Stochastic 2D Euler Equations

• Stochastic 2D Euler equations are nonlinear PDEs modeling the evolution of incompressible fluid flows under stochastic perturbations with various noise types.
• Recent advances establish existence, uniqueness, invariant measures, and ergodic properties using analytic, geometric, and numerical techniques.
• Efficient stochastic model reduction and numerical schemes accurately capture large-scale energy spectra and coherent structures in turbulent flows.

Stochastic 2D Euler equations are nonlinear partial differential equations describing the time evolution of an incompressible, inviscid two-dimensional fluid subject to stochastic perturbations. Their significance lies in modeling turbulence, uncertainty, and statistical behaviors in fluid dynamics by coupling classical Euler mechanics with stochastic processes such as additive noise, transport-type noise, or multiplicative noise. The equations may be posed on periodic domains or bounded regions, and are formulated in velocity or vorticity variables. Advances in mathematical analysis, geometric theory, and numerical methodology have clarified existence, uniqueness, invariant measures, ergodic properties, and practical model reduction for these systems.

1. Mathematical Formulations and Stochastic Forcings

The canonical velocity form on the periodic domain $D = \mathbb{T}^2$ is

$$\mathrm{d} u + (u \cdot \nabla)u\,\mathrm{d}t + \nabla p\,\mathrm{d}t + \gamma u\,\mathrm{d}t = \mathrm{d}W_t,\quad \nabla \cdot u=0$$

where $u(x,t)$ is the velocity, $p(x,t)$ the pressure, $\gamma>0$ a linear damping coefficient, and $W_t$ is an $H$-valued Wiener process in the space of divergence-free, zero-mean vector fields. Vorticity formulation is usually preferred, yielding

$$\mathrm{d}\omega + (u \cdot \nabla)\omega\,\mathrm{d}t + \gamma \omega\,\mathrm{d}t = \mathrm{d}W_\text{curl}, \quad u = K*\omega$$

with $K$ the Biot–Savart kernel, and $\omega=\mathrm{curl}\,u$. Generalizations include additive noise (purely stochastic external forcing), multiplicative noise (linear or nonlinear in $u$ or $\omega$), and transport-type noise (entered as $$\sum_k (\sigma_k \cdot \nabla) u \circ \mathrm{d}W^k$$ in the Stratonovich sense or its Itô variant).

Transport noise preserves geometric and statistical conservation laws (such as circulation, enstrophy, and Casimirs), and is central for variational and data-driven model derivations (Cruzeiro et al., 2013, Brzeźniak et al., 2014). Linear multiplicative noise in velocity leads to stochastic weak attractors and reveals unique attractor structures (Kinra et al., 2023).

2. Well-posedness, Existence, and Regularity

Existence and uniqueness results depend crucially on the function space and regularity of the initial data. For bounded vorticity initial data $\omega_0 \in L^\infty$, the stochastic 2D Euler equation with transport-type multiplicative noise admits a unique strong pathwise solution, with the vorticity remaining bounded in $L^\infty$ for all time (Brzeźniak et al., 2014, Bessaih et al., 2019). When the noise is less regular (e.g., Kraichnan-type noise), pathwise uniqueness can be extended down to initial data in $L^1 \cap L^p$, $p>1$, via spectral energy estimates and optimal control on the nonlinearity (Jiao et al., 2024). On bounded domains or the torus, weak existence holds in $H^1$ and is established via Faedo–Galerkin approximations, tightness, and martingale characterizations (Cruzeiro et al., 2013). For measure-valued vorticity (vortex sheets, signed measures), Skorokhod–Jakubowski compactness and two-point function continuity give non-negative measure-valued weak solutions, extending Delort's deterministic theory to the stochastic setting (Brzeźniak et al., 2019).

Critical for strong results are precise summability and regularity conditions on the vector fields $\sigma_k$ driving the noise; smooth, divergence-free, and trace-class vector fields ensure conservation and a priori bounds in Sobolev spaces. Transport-type noise and commutator estimates guarantee the preservation of crucial regularity features, including $L^2$ or higher Sobolev norms (Crisan et al., 2019).

