They introduce the damped mode $w-u$ to recover low-frequency information, and prove time-decay results (under an extra $L^1$-type condition on initial velocities) for $w,u$, and their difference, with dimension-dependent rates obtained through Nash-type Lyapunov arguments and propagation of negative regularity. The paper further extends decay and regularity results to higher Besov spaces, providing a comprehensive well-posedness and long-time behavior theory for this hybrid Euler–Navier–Stokes system. These results contribute to the mathematical understanding of hybrid compressible–incompressible fluid models and their long-time dynamics in critical regularity settings.
Abstract
In this paper, we study the well-posedness of the pressureless Euler-Navier-Stokes system in $\mathbb{R}^d$ (with $d\geq 2$) in the critical regularity setting for a density close to $0$. We prove a global existence result for small data for this system, and then give optimal time decay estimates.