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Stochastic Compressible Euler Equations with Frictional Damping: Existence of $L^\infty$ Martingale Solutions and Asymptotic Porous Medium-Like Behavior

Rongyi Dai, Jeffrey Kuan, Krutika Tawri, Sunčica Čanić, Konstantina Trivisa

Abstract

We study the one-dimensional isentropic compressible Euler equations with linear (frictional) damping, subject to multiplicative, white-in-time stochastic forcing. The system is posed on a bounded interval with $L^\infty$ initial data and Dirichlet boundary conditions imposed on the momentum. We establish the global-in-time existence of $L^\infty$ martingale solutions that satisfy an appropriate entropy inequality. Then, we analyze the long-time behavior of these solutions and show that, under suitable assumptions on the noise, they converge almost surely and exponentially fast to a constant steady state of the system. The limiting density is well-approximated by the asymptotic solution of the deterministic porous medium equation, while the momentum exhibits the asymptotic behavior predicted by Darcy's law. The analysis in the stochastic setting is delicate, as temporal white-noise perturbations can significantly influence the long-time statistics of the solution. Our approach hinges on deriving sharp moment estimates for the entropy, which enable us to quantify and ultimately prove the decay of stochastic effects. To the best of our knowledge, this work provides the first rigorous pathwise convergence result for the long-time behavior of solutions to the stochastic isentropic compressible Euler equations with linear damping.

Stochastic Compressible Euler Equations with Frictional Damping: Existence of $L^\infty$ Martingale Solutions and Asymptotic Porous Medium-Like Behavior

Abstract

We study the one-dimensional isentropic compressible Euler equations with linear (frictional) damping, subject to multiplicative, white-in-time stochastic forcing. The system is posed on a bounded interval with initial data and Dirichlet boundary conditions imposed on the momentum. We establish the global-in-time existence of martingale solutions that satisfy an appropriate entropy inequality. Then, we analyze the long-time behavior of these solutions and show that, under suitable assumptions on the noise, they converge almost surely and exponentially fast to a constant steady state of the system. The limiting density is well-approximated by the asymptotic solution of the deterministic porous medium equation, while the momentum exhibits the asymptotic behavior predicted by Darcy's law. The analysis in the stochastic setting is delicate, as temporal white-noise perturbations can significantly influence the long-time statistics of the solution. Our approach hinges on deriving sharp moment estimates for the entropy, which enable us to quantify and ultimately prove the decay of stochastic effects. To the best of our knowledge, this work provides the first rigorous pathwise convergence result for the long-time behavior of solutions to the stochastic isentropic compressible Euler equations with linear damping.
Paper Structure (26 sections, 32 theorems, 278 equations)

This paper contains 26 sections, 32 theorems, 278 equations.

Table of Contents

  1. Introduction
  2. Problem Definition
  3. Summary of Main Results.
  4. Literature review
  5. Existence of global solution
  6. Long time behavior
  7. Organization of the paper
  8. Construction of approximate solutions in $\epsilon$ and $\tau = \Delta t$.
  9. The $\epsilon$-approximation via parabolic regularization
  10. The $\tau$-approximation via a time-splitting scheme
  11. Existence of solutions to the the deterministic and stochastic subproblems
  12. Passage to the limit as $\tau \to 0$
  13. Uniform $L^{\infty}(Q_{T})$ estimates on approximate solutions
  14. Time regularity estimates
  15. Uniform entropy dissipation estimates
  16. ...and 11 more sections

Key Result

Theorem 1.1

Assume that the initial data $U_0=(\rho_0, m_0)$ satisfies and that the noise coefficient $\sigma(x, U)$ satisfies the following conditions: Then there exists a global-in-time martingale $L^\infty$ weak entropy solution in the sense of Definition martingale_soln_defn to problem with initial data $U_0$.

Theorems & Definitions (70)

  • Definition 1.1: Martingale $L^\infty$ Weak Entropy Solution
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.1: Existence
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.2: Exponential decay to constant state
  • Theorem 1.3: Porous Medium Approximation of the Long-Time Behavior
  • Remark 1.6
  • ...and 60 more