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

2025/11/04 by Li, Fucai, Ni, Jinkai, Zhang, Zhipeng
Engineering · Mathematics · #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations

paper · doi:10.48550/arxiv.2511.02321

openalex publication_date 2025/11/04 · openalex created_date 2025/11/06 · openalex updated_date 2026/07/28

Abstract

This paper investigates the Cauchy problem for the compressible pressureless Navier-Stokes system in ℝ^d with d ≥ 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 B_2,1^(d)/(2) × B_2,1^(d)/(2)-1. Then, under the additional assumption that the low-frequency component of the initial density belongs to B_2,∞^σ₀+1 and that the initial velocity is sufficiently small in B_2,∞^σ₀ with σ₀ ∈ (-(d)/(2), (d)/(2)-1], we overcome the challenge of derivative loss caused by nonlinearity and establish optimal decay estimates for u in B_2,1^σ with σ∈ (σ₀, (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.

Citations

• Large-friction and incompressible limits for pressureless Euler-Navier-Stokes flows
• Fujita-Kato solution for the 3D compressible pressureless Navier-Stokes equations with discontinuous and large-variation density
• Global well-posedness of inhomogeneous Navier-Stokes equations with bounded density
• Global existence and optimal time-decay rates of the compressible Navier-Stokes-Euler system
• Global Fujita-Kato solution of 3-D inhomogeneous incompressible Navier-Stokes system
• Characterization of solutions to dissipative systems with sharp algebraic decay
• Fourier analysis methods for the compressible Navier-Stokes equations
• From the highly compressible Navier-Stokes equations to the Porous\n Medium equation - rate of convergence
• Global Well-Posedness of Classical Solutions with Large Oscillations and Vacuum to the Three-Dimensional Isentropic Compressible Navier-Stokes Equations

Related

• The Pressureless Euler--Navier--Stokes System
• Optimal decay for the full compressible Navier-Stokes system in critical L^p Besov spaces
• Global existence and optimal time-decay rates of the compressible Navier-Stokes-Euler system
• Uniform regularity and vanishing dissipation limit for the full compressible Navier-Stokes system in 3-D bounded domain
• Optimal time-decay estimates for the compressible Navier-Stokes-Poisson equations without additional smallness assumptions