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

1. Problem Definition

The authors investigate the one-dimensional, isentropic, compressible Euler equations subject to linear (frictional) damping and multiplicative, white-in-time stochastic forcing. The system describes the evolution of fluid density $\rho$ and momentum $m = \rho u$ on a bounded interval $x \in [0, 1]$.

The governing stochastic partial differential equations (SPDEs) are:
$$\begin{cases} d\rho + \partial_x m \, dt = 0 \\ dm + \partial_x \left( \frac{m^2}{\rho} + p(\rho) \right) dt = -\alpha m \, dt + \sigma(x, \rho, m) \, dW(t) \end{cases}$$
where:

• $p(\rho) = \kappa \rho^\gamma$ is the pressure law for a polytropic gas ($\gamma > 1$).
• $\alpha > 0$ is the frictional damping coefficient.
• $W(t)$ is a standard Brownian motion.
• $\sigma(x, \rho, m)$ is a multiplicative noise coefficient satisfying specific Lipschitz and compact support conditions.
• Boundary Conditions: Dirichlet conditions are imposed on the momentum: $m(t, 0) = m(t, 1) = 0$.
• Initial Data: $L^\infty$ initial data $(\rho_0, m_0)$ with bounded density and velocity.

The primary challenges addressed are the lack of global regularity for compressible Euler equations, the presence of vacuum states, the nonlinearity of the stochastic forcing, and the need to establish long-time asymptotic behavior in a stochastic setting.

2. Methodology

The proof strategy is divided into two main phases: Existence and Long-Time Asymptotics.

A. Existence of Global-in-Time $L^\infty$ Martingale Solutions

To prove the existence of solutions, the authors employ a two-level approximation scheme:

1. Parabolic Regularization ($\epsilon$-level):

   • The hyperbolic system is augmented with artificial viscosity terms ($\epsilon \partial_{xx} \rho$ and $\epsilon \partial_{xx} m$) to create a parabolic system.
   • Initial data and noise coefficients are regularized to ensure smoothness and compatibility with boundary conditions.

2. Time-Splitting Scheme ($\tau$-level):

   • The time interval $[0, T]$ is discretized into steps of size $\tau$.
   • A Lie-Trotter splitting scheme is used to separate the deterministic and stochastic components.
     • Deterministic Subproblem: Solves the viscous, damped Euler equations over a time step.
     • Stochastic Subproblem: Updates the momentum via the stochastic integral while keeping density constant.
   • A continuous interpolant is constructed between these sub-steps.

3. Compactness and Limit Passage:

   • Uniform $L^\infty$ Bounds: The authors utilize the theory of invariant regions (using Riemann invariants) and the compact support of the noise to derive uniform $L^\infty$ bounds for the approximate solutions, independent of $\epsilon$ and $\tau$.
   • Entropy Estimates: Sharp estimates on higher moments of the entropy are derived to control the stochastic terms.
   • Tightness & Skorohod Representation: Tightness of the laws of the approximate solutions is established in appropriate function spaces (e.g., $C_w,loc \cap H^{-2}$). The Skorohod Representation Theorem is used to obtain almost sure convergence on a new probability space.
   • Compensated Compactness: The Young measure theory and stochastic compensated compactness (extending the deterministic theory of Lions, Perthame, and Souganidis) are used to pass to the limit $\epsilon \to 0$ in the nonlinear terms, recovering the entropy inequality for the original hyperbolic system.
   • Pathwise Uniqueness: The Gyöngy-Krylov theorem is invoked to upgrade the martingale solution to a pathwise solution on the original probability space.

B. Long-Time Asymptotic Behavior

To analyze the behavior as $t \to \infty$, the authors focus on the decay of the solution toward a constant steady state.

1. Entropy Dissipation Method:

   • They define a modified entropy functional $\eta^*$ related to the mechanical energy relative to the steady state.
   • By applying Itô's formula to the regularized system, they derive an entropy balance equation (equality holds for the regularized system, inequality for the limit).