3. Markov Semigroups, Invariant Measures, and Ergodicity

Stochastic 2D Euler flows induce nontrivial Markov semigroups acting on appropriate function spaces. For bounded vorticity, the Markov property extends to the bounded weak* topology $T_{bw*}$ in $L^\infty$, and measurable functionals $\phi \in B_b(L^\infty, T_{w*})$ have Feller continuity (Bessaih et al., 2019). Krylov–Bogoliubov-type averaging, Banach–Alaoglu compactness, and Jakubowski's generalization of Prokhorov's theorem yield the existence of invariant probability measures in $(L^\infty, T_{bw*})$ for stochastically forced, damped Euler (Bessaih et al., 2019).

With fractional dissipation $\Lambda^\gamma$ and stochastic excitation, unique ergodicity holds for the broad family of fractionally dissipated stochastic Euler equations, provided the forcing is sufficiently nondegenerate and regular (Constantin et al., 2013). The invariant measures are supported on spaces of high regularity (Gevrey class), and the unique ergodic measure enjoys exponential mixing, provided the noise acts on enough Fourier modes.

Stochastic stability of invariant measures supplies a selection principle: among the infinitely many deterministic invariants (Casimirs), only those maximizing stochastic stability, as determined by boundary conditions in mode space, persist in the limit $\varepsilon \to 0$ of vanishing noise (Cipriano et al., 2018). Stochastic Euler flows can spontaneously select physically relevant coherent structures (e.g., monopoles, dipoles, jets) predicted by small-noise variational theory and observed in numerical simulation.

4. Model Reduction, Numerical Simulation, and Statistical Closure

Direct numerical simulation (DNS) of 2D Euler at high resolution is computationally demanding due to the nonlinear advection and extensive mode coupling. Sparse-stochastic model reduction techniques, such as Zeitlin su($N$) discretization with projection onto low-degree spherical harmonics and closure using stochastic corrections (SALT-type enstrophy-preserving or energy-preserving noise), yield efficient reduced models capable of reproducing large-scale dynamics (Cifani et al., 2023).

Key strategies:

• Orthogonal projection separates large-scale ($l \leq \bar l$) and small-scale ($l > \bar l$) modes, with unresolved modes modeled as band-limited Gaussian noise in modal coefficients.
• Stratonovich calculus ensures preservation of geometric invariants.
• Empirical closure parameters are statistically fitted to DNS data using quantile- and correlation-matching techniques.
• Stochastic Heun schemes provide robust time stepping.

Reduced models capture both the stationary energy spectrum and mode-to-mode energy fluxes, with computational cost savings up to $20\times$ versus full DNS. SALT closures accurately reproduce spectral properties and energy transfer, with nonzero but weak small-to-large scale energy exchange (Cifani et al., 2023).

5. Geometric and Kinetic Approaches, Fluctuations, and Turbulent Structures

The stochastic Euler equations possess geometric interpretations as stochastic geodesic flows on the group of volume-preserving diffeomorphisms, with noise encoded as stochastic parallel transport (Cruzeiro et al., 2013, Maurelli et al., 2019). In 2D, stochastic dynamics preserve vorticity along particle trajectories, maintain measure-preserving flows, and conserve Casimir-type invariants.

Kinetic theory and multi-scale averaging approach large-scale jet and flow formation in stochastically forced Euler. Reynolds stress fluctuations are analytically tractable and display Gaussian statistics; velocity covariance matrices reveal enhancement of fluctuations away from stationary points of jet profiles (Nardini et al., 2016). Gaussian fluctuations show $O(1)$ variance where local shear is nonzero, decaying to $O(\alpha)$ near extrema due to vorticity depletion and the inviscid Orr mechanism.

Stochastic model reduction using Stratonovich transport noise (SALT) is justified mathematically as a large-scale limit of a three-time-scale Euler system: fine-scale deterministic equations, intermediate Ornstein–Uhlenbeck reduction, and large-scale white-noise limit, with weak convergence of vorticity and strong convergence of velocity fields established under broad assumptions (Flandoli et al., 2021).

