Uniform large deviations and long-time dynamics for the 2D anisotropic Navier-Stokes equations on the torus

Technical Summary: Uniform Large Deviations and Long-Time Dynamics for the 2D Anisotropic Navier-Stokes Equations on the Torus

Problem Statement
This paper investigates the two-dimensional stochastic incompressible Navier-Stokes equations on the torus $\mathbb{T}^2$ subject to horizontal dissipation only and additive noise. The system is defined by:
$$du^\varepsilon + (u^\varepsilon \cdot \nabla)u^\varepsilon dt + \nabla p^\varepsilon dt - \partial_1^2 u^\varepsilon dt = \sqrt{\varepsilon} Q^{1/2} dW, \quad \nabla \cdot u^\varepsilon = 0,$$
where the dissipation operator $A = -P\partial_1^2$ acts solely in the horizontal direction. The absence of vertical dissipation creates an infinite-dimensional kernel of steady states (purely vertical shear modes) that are immune to the dissipative mechanism. The authors address three primary challenges: establishing a uniform Large Deviation Principle (LDP) for solution paths in the energy space without relying on exponential tightness; analyzing the long-time mixing properties; and characterizing the structure of invariant measures in the presence of this degenerate dissipation.

Methodology
The analysis proceeds through three distinct methodological pillars:

1. Uniform Large Deviation Principle (LDP):
   The authors employ the weak convergence approach combined with a uniform contraction principle.

   • Linear Analysis: They first establish an LDP for the linear Ornstein-Uhlenbeck process $z^\varepsilon$ in the auxiliary space $G = C([0, T]; H) \cap L^2(0, T; \tilde{H}^{1,1})$. This relies on proving a priori estimates for the skeleton equation and the compactness of the solution map for bounded controls.
   • Nonlinear Mapping: A crucial step involves constructing a nonlinear map $F_x: G \to C([0, T]; H)$ that sends the linear orbit to the solution of the full stochastic equation. The authors prove a uniform Lipschitz estimate for this map on bounded sets of initial data. This estimate utilizes a novel algebraic manipulation of anisotropic Gagliardo-Nirenberg inequalities and the divergence-free condition to control the nonlinear term $(u \cdot \nabla)u$.
   • Contraction Principle: By applying Wang's uniform contraction principle, they derive the LDP for the nonlinear system. This approach circumvents the need for the exponential tightness condition typically required in classical LDP proofs for SPDEs, as the Lipschitz continuity is established on the entire auxiliary space $G$.

2. Long-Time Dynamics (Exponential Mixing):
   The authors analyze the failure of exponential mixing by identifying a specific deterministic steady state $u_0 = (\sin x_2, 0)$ which lies in the kernel of the horizontal Stokes operator. Since this mode has zero horizontal derivative, the dissipation term vanishes, and the nonlinear term also vanishes for this specific profile. Consequently, the $L^2$-norm of the difference between this steady state and the zero solution remains constant in the deterministic limit. The authors use a contradiction argument to show that this undamped mode prevents the stochastic system from satisfying a uniform exponential mixing estimate, even with small noise.

3. Invariant Measures:
   The paper investigates the existence and structure of invariant measures by contrasting the deterministic and stochastic cases:

   • Deterministic Case: The infinite-dimensional kernel $N$ of the operator $A$ consists entirely of steady states. Thus, every Dirac measure $\delta_u$ for $u \in N$ is an invariant measure, resulting in infinitely many distinct invariant measures.
   • Stochastic Case: The authors construct a specific degenerate noise where the noise covariance range lies entirely within the kernel $N$. They prove that any invariant measure for this system must be orthogonal to $N$ (i.e., $\mu(N) = 0$). The proof demonstrates that if an invariant measure charged $N$, the restriction of the dynamics to $N$ would behave like a Gaussian process with linearly growing variance in an infinite-dimensional space, which admits no invariant probability measure.

Key Results

• Uniform LDP: The paper proves a uniform Freidlin-Wentzell LDP for the solution paths in the energy space $C([0, T]; H)$, uniformly over bounded sets of initial data in $\tilde{H}^{0,1}$. This is the first such result for anisotropic Navier-Stokes equations in the energy space, achieved without requiring exponential tightness or additional damping.
• Failure of Exponential Mixing: It is rigorously proven that the anisotropic Navier-Stokes equations on the torus do not exhibit exponential mixing. The existence of the infinite-dimensional kernel of vertical shear modes, which are preserved by periodic boundary conditions, prevents the system from forgetting initial conditions exponentially fast.
• Invariant Measure Structure:
  • In the deterministic limit, the system possesses infinitely many Dirac invariant measures corresponding to the kernel of the horizontal Laplacian.
  • Under degenerate additive noise acting within the kernel, any existing invariant measure must be orthogonal to the kernel. The paper further demonstrates that under such noise, the kernel itself supports no invariant measure.

Significance and Claims
The authors position this work as a sharp distinction between anisotropic equations on the torus and both fully dissipative Navier-Stokes equations and anisotropic equations with no-slip (Dirichlet) boundaries.

• Contrast with Dirichlet Boundaries: On bounded domains with no-slip conditions, the anisotropic Poincaré inequality holds, the kernel is trivial, and unique ergodicity (exponential mixing) has been established (citing Sun, Qiu, and Tang). On the torus, the periodic boundary conditions preserve the infinite-dimensional kernel, fundamentally altering the long-time behavior.
• Methodological Advance: The derivation of the uniform LDP without exponential tightness is highlighted as a significant technical contribution, relying on the specific regularity of the linear process in the auxiliary space $G$ and the uniform Lipschitz continuity of the nonlinear map.
• Conceptual Framework: The paper provides a framework for understanding partially dissipative systems, identifying the kernel of the dissipation operator as the primary obstruction to statistical stability. It suggests that while the undamped vertical modes prevent exponential mixing, the nonlinear term remains the only potential mechanism for energy transfer to the dissipative horizontal modes, though the existence of a global invariant measure for the general stochastic case remains an open question tied to the efficiency of this energy cascade.

The authors conclude that their results clarify the role of boundary conditions in determining the qualitative long-time dynamics of anisotropic fluid models and offer a structural perspective applicable to future studies of partially dissipative systems, such as anisotropic MHD and primitive equations.