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Global Existence for General Systems of Isentropic Gas Dynamics via a Weighted Pressure Perturbation Approach

Kewang Chen

TL;DR

This work tackles global existence for the 1D isentropic Euler equations with a general pressure law in the presence of vacuum. It introduces a mass-conserving regularization through a weighted pressure perturbation $\tilde{P}(\rho,\delta)$ that vanishes at the cut-off density $\delta$, ensuring a degenerate hyperbolic boundary while preserving convexity and genuine nonlinearity. The approach combines vanishing viscosity, invariant-region estimates, and $H^{-1}$ compactness of entropy dissipation to obtain global weak entropy solutions for fixed $\delta$, then uses a singular perturbation framework and a near-vacuum structural condition to pass to the vacuum limit $\delta\to0$, yielding a global weak entropy solution to the original system that may admit vacuum. The results provide a physically natural, mass-conserving regularization that handles vacuum for general equations of state and establishes a robust pathway to vacuum limits via compensated compactness.

Abstract

This paper establishes the global existence of bounded entropy solutions to the one-dimensional system of isentropic gas dynamics with general pressure laws, allowing for the presence of vacuum states. We introduce a novel regularization scheme based on a weighted pressure perturbation of the form $\tilde{P}(ρ,δ) = \int_δ^ρs^{-2}(s^2P'(s) - δ^2 P'(δ)) \, ds$. Unlike previous methods that modify the continuity equation, our approach strictly preserves mass conservation while creating a degenerate hyperbolic boundary at the cut-off density $ρ=δ$. We provide a rigorous derivation of the eigenstructure and Riemann invariants for the perturbed system. A key theoretical contribution is the proof that the perturbed pressure exactly inherits the convexity properties of the original state equation ($2\tilde{P}'+ρ\tilde{P}'' = 2P'+ρP''$), thereby ensuring the strict convexity of the mechanical energy entropy without additional assumptions. Using the method of vanishing viscosity, we derive uniform $L^\infty$ bounds via invariant regions and establish the $H^{-1}_{\mathrm{loc}}$ compactness of entropy dissipation measures. The global existence for the vacuum-free limit is then proved using the theory of compensated compactness and singular perturbation analysis of the entropy equations.

Global Existence for General Systems of Isentropic Gas Dynamics via a Weighted Pressure Perturbation Approach

TL;DR

This work tackles global existence for the 1D isentropic Euler equations with a general pressure law in the presence of vacuum. It introduces a mass-conserving regularization through a weighted pressure perturbation that vanishes at the cut-off density , ensuring a degenerate hyperbolic boundary while preserving convexity and genuine nonlinearity. The approach combines vanishing viscosity, invariant-region estimates, and compactness of entropy dissipation to obtain global weak entropy solutions for fixed , then uses a singular perturbation framework and a near-vacuum structural condition to pass to the vacuum limit , yielding a global weak entropy solution to the original system that may admit vacuum. The results provide a physically natural, mass-conserving regularization that handles vacuum for general equations of state and establishes a robust pathway to vacuum limits via compensated compactness.

Abstract

This paper establishes the global existence of bounded entropy solutions to the one-dimensional system of isentropic gas dynamics with general pressure laws, allowing for the presence of vacuum states. We introduce a novel regularization scheme based on a weighted pressure perturbation of the form . Unlike previous methods that modify the continuity equation, our approach strictly preserves mass conservation while creating a degenerate hyperbolic boundary at the cut-off density . We provide a rigorous derivation of the eigenstructure and Riemann invariants for the perturbed system. A key theoretical contribution is the proof that the perturbed pressure exactly inherits the convexity properties of the original state equation (), thereby ensuring the strict convexity of the mechanical energy entropy without additional assumptions. Using the method of vanishing viscosity, we derive uniform bounds via invariant regions and establish the compactness of entropy dissipation measures. The global existence for the vacuum-free limit is then proved using the theory of compensated compactness and singular perturbation analysis of the entropy equations.
Paper Structure (34 sections, 6 theorems, 82 equations, 1 table)

This paper contains 34 sections, 6 theorems, 82 equations, 1 table.

Table of Contents

  1. Introduction
  2. Structural Analysis of the Perturbed System
  3. Properties of the Weighted Pressure
  4. Eigenstructure and Riemann Invariants
  5. Entropy Analysis
  6. Construction of the Mechanical Energy Entropy Pair
  7. Viscous Regularization and A Priori Estimates
  8. Evolution of Riemann Invariants
  9. Step 1: Rewrite the momentum equation.
  10. Step 2: Compute $w_t+\lambda_2 w_x$.
  11. Step 3: Exact cancellation of inviscid terms.
  12. Step 4: Express $u_{xx}$ via $w_{xx}$.
  13. Step 5: Final evolution equation.
  14. Step 6: Positivity of $K(\rho)$.
  15. Invariant Regions and $L^\infty$ Bounds
  16. ...and 19 more sections

Key Result

Lemma 2.1

If $P(\rho)$ satisfies conditions eq:pressure_assumptions, then for all $\rho > \delta$:

Theorems & Definitions (20)

  • Remark 1.1
  • Lemma 2.1: Inherited Convexity
  • proof
  • Remark 2.2: Consistency with Original Riemann Invariants
  • Proposition 2.3: Explicit Riemann Invariants for Perturbed Polytropic Gas
  • proof
  • Remark 2.4: On the Conservation of Mass and Eigenvalue Degeneracy
  • proof : Verification of the Entropy Pair
  • Theorem 3.1: Strict Convexity
  • proof
  • ...and 10 more