Navier-Stokes Equations with Transport Noise

• The paper shows that stochastic transport noise yields effective eddy viscosity, converging the dynamics of Navier-Stokes equations toward a deterministic limit.
• It demonstrates the impact of noise types, including Itô, Stratonovich, Marcus, and Lévy, on energy estimates, statistical mixing, and solution uniqueness.
• Rigorous techniques such as Galerkin approximations, energy estimates, and tightness in Skorokhod space underpin the analysis of convergence and regularity.

The Navier-Stokes equations with transport noise encompass a family of stochastic partial differential equations where the fluid velocity (or its vorticity) evolves under deterministic nonlinear dynamics, conventional viscous dissipation, and an additional transport-type stochastic perturbation. This noise is typically modeled by random Lie derivatives or advection operators acting on the solution, driven by Brownian or more general Lévy processes, and interpreted in Stratonovich, Itô, Marcus, or rough path senses. The central role of such noise is to model unresolved small-scale motions and to analyze their impact on dissipation, regularity properties, statistical mixing, and (in some regimes) the uniqueness of solutions.

1. Mathematical Formulations and Types of Transport Noise

The class of Navier-Stokes equations with transport noise takes several forms, depending on domain, spatial dimension, compressibility, and the nature of the random perturbation.

1.1 Incompressible 2D and 3D Navier-Stokes Systems

For incompressible flow on the torus $\mathbb T^d$, the velocity formulation with Stratonovich transport noise reads: $$du + (u\cdot\nabla)u\,dt + \nabla p\,dt = \nu\Delta u\,dt + \sum_{k}(\sigma_k\cdot\nabla)u\circ dW^k_t,$$ with divergence-free $\sigma_k$, typically expressed in terms of Fourier basis elements or spatial modes. In the vorticity formulation (especially in 2D), the stochastic Euler equation with Lévy transport noise in Marcus form is

$$d\xi + (u\cdot\nabla\xi)\,dt + \sum_k \theta_k\,\sigma_k\cdot\nabla\xi \diamond dZ^k_t = 0,$$

where $Z^k_t$ are Lévy processes and $\diamond$ denotes the Marcus integral, which preserves the Newton-Leibniz chain rule for jump SDEs.

1.2 Compressible Systems and Generalizations

For compressible fluids (density $\rho$, velocity $u$), transport noise enters both the continuity and momentum equations: $$\begin{aligned} & d\rho + \mathrm{div}(\rho u)\,dt = -\sum_k \mathrm{div}(\rho Q_k) \circ dW^k_t, \ & d(\rho u) + \mathrm{div}(\rho u\otimes u)\,dt + \nabla p(\rho)\,dt = \mathrm{div}S(\nabla u)\,dt - \sum_k \mathrm{div}(\rho u\otimes Q_k)\circ dW^k_t, \end{aligned}$$ where $Q_k$ are divergence-free, smooth vector fields representing spatial noise structure.

1.3 Marcus and Lévy-Type Transport Noise

The formulation with Marcus/Lévy noise extends the classical Gaussian noise by allowing finite or infinite activity jumps with prescribed scaling, preserving an explicit chain rule. The stochastic evolution then captures instantaneous strong random stirring events—mathematically, discontinuous stochastic flows.

2. Scaling Regimes, Corrector Limits, and Emergence of Eddy Viscosity

A foundational result, first in the Wiener (Gaussian) and then in the Lévy jump context (Luo et al., 15 Oct 2025, Flandoli et al., 2019), is that under "high-frequency" or "vanishing-amplitude" scaling of transport noise coefficients, solutions to the stochastic Euler equations converge in law to those of deterministic Navier–Stokes equations with an effective ("eddy") viscosity: $$\partial_t \xi + u\cdot\nabla\xi = \kappa\Delta\xi,$$ where $\kappa$ is determined by the noise's effective quadratic variation or second moment.

Scaling Regime:

• Take coefficients $\theta^n$ satisfying $\|\theta^n\|_{\ell^2}=1$, $\|\theta^n\|_{\ell^\infty} \to 0$ as $n\to\infty$.
• For Lévy noise, the eddy viscosity is given by:

$$\kappa = C_2 \int_{|z|\leq1} z^2\,\nu(dz),\qquad \sum_k (\theta_k^n)^2 (\sigma_k\otimes\sigma_k) = 2C_2 I_2.$$

• In the scaling limit, the nonlocal Lévy-corrector term imparts a Laplacian effect of the correct magnitude, and the random martingale terms vanish.
• Existence, uniqueness, and tightness of solutions under this scaling are proved via energy methods, tightness in Skorokhod space, and identification of limit points.

This mechanism rigorously extends the Itô–Stratonovich corrector phenomenon to the discontinuous case and provides a precise pathway by which unresolved stochastic advection at small scales emerges in the macroscopic description as eddy viscosity.

