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Sufficient Conditions for the Energy Balance for the Stochastic Incompressible Euler Equations with Additive Noise in two Space Dimensions

Tobias Rohner, Franziska Weber

TL;DR

The paper addresses the problem of when the stochastic two-dimensional incompressible Euler equations with additive noise satisfy an energy balance in mean, viewed as the vanishing viscosity limit of stochastic Navier–Stokes solutions. It develops a probabilistic compactness framework: a uniform modulus-of-continuity bound on time-integrated structure functions yields tightness of NS laws in $L^2$, enabling a Skorokhod construction to obtain a martingale Euler solution that honors energy balance in mean. The authors establish both directions of a tightness–energy-balance correspondence, adapting deterministic tools to the stochastic setting via stopping times and enhanced Poincaré-type lemmas, and they validate the theory with numerical experiments (flat vortex sheet and fractional Brownian bridge initial data) showing energy input matching the theoretical predictions. The results illuminate when energy dissipation artifacts vanish in the zero-viscosity limit and provide practical guidelines for numerical approximations of stochastic Euler in 2D. This advances understanding of energy conservation in stochastic turbulence models and informs the design of vanishing-viscosity schemes under additive noise.

Abstract

We consider vanishing viscosity approximations to solutions of the stochastic incompressible Euler equations in two space dimensions with additive noise. We identify sufficient and necessary conditions under which martingale solutions of the stochastic Euler equations satisfy an exact energy balance in mean. We find that the tightness of the laws of the approximating sequence of solutions of the stochastic Navier-Stokes equations in $L^2([0,T]\times D)$ is equivalent to the limiting martingale solution satisfying an energy balance in mean. Numerical simulations illustrate the theoretical findings.

Sufficient Conditions for the Energy Balance for the Stochastic Incompressible Euler Equations with Additive Noise in two Space Dimensions

TL;DR

The paper addresses the problem of when the stochastic two-dimensional incompressible Euler equations with additive noise satisfy an energy balance in mean, viewed as the vanishing viscosity limit of stochastic Navier–Stokes solutions. It develops a probabilistic compactness framework: a uniform modulus-of-continuity bound on time-integrated structure functions yields tightness of NS laws in , enabling a Skorokhod construction to obtain a martingale Euler solution that honors energy balance in mean. The authors establish both directions of a tightness–energy-balance correspondence, adapting deterministic tools to the stochastic setting via stopping times and enhanced Poincaré-type lemmas, and they validate the theory with numerical experiments (flat vortex sheet and fractional Brownian bridge initial data) showing energy input matching the theoretical predictions. The results illuminate when energy dissipation artifacts vanish in the zero-viscosity limit and provide practical guidelines for numerical approximations of stochastic Euler in 2D. This advances understanding of energy conservation in stochastic turbulence models and informs the design of vanishing-viscosity schemes under additive noise.

Abstract

We consider vanishing viscosity approximations to solutions of the stochastic incompressible Euler equations in two space dimensions with additive noise. We identify sufficient and necessary conditions under which martingale solutions of the stochastic Euler equations satisfy an exact energy balance in mean. We find that the tightness of the laws of the approximating sequence of solutions of the stochastic Navier-Stokes equations in is equivalent to the limiting martingale solution satisfying an energy balance in mean. Numerical simulations illustrate the theoretical findings.

Paper Structure

This paper contains 8 sections, 8 theorems, 163 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminary results
  3. Energy balance
  4. A sufficient condition for an energy balance in mean
  5. Numerical Experiments
  6. Flat Vortex Sheet
  7. Fractional Brownian Bridges
  8. Auxiliary results

Key Result

Theorem 2.10

Let $\{v_n\}_{n\in \mathbb{N}}$ be a bounded sequence of functions in $L^2(D)$ where $D\subset \mathbb{R}^d$ is bounded, $d\in \mathbb{N}$. Assume that for all $n\in \mathbb{N}$ for some modulus of continuity $\phi$. Then $\{v_n\}_{n\in \mathbb{N}}$ has a subsequence that is convergent in $L^2(D)$.

Figures (4)

  • Figure 1: Behavior of key quantities of interest for the flat vortex sheet experiment. The top row shows the results for $\sigma = 0.01$, while the bottom row does so for $\sigma = 0.1$. The plots depict from left to right: Second-order time-integrated structure function, measured and predicted energy input due to forcing, total kinetic energy, energy dissipation due to viscosity. The shaded region in the plots depicts one standard deviation in the Monte Carlo approximation of the mean. One can see that the structure functions clearly have a uniform modulus of continuity. Therefore, the experiments are expected to satisfy the energy balance equation. This is verified by the plots in the second column, demonstrating that the actual energy input perfectly follows the predicted one.
  • Figure 2: Results of the fractional Brownian bridge experiment with Hurst index $H = 0.15$. The plots in the top row correspond to forcing strength $\sigma = 0.01$, and the bottom row to $\sigma = 0.1$. Furthermore, the depicted quantities from left to right are: Second-order time-integrated structure functions, measured and predicted energy input due to forcing, total kinetic energy, energy dissipation due to viscosity. The shaded region depicts one standard deviation around the Monte Carlo approximation of the mean.
  • Figure 3: Results of the fractional Brownian bridge experiment with Hurst index $H = 0.50$. The plots in the top row correspond to forcing strength $\sigma = 0.01$, and the bottom row to $\sigma = 0.1$. Furthermore, the depicted quantities from left to right are: Second-order time-integrated structure functions, measured and predicted energy input due to forcing, total kinetic energy, energy dissipation due to viscosity. The shaded region depicts one standard deviation around the Monte Carlo approximation of the mean.
  • Figure 4: Results of the fractional Brownian bridge experiment with Hurst index $H = 0.75$. The plots in the top row correspond to forcing strength $\sigma = 0.01$, and the bottom row to $\sigma = 0.1$. Furthermore, the depicted quantities from left to right are: Second-order time-integrated structure functions, measured and predicted energy input due to forcing, total kinetic energy, energy dissipation due to viscosity. The shaded region depicts one standard deviation around the Monte Carlo approximation of the mean.

Theorems & Definitions (27)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3: Solutions of the stochastic Navier-Stokes equations Kuksin2012
  • Remark 2.4
  • Definition 2.5: Solutions of the stochastic Euler equations in the probabilistically strong sense Kuksin2012
  • Definition 2.6: Martingale solutions of Euler equations
  • Remark 2.7
  • Definition 2.8: Structure functions
  • Definition 2.9: Modulus of continuity
  • Theorem 2.10: Fréchet-Kolmogorov theorem
  • ...and 17 more