2. Decay Estimates:

   • Expectation Level: They first prove exponential decay of the $L^2$ norm of the solution in expectation ($\mathbb{E}[\|\cdot\|_{L^2}^2]$) using Gronwall's lemma, provided the noise intensity is sufficiently small relative to the damping.
   • Pathwise Level: To move from expectation to almost sure convergence, they derive fourth-moment estimates for the entropy.
   • Stochastic Integral Control: A crucial step involves bounding the stochastic integral term in the energy equality. They utilize a recent result by Yuskovych [58] regarding the almost sure decay of stochastic integrals. By showing the integrand satisfies specific growth conditions (via the fourth-moment bounds), they prove the stochastic noise contribution vanishes almost surely as $t \to \infty$.

3. Porous Medium Approximation:

   • The authors demonstrate that the long-time limit of the stochastic Euler system is well-approximated by the solution to the deterministic Porous Medium Equation (PME):
     $$\partial_t \rho = \partial_{xx} p(\rho)$$
   • The momentum converges to the profile predicted by Darcy's Law ($m \approx -\frac{1}{\alpha} \partial_x p(\rho)$).

3. Key Contributions

1. First Pathwise Convergence Result: This is the first rigorous proof of almost sure (pathwise) exponential convergence to a steady state for the stochastic isentropic compressible Euler equations with linear damping. Previous literature on stochastic compressible fluids mostly focused on the existence of invariant measures or statistical stationarity without proving trajectory convergence.
2. $L^\infty$ Martingale Solutions: The authors establish the existence of global-in-time martingale solutions with $L^\infty$ bounds on a bounded domain with Dirichlet boundary conditions. This is a significant improvement over previous works (e.g., Berthelin & Vovelle) which often dealt with $L^2$ data or periodic domains.
3. Novel Technical Framework:
   • Development of a continuous Lie-Trotter splitting scheme adapted for stochastic PDEs.
   • Derivation of sharp higher-moment entropy estimates to control stochastic noise in the long-time limit.
   • Application of Yuskovych's theorem to prove the vanishing of stochastic effects in the asymptotic regime.
4. Connection to Porous Medium Equation: The work rigorously justifies that the long-time dynamics of the stochastically damped Euler system are governed by the deterministic porous medium equation, extending deterministic results (Pan & Zhao) to the stochastic setting.

4. Main Results

• Theorem 1.1 (Existence): Under assumptions of bounded initial data and a noise coefficient with compact support and small Lipschitz constant, there exists a global-in-time martingale $L^\infty$ weak entropy solution to the system.
• Theorem 1.2 (Exponential Decay): The solution $(\rho, m)$ converges almost surely and exponentially fast to a constant steady state $(\rho^*, 0)$, where $\rho^*$ is the total initial mass. Specifically:
  $$\int_0^1 \left( (\rho(t) - \rho^*)^2 + m(t)^2 \right) dx \leq C_0 e^{-rt} \quad \text{a.s.}$$
• Theorem 1.3 (Porous Medium Approximation): The stochastic solution is asymptotically well-approximated by the classical solution of the deterministic porous medium equation with the same initial mass. The error between the stochastic Euler solution and the porous medium solution decays exponentially in time.

5. Significance

This work bridges a critical gap between deterministic fluid dynamics and stochastic analysis.

• Physical Relevance: It provides a rigorous mathematical foundation for understanding how random fluctuations (e.g., in blood flow or atmospheric models) affect the long-term stability of compressible flows. It confirms that strong frictional damping can suppress stochastic perturbations, leading to a deterministic-like equilibrium.
• Mathematical Advancement: The techniques developed for handling the interplay between $L^\infty$ bounds, stochastic noise, and long-time decay offer a new toolkit for analyzing other stochastic hyperbolic systems. The result challenges the notion that stochastic systems only possess statistical equilibria, demonstrating that pathwise convergence to a deterministic state is possible under specific damping-noise balance conditions.