[Paper Review] Exponential Mixing for 3D Stochastic Primitive Equations

Zhao Dong, Jianliang Zhai+1 more|arXiv (Cornell University)|Jun 29, 2015

Stochastic processes and financial applications18 references4 citations

TL;DR

This paper establishes exponential mixing for weak solutions of 3D stochastic primitive equations under mild conditions: viscosity is strictly positive, and noise is smooth and non-degenerate. Using the coupling method and uniqueness of strong solutions, it proves that all weak solutions derived from Galerkin approximations share the same invariant measure, ensuring uniqueness of the invariant measure for strong solutions—resolving an open problem in stochastic geophysical fluid dynamics.

ABSTRACT

In this paper, we prove that weak solutions of 3D stochastic primitive equations have exponential mixing property if the noise is sufficiently smooth and non-degenerate. With the help of uniqueness of strong solution of 3D stochastic primitive equations, we obtain that all weak solutions which are limitations of Galerkin approximations share the same invariant measure. In particular, the invariant measure of strong solution is unique. The coupling method plays a key role.

Motivation & Objective

• To establish the exponential mixing property for weak solutions of 3D stochastic primitive equations under minimal viscosity and noise conditions.
• To show that all weak solutions obtained as limits of Galerkin approximations share the same invariant measure.
• To prove the uniqueness of the invariant probability measure for strong solutions, extending prior results.
• To overcome technical challenges from higher-order nonlinearities in the primitive equations compared to Navier-Stokes.

Proposed method

• Adopts the coupling method introduced by Odasso to construct a coupling of Galerkin approximations with a uniform lower bound of 3/4 on meeting probability.
• Imposes a small-ball condition in the H₃ norm to control the nonlinear term, which is one order higher than in 3D stochastic Navier-Stokes equations.
• Uses the strong Markov property and estimates on exit times from small balls to establish exponential moments for hitting times.
• Employs high-order Sobolev norm estimates (L⁶, H², H³) to control the nonlinear and stochastic terms in the equations.
• Relies on the uniqueness of strong solutions to deduce the uniqueness of the invariant measure for the entire system.
• Applies Freidlin-Wentzell-type large deviation principles and ergodic theory tools to analyze long-time behavior.

Experimental results

Research questions

• RQ1 Does the weak solution of the 3D stochastic primitive equations exhibit exponential mixing under minimal regularity assumptions on viscosity and noise?
• RQ2 Do all weak solutions derived as limits of Galerkin approximations converge to the same invariant measure?
• RQ3 Is the invariant measure of the strong solution to the 3D stochastic primitive equations unique?
• RQ4 Can the coupling method be adapted to handle the higher-order nonlinearity in the primitive equations compared to Navier-Stokes?
• RQ5 What role does the uniqueness of strong solutions play in establishing the uniqueness of the invariant measure?

Key findings

• Exponential mixing holds for weak solutions of 3D stochastic primitive equations when the noise is sufficiently smooth and non-degenerate, even with only strictly positive viscosity.
• All weak solutions obtained as limits of Galerkin approximations share the same invariant measure, implying statistical stationarity is unique across solution types.
• The invariant measure of the strong solution is uniquely determined, resolving a key open question in the stochastic primitive equations literature.
• The coupling method succeeds in constructing a coupling with a uniform lower bound of 3/4 on the meeting probability, despite the higher-order nonlinearity.
• The proof requires novel high-order Sobolev norm estimates (L⁶, H², H³) to control the nonlinear term involving vertical derivatives.
• The existence of an exponential moment for the time to enter a small ball in H₃ is established, a critical step in proving exponential ergodicity.