Regularity and dynamics of weak solutions for one-dimensional compressible Navier-Stokes equations with vacuum

2025/09/01 by Jin Tan, Tan, Jin, Yan-Lin Wang +3
Engineering · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2509.01196

openalex publication_date 2025/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In the spirit of D. Hoff's weak solution theory for the compressible Navier-Stokes equations (CNS) with bounded density, in this paper we establish the global existence and regularity properties of finite-energy weak solutions to an initial boundary value problem of one-dimensional CNS with general initial data and vacuum. The core of our proof is a global in time a priori estimate for one-dimensional CNS that holds for any H¹ initial velocity and bounded initial density not necessarily strictly positive: it could be a density patch or a vacuum bubble. We also establish that the velocity and density decay exponentially to equilibrium. As a by-product, we obtain the quantitative dynamics of aforementioned two vacuum states.

Citations

• Global well-posedness of the 1d compressible Navier-Stokes system with rough data
• Local well-posedness of the 1d compressible Navier-Stokes system with rough data
• Compressible Navier-Stokes equations with ripped density
• Global Existence of Weak Solutions for Compresssible Navier--Stokes Equations: Thermodynamically unstable pressure and anisotropic viscous stress tensor
• Existence and Blowup Behavior of Global Strong Solutions to the Two-Dimensional Baratropic Compressible Navier-Stokes System with Vacuum and Large Initial Data
• Global well-posedness of 2D compressible Navier-Stokes equations with large data and vacuum
• Global Well-Posedness of Classical Solutions with Large Oscillations and Vacuum to the Three-Dimensional Isentropic Compressible Navier-Stokes Equations

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