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Inertial Limit of global weak solutions for Compressible Navier--Stokes

Cheng Yu

Abstract

We investigate the inertial limit of the compressible Navier--Stokes system posed on the $3$-dimensional torus, and allowing for regions of vacuum. Considering global-in-time finite-energy weak solutions of a scaled system, we rigorously establish convergence to a limiting system in which the momentum equation reduces to a stationary elliptic balance between pressure and viscous forces. In this limit, the scaled kinetic energy vanishes, reflecting an overdamped regime, and the limiting weak solution satisfies an exact energy equality. Our analysis relies on uniform a priori estimates, renormalized techniques, and compactness arguments in the Lions--Feireisl framework, providing a mathematically rigorous analysis for the overdamped dynamics arising from vanishing inertia in compressible viscous flows.

Inertial Limit of global weak solutions for Compressible Navier--Stokes

Abstract

We investigate the inertial limit of the compressible Navier--Stokes system posed on the -dimensional torus, and allowing for regions of vacuum. Considering global-in-time finite-energy weak solutions of a scaled system, we rigorously establish convergence to a limiting system in which the momentum equation reduces to a stationary elliptic balance between pressure and viscous forces. In this limit, the scaled kinetic energy vanishes, reflecting an overdamped regime, and the limiting weak solution satisfies an exact energy equality. Our analysis relies on uniform a priori estimates, renormalized techniques, and compactness arguments in the Lions--Feireisl framework, providing a mathematically rigorous analysis for the overdamped dynamics arising from vanishing inertia in compressible viscous flows.
Paper Structure (6 sections, 5 theorems, 68 equations)

This paper contains 6 sections, 5 theorems, 68 equations.

Table of Contents

  1. Introduction and overview
  2. Definitions
  3. Main Result
  4. Weak compactness as $\varepsilon\to0$
  5. Energy equality for the limit system
  6. Proof to the Main theorem

Key Result

Theorem 1.1

For any $\gamma>\frac{3}{2}$, let $(\rho_\varepsilon, u_\varepsilon)$ be a finite-energy weak solution of eq:CNS-scaled-initial data on $[0,T]\times \mathbb{T}^3$. Then there exists a subsequence (still denoted by $\varepsilon\to0$) such that and the weak limit $(\rho, u)$ is a weak solution of system of limit and initial data for limit system on $[0,T]\times\mathbb{T}^3$, and the weak solution s

Theorems & Definitions (11)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.1: Inertial limit of finite-energy weak solutions
  • Remark 1.1
  • Lemma 2.1
  • Proposition 2.1
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.1
  • ...and 1 more