Anisotropic Inviscid Limit for the Navier-Stokes Equations with Transport Noise Between Two Plates

1. Problem Statement

The paper investigates the inviscid limit (vanishing viscosity) of the 3D stochastic Navier-Stokes equations posed on a domain $\mathcal{O} = \mathbb{T}^2 \times (0, 1)$ (a channel between two horizontal plates) with Dirichlet no-slip boundary conditions ($u=0$ on $\partial\mathcal{O}$).

The core novelty lies in the anisotropic scaling of the viscosity and the noise:

• Viscosity: The turbulent viscosity is split into horizontal ($\nu_h$) and vertical ($\nu_z$) components. The limit is taken such that $\nu_h \to 0$ and the ratio $\nu_z/\nu_h \to 0$. This models geophysical flows where horizontal diffusion dominates vertical diffusion due to the Coriolis force.
• Noise: The system is driven by transport-stretching noise (Stratonovich integral), derived from the Stochastic Advection by Lie Transport (SALT) framework. Crucially, the noise is scaled anisotropically: the horizontal components of the noise are scaled by $\sqrt{\nu_h}$ and the vertical components by $\sqrt{\nu_z}$.
• The Challenge: The anisotropic scaling of the noise destroys the divergence-free property of the spatial correlation functions ($\nabla \cdot \tilde{\xi}_i \neq 0$). In standard stochastic fluid dynamics, the divergence-free property is essential for energy estimates to cancel out the Itô-Stratonovich correction terms. Its absence creates a significant analytical hurdle.

The goal is to prove that a sequence of weak martingale solutions to this stochastic system converges to the strong solution of the deterministic Euler equation (with slip boundary condition $w \cdot n = 0$) on the lifetime of the Euler solution.

2. Methodology

The proof employs a combination of stochastic analysis, functional analysis, and careful energy estimates adapted to the anisotropic setting.

A. Mathematical Framework

• Equation Conversion: The Stratonovich equation is converted to Itô form. This introduces an Itô-Stratonovich correction term involving the operator $\tilde{G}^2$. Due to the anisotropic noise, this term generates cross-terms between horizontal and vertical derivatives.
• Boundary Corrector: Following Masmoudi's deterministic work [55], the author constructs a boundary corrector $B$. This function is supported near the plates ($z=0$ and $z=1$) and is designed such that $w + B$ satisfies the no-slip boundary condition. The difference $v = u - w - B$ is then analyzed.
• Galerkin Approximation: Since weak solutions to 3D Navier-Stokes are not unique, the author constructs martingale weak solutions via Galerkin approximations. To handle the lack of a single probability space for all viscosity parameters, the author constructs a product probability space supporting a sequence of solutions $(u^k, W^k)$.

B. Handling the Non-Divergence-Free Noise

The most significant technical innovation is addressing the term $\langle (\tilde{\xi}_i \cdot \nabla) f, g \rangle$.

• In standard cases where $\nabla \cdot \xi_i = 0$, the term $\langle (\xi_i \cdot \nabla)(\xi_i \cdot \nabla)u, u \rangle + \|(\xi_i \cdot \nabla)u\|^2$ vanishes.
• Here, because $\nabla \cdot \tilde{\xi}_i \neq 0$, this term does not vanish. The author derives a bound using Young's Inequality:
  $$\langle (\tilde{\xi}_i \cdot \nabla)(\tilde{\xi}_i \cdot \nabla)u, u \rangle + \|(\tilde{\xi}_i \cdot \nabla)u\|^2 \leq C(\nu_h + \nu_z)\|u\|^2 + \delta(\nu_h \|\nabla_h u\|^2 + \nu_z \|\partial_z u\|^2)$$
  The terms involving $\delta$ are absorbed into the viscous dissipation terms ($\nu_h \Delta_h$ and $\nu_z \Delta_z$), while the remaining term vanishes as $\nu \to 0$.

C. Energy Estimates

The proof relies on estimating the energy of the difference $v_t = u_t - w_t - B_t$. The author derives an energy identity involving:

1. Nonlinear terms: Handled via Gagliardo-Nirenberg inequalities and the specific structure of the boundary corrector.
2. Viscous terms: Used to absorb the "bad" terms arising from the non-divergence-free noise.
3. Stochastic terms: Controlled using the Burkholder-Davis-Gundy inequality. The anisotropic scaling ensures that the noise terms vanish in the limit.

3. Key Contributions

1. Anisotropic Noise Scaling: The paper introduces and rigorously analyzes a transport noise model where the noise intensity scales with the square root of the respective directional viscosities ($\sqrt{\nu_h}$ and $\sqrt{\nu_z}$). This is a novel setup in stochastic fluid dynamics.
2. Overcoming Non-Divergence-Free Noise: The work provides the first rigorous treatment of transport-stretching noise where the spatial correlation functions are not divergence-free. It demonstrates that the resulting "error" terms can be controlled by the anisotropic viscosity, provided the ratio $\nu_z/\nu_h \to 0$.
3. Extension of Kato's Criterion: The paper extends the deterministic results of Masmoudi [55] (which established the inviscid limit for anisotropic Navier-Stokes without noise) to the stochastic setting. It confirms that the convergence to the Euler equation holds even with transport noise, provided the noise vanishes at the correct rate.
4. Construction of Martingale Solutions: It establishes the existence of martingale weak solutions for the anisotropic stochastic Navier-Stokes equations with non-divergence-free noise, a result not previously available in the literature.

4. Main Results

Theorem 3.2 states that for any sequences of viscosities $(\nu_h^k)$ and $(\nu_z^k)$ such that $\nu_h^k \to 0$ and $\nu_z^k/\nu_h^k \to 0$:

• There exists a filtered probability space and a sequence of martingale weak solutions $(u^k, W^k)$ to the stochastic Navier-Stokes equations.
• Convergence: $u^k$ converges to the strong solution $w$ of the deterministic Euler equation in the space $L^2(\Omega; L^\infty([0, T]; L^2_\sigma))$.
• Gradient Convergence: The scaled gradients converge to zero:
  $$\nu_h^k \sum_{j=1}^2 \|\partial_j u^k\|_{L^2} \to 0 \quad \text{and} \quad \nu_z^k \|\partial_z u^k\|_{L^2} \to 0$$
  This implies that the energy dissipation in the boundary layer vanishes in the limit, satisfying the stochastic analogue of Kato's criterion.

5. Significance

• Geophysical Fluid Dynamics: The model is highly relevant for geophysical flows (e.g., ocean or atmospheric dynamics) where rotation induces strong anisotropy (horizontal mixing >> vertical mixing). The result validates the use of the Euler equation as a limit for such stochastic models.
• Stochastic PDE Theory: The paper resolves a critical technical gap regarding transport noise with non-divergence-free correlations. It shows that the "divergence-free" assumption, often taken for granted in SALT theory, can be relaxed if the noise is scaled appropriately relative to the viscosity.
• Boundary Layer Theory: It reinforces the understanding that in the inviscid limit with no-slip conditions, the convergence depends entirely on the behavior of the solution in a thin boundary layer. The anisotropic scaling ensures this layer remains controlled even in the presence of stochastic forcing.

In summary, this paper successfully bridges the gap between deterministic anisotropic fluid limits and stochastic analysis, proving that the Euler equation remains the correct inviscid limit for anisotropic Navier-Stokes equations even when driven by transport noise that breaks the divergence-free symmetry of the correlation functions.