[Paper Review] Stochastic 3D Navier-Stokes equations with nonlinear damping: martingale solution, strong solution and small time large deviation principles

Hui Liu, Hui Gao|arXiv (Cornell University)|Aug 29, 2016

Navier-Stokes equation solutions22 references3 citations

TL;DR

This paper establishes the existence of martingale and strong solutions for the stochastic 3D Navier-Stokes equations with nonlinear damping, proving global existence and uniqueness for β > 3 and α ≥ 1/2 when β = 3. It further derives a small time large deviation principle (LDP) for the same system using stochastic analysis and compactness methods.

ABSTRACT

In this paper, by using classical Faedo-Galerkin approximation and compactness method, the existence of martingale solutions for the stochastic 3D Navier-Stokes equations with nonlinear damping is obtained. The existence and uniqueness of strong solution are proved for $β> 3$ with any $α>0$ and $α\geq \frac12$ as $β= 3$. Meanwhile, a small time large deviation principle for the stochastic 3D Navier-Stokes equation with damping is proved for $β> 3$ with any $α>0$ and $α\geq \frac12$ as $β= 3$.

Motivation & Objective

• To establish the existence of martingale solutions for stochastic 3D Navier-Stokes equations with nonlinear damping using Faedo-Galerkin approximation and compactness methods.
• To prove the existence and uniqueness of strong solutions under the condition β > 3 for any α > 0, and α ≥ 1/2 when β = 3.
• To derive a small time large deviation principle (LDP) for the stochastic 3D Navier-Stokes equation with damping, extending Freidlin-Wentzell theory to the damped, stochastic 3D case.
• To improve upon prior deterministic results by relaxing the α ≥ 1/2 condition for β ≥ 3 through refined analysis of nonlinear damping effects.
• To provide a rigorous framework for studying rare events and asymptotic behavior in damped, stochastic fluid dynamics via LDP and ergodicity estimates.

Proposed method

• Employs the Faedo-Galerkin approximation method to construct approximate solutions in finite-dimensional subspaces.
• Applies compactness arguments and a priori estimates to pass to the limit and obtain martingale solutions in the weak sense.
• Uses energy estimates and Itô's formula to control nonlinear terms, especially ∫₀ᵗ∫_D |u·∇u|² dx ds, via the damping term |u|^{β−1}u.
• Applies the weak convergence method and Skorokhod’s embedding theorem to establish tightness of laws for the LDP analysis.
• Implements the weak convergence approach of Freidlin-Wentzell to derive the small time LDP, relying on exponential estimates and large deviation bounds.
• Uses stopping times τ_ε,M^n and exponential moment estimates to control pathwise differences between solutions and their approximations.

Experimental results

Research questions

• RQ1 Under what conditions does a martingale solution exist for the stochastic 3D Navier-Stokes equations with nonlinear damping?
• RQ2 For which values of β and α does a unique strong solution exist globally in time?
• RQ3 Can a small time large deviation principle be established for the stochastic 3D Navier-Stokes equation with nonlinear damping?
• RQ4 How does the nonlinear damping term |u|^{β−1}u improve regularity and enable stronger existence and uniqueness results compared to the standard case?
• RQ5 What is the role of the α ≥ 1/2 condition in the existence of strong solutions, and can it be relaxed under refined analysis?

Key findings

• A martingale solution exists for the stochastic 3D Navier-Stokes equations with nonlinear damping for any β > 1.
• Strong solution exists and is unique for β > 3 and any α > 0, and for β = 3 when α ≥ 1/2.
• A small time large deviation principle is established for β > 3 and any α > 0, and for β = 3 when α ≥ 1/2.
• The exponential estimates in the LDP proof show that P(supₜ|u^ε(t)−v^ε(t)|² > δ) ≤ 3e^{−R/ε} for arbitrary R > 0, implying lim_{ε→0} ε log P(...) = −∞.
• The analysis improves upon earlier deterministic results by relaxing the α ≥ 1/2 condition for β ≥ 3 through deeper exploitation of the damping structure.
• The compactness and tightness arguments, combined with stopping time controls, ensure convergence of approximating sequences and validity of the LDP.