6. Point Vortex Dynamics, Regularization by Noise, and Measure-Valued Solutions

Stochastic perturbations of point vortex dynamics on periodic domains remove collision singularities and restore well-posedness for all initial configurations, contrasting with the deterministic case where coalescence occurs for degenerate initial data (Flandoli et al., 2010). Hypoelliptic, divergence-free noise ensures global existence, uniqueness, and no-collision property for the associated SDE system. Regularization by transport noise is a recurring phenomenon, also observed in measure-valued vorticity, vortex sheets, and stochastic extensions of Delort's theorem (Brzeźniak et al., 2019). Measure-valued solutions are constructed using compactness arguments, and convergence of the nonlinear terms is ensured by the continuity of bilinear forms on atomless, nonnegative measures.

7. Kolmogorov Equations, Statistical Solutions, and Open Problems

The Kolmogorov (backward) equation and its associated generator describe statistical evolution of functionals under the stochastic Euler flow (Flandoli et al., 2018). Galerkin truncation, Wiener chaos expansion, and gradient estimates supply existence and regularity of probability densities in $L^2(\mu)$ (enstrophy measure). For noise roughness above the critical threshold ($\gamma>2$), the statistical density evolution is well-posed with a priori bounds; at the roughness threshold, densities trivialize.

White-noise measure solutions, constructed as limits of random point vortex approximations, satisfy continuity equations in weak form, with a priori $L^2$ gradient bounds in the noise directions (Flandoli et al., 2017). Uniqueness in the infinite-dimensional continuity equation remains open, with regularity obtained via stochastic transport but drift terms remaining only distributional.

Key open problems include mixing rates and uniqueness of invariant measures, extension to more general noise types or 3D flows, long-time statistical properties (ergodicity, attractor characterization), and the rigorous derivation and justification of various stochastic closures and reductions.

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References

• Invariant measures for stochastic damped 2D Euler equations (Bessaih et al., 2019)
• Sparse-stochastic model reduction for 2D Euler equations (Cifani et al., 2023)
• Fluctuations of large-scale jets in the stochastic 2D Euler equation (Nardini et al., 2016)
• Existence and uniqueness for stochastic 2D Euler flows with bounded vorticity (Brzeźniak et al., 2014)
• On a 2D stochastic Euler equation of transport type: existence and geometric formulation (Cruzeiro et al., 2013)
• Scaling limit of stochastic 2D Euler equations with transport noises to the deterministic Navier-Stokes equations (Flandoli et al., 2019)
• Data-driven stochastic Lie transport modelling of the 2D Euler equations (Ephrati et al., 2022)
• Kolmogorov equations associated to the stochastic 2D Euler equations (Flandoli et al., 2018)
• $ρ$-white noise solution to 2D stochastic Euler equations (Flandoli et al., 2017)
• Unique Ergodicity for Fractionally Dissipated, Stochastically Forced 2D Euler Equations (Constantin et al., 2013)
• Full well-posedness of point vortex dynamics corresponding to stochastic 2D Euler equations (Flandoli et al., 2010)
• Existence for stochastic 2D Euler equations with positive $H^{-1}$ vorticity (Brzeźniak et al., 2019)
• Stationary Solutions of Damped Stochastic 2-dimensional Euler's Equation (Grotto, 2019)
• Theory of weak asymptotic autonomy of pullback stochastic weak attractors and its applications to 2D stochastic Euler equations driven by multiplicative noise (Kinra et al., 2023)
• On the pathwise uniqueness of stochastic 2D Euler equations with Kraichnan noise and $L^p$-data (Jiao et al., 2024)
• Stochastic stability of invariant measures: The 2D Euler equation (Cipriano et al., 2018)
• Incompressible Euler equations with stochastic forcing: a geometric approach (Maurelli et al., 2019)
• Well-posedness for a stochastic 2D Euler equation with transport noise (Crisan et al., 2019)