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Global Martingale Entropy Solutions to the Stochastic Isentropic Euler Equations

Gui-Qiang G. Chen, Feimin Huang, Danli Wang

TL;DR

The paper proves the global existence of martingale entropy solutions with finite relative-energy for the stochastic isentropic Euler equations under a general pressure law, addressing the lack of uniform $L^{\infty}$ bounds caused by stochastic forcing via a stochastic $L^p$ compensated compactness framework. The authors construct solutions by solving a stochastic parabolic regularization, obtaining uniform energy and higher-integrability estimates, and passing to the limit through Jakubowski-Skorokhod representations and random Young measures, with a stochastic div-curl argument yielding a Tartar commutation relation. In the polytropic (gamma-law) case, sharper results are achieved, including a local mechanical energy inequality and higher-order relative-energy control, enabling entropy inequalities for a broader class of entropies and compactness of the solution sequence. The framework developed for the stochastic setting generalizes known deterministic compensated compactness techniques and provides tools potentially extensible to multi-dimensional radial settings and other stochastic hyperbolic systems. Overall, the work significantly advances the stochastic theory for compressible flows by delivering rigorous global existence, energy, and entropy structure under general pressure laws.

Abstract

We establish the existence and compactness of global martingale entropy solutions with finite relative-energy for the stochastically forced system of isentropic Euler equations governed by a general pressure law. To achieve these, a stochastic compensated compactness framework in $L^p$ is developed to overcome the difficulty that the uniform $L^{\infty}$ bound for the stochastic approximate solutions is unavailable, owing to the stochastic forcing term. The convergence of the vanishing viscosity method is established by employing the stochastic compactness framework, along with careful uniform estimates of the stochastic approximate solutions, to obtain the existence of global martingale entropy solutions with finite relative-energy. In particular, in the polytropic pressure case for all adiabatic exponents, we prove that the global solutions satisfy the local mechanical energy inequality when the initial data are only required to have finite relative-energy (while the higher moment estimates for entropy are not required here, as needed in the earlier work). Higher-order relative energy estimates for approximate solutions are also derived to establish the entropy inequality for more convex entropy pairs and to then prove the compactness of solutions to the stochastic isentropic Euler system. The stochastic compensated compactness framework and the uniform estimate techniques for approximate solutions developed in this paper should be useful in the study of other similar problems.

Global Martingale Entropy Solutions to the Stochastic Isentropic Euler Equations

TL;DR

The paper proves the global existence of martingale entropy solutions with finite relative-energy for the stochastic isentropic Euler equations under a general pressure law, addressing the lack of uniform bounds caused by stochastic forcing via a stochastic compensated compactness framework. The authors construct solutions by solving a stochastic parabolic regularization, obtaining uniform energy and higher-integrability estimates, and passing to the limit through Jakubowski-Skorokhod representations and random Young measures, with a stochastic div-curl argument yielding a Tartar commutation relation. In the polytropic (gamma-law) case, sharper results are achieved, including a local mechanical energy inequality and higher-order relative-energy control, enabling entropy inequalities for a broader class of entropies and compactness of the solution sequence. The framework developed for the stochastic setting generalizes known deterministic compensated compactness techniques and provides tools potentially extensible to multi-dimensional radial settings and other stochastic hyperbolic systems. Overall, the work significantly advances the stochastic theory for compressible flows by delivering rigorous global existence, energy, and entropy structure under general pressure laws.

Abstract

We establish the existence and compactness of global martingale entropy solutions with finite relative-energy for the stochastically forced system of isentropic Euler equations governed by a general pressure law. To achieve these, a stochastic compensated compactness framework in is developed to overcome the difficulty that the uniform bound for the stochastic approximate solutions is unavailable, owing to the stochastic forcing term. The convergence of the vanishing viscosity method is established by employing the stochastic compactness framework, along with careful uniform estimates of the stochastic approximate solutions, to obtain the existence of global martingale entropy solutions with finite relative-energy. In particular, in the polytropic pressure case for all adiabatic exponents, we prove that the global solutions satisfy the local mechanical energy inequality when the initial data are only required to have finite relative-energy (while the higher moment estimates for entropy are not required here, as needed in the earlier work). Higher-order relative energy estimates for approximate solutions are also derived to establish the entropy inequality for more convex entropy pairs and to then prove the compactness of solutions to the stochastic isentropic Euler system. The stochastic compensated compactness framework and the uniform estimate techniques for approximate solutions developed in this paper should be useful in the study of other similar problems.
Paper Structure (41 sections, 75 theorems, 460 equations)

This paper contains 41 sections, 75 theorems, 460 equations.

Table of Contents

  1. Introduction
  2. Mathematical Problems and Main Theorems
  3. Stochastic Isentropic Euler Equations
  4. Entropy Functions
  5. Properties of the General Pressure Function and the Relative Internal Energy Function
  6. Stochastic Forcing
  7. Martingale Solutions and Main Theorems
  8. Martingale solutions with finite relative-energy
  9. Main Theorems: Theorem \ref{['thm-well-posedness-for-euler-on-whole-space-deterministic-initial-data-general-pressure-law']} and Theorem \ref{['thm-better-well-posedness-for-euler-on-whole-space-deterministic-initial-data-general-pressure-law']}
  10. Stochastic Parabolic Approximation
  11. Well-Posedness of the Stochastic Parabolic Approximation
  12. Uniform Estimates of the Stochastic Parabolic Approximation
  13. Higher integrability of the density
  14. Higher integrability of the velocity
  15. Tightness of the Parabolic Approximation
  16. ...and 26 more sections

Key Result

Lemma 2.1

For sufficiently small $\rho_{\star}$ in eq-general-pressure-law-1 and sufficiently large $\rho^{\star}$ in eq-general-pressure-law-2, the following properties hold : where, for $i=1,2$, $\,\,\theta_i=\frac{\gamma_i-1}{2}$, the positive constants $\underline{C}_i$ and $\overline{C}_i$ depend only on $\gamma_i$ respectively, and $C=C(\gamma_1,\gamma_2,\kappa_1,\kappa_2,\rho_{\star},\rho^{\star})>0

Theorems & Definitions (128)

  • Lemma 2.1: chenhuangLiWangWang24CMP, Lemma 3.1
  • Lemma 2.2
  • Remark 2.1
  • Remark 2.2
  • Definition 2.1
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.1
  • Remark 2.6
  • ...and 118 more