3. Analytical Techniques and Proof Strategies

The theories developed to analyze these systems rely on advanced stochastic and deterministic PDE techniques:

• Energy Estimates: The skew-adjoint, divergence-free structure of transport operators guarantees the preservation or dissipation of $L^2$-norm, enabling uniform energy bounds even in the presence of strong nonlinearity and singular noise (Luo et al., 15 Oct 2025).
• Compactness: The Aldous condition (for Skorokhod path space) and Sobolev embeddings control increments and ensure tightness of laws in suitable topology.
• Identification of Limits: By exploiting Taylor expansions and cancellations in the flow-pullback structure, Laplacian terms emerge as leading-order correctors in the noise-induced drift. Remaining remainder terms are controlled by the vanishing amplitude of $\theta^n$.
• Weak/Strong Convergence: Prokhorov's and Skorokhod's theorems enable passage from tightness to convergence in law and, under uniqueness of limit equations, to convergence in probability.
• Marcus/Itô/Stratonovich Calculus for Jumps: In the presence of Lévy noise, the Marcus formulation ensures pathwise uniqueness and the correct chain rule, crucial for nonlinear transfer to the deterministic limit.

4. Implications for Fluid Dynamics: Physical and Modeling Aspects

These results provide foundational justification for viewing the effect of small-scale random stirring as enhanced viscosity at macroscopic scales, rigorously confirming the Boussinesq hypothesis in the context of physically motivated stochastic models.

• Eddy Viscosity Generation: Both continuous and jump (Lévy) transport noise imparts a positive-definite dissipative correction, quantifiable via explicit formulae involving the noise's distributional parameters.
• Recovery of Deterministic Navier–Stokes: In the high-frequency or vanishing-noise limit, stochastic kinematic effects vanish, leaving only the enhanced dissipation.
• Structure-Preserving Properties: Incompressible transport noise preserves critical invariants (e.g., Casimirs in 2D Euler) at each realization, and only introduces dissipation at the collective scale.
• Robustness: The method is robust to the precise mode structure of noise, relying only on isotropy and suitable normalization of the energy distribution in the noise coefficients.

5. Comparison to Gaussian/Itô Case and Broader Stochastic Models

While previous studies focused on diffusion-type (Itô/Stratonovich) noise, the extension to Lévy and Marcus settings introduces several new phenomena:

• Discontinuous (Jump) Effects: Pure-jump Lévy noise, with finite second moment, achieves the same macroscopic limit under suitable scaling as the Gaussian case.
• Energy-conservative Structure: The Marcus calculus preserves the stochastic chain rule, ensuring proper energy balances carry over.
• General Applicability: The approach applies to systems driven by both Brownian and heavy-tailed noise, indicating universality in the passage from fine-scale random transport to viscous dissipation.

This suggests that the connection between stochastic small-scale advection and effective macroscopic viscosity does not depend on specific features of the noise, provided the basic symmetry and moment conditions hold.

6. Assumptions, Technical Conditions, and Limitations

The rigorous convergence relies on:

• Initial data in $L^2(\mathbb T^2)$.
• Noise coefficients smooth, divergence-free, finitely many Fourier modes (for technical construction).
• Radially symmetric coefficient sequences $\theta^n$ with normalization and amplitude decay.
• Lévy measure $\nu$ with support in $\{|z|\le1\}$ and finite second moment.
• Solutions constructed by Galerkin approximation and compactness arguments in Skorokhod space.
• Validity of the Marcus calculus for jumps; Newton-Leibniz chain rule in the stochastic setting.

Table: Key Features of the Model and Convergence Result

Component	Description/Assumption	Role in Analysis
Initial Data	$\xi_0\in L^2(\mathbb T^2)$	Ensures energy estimates; well-posedness
Noise Profile	Smooth, divergence-free Fourier modes $\sigma_k$	Skew-symmetric advection, maintains structure
Lévy Process	Pure-jump, i.i.d., finite 2nd moment ($\nu(\{0\})=0$)	Controls dissipation via jump intensity
Scaling	$\|\theta^n\|_{\ell^2}=1$, $\|\theta^n\|_{\ell^\infty}\to0$	High-frequency, vanishing amplitude limit
Limit Equation	Deterministic 2D Navier-Stokes with eddy viscosity $\kappa$	Edifies macroscopic dissipation
Key Mechanism	Taylor expansion and Laplacian emergence in Lévy corrector	Converts quadratic noise effect to diffusion

7. Broader Context and Future Directions

The work on Navier–Stokes with transport noise fundamentally extends the physical understanding of viscosity as an emergent, statistical property induced by underlying random, small-scale particle motions.

• Extension to Anisotropic/Non-Gaussian Models: The methods suggest similar eddy viscosity effects should arise under anisotropic or heavier-tailed noise, subject to symmetry and moment conditions.
• Beyond 2D, to 3D and Compressible Regimes: The strategy motivates further paper in higher dimensions and compressible models where compactness and regularity properties are more challenging.
• Numerics and Modeling: The rigorous analysis provides justification for parameterizing subgrid-scale transport in simulations and for the calibration of stochastic models in data assimilation and geophysical applications.

Future work may consider the impact of boundary layers, extension to non-torus domains, coupling with diffusion of other conserved quantities, and the relation to rough path and regularity structure approaches for more singular stochastic fluid systems.