Three-Dimensional stochastic Navier-Stokes equations with Markov switching

TL;DR

This paper introduces a novel stochastic Navier-Stokes model with Markov switching to account for different noise regimes, establishing existence and uniqueness of solutions through regularization and convergence methods.

Contribution

It develops a new framework for stochastic Navier-Stokes equations incorporating Markov switching, with rigorous proof of solution existence and uniqueness.

Findings

Existence of unique strong solutions for regularized systems.

Convergence of regularized solutions to the Markov switching system.

Framework for analyzing stochastic PDEs with regime switching.

Abstract

A finite-state Markov chain is introduced in the noise terms of the three-dimensional stochastic Navier-Stokes equations in order to allow for transitions between two types of multiplicative noises. We call such systems as stochastic Navier-Stokes equations with Markov switching. To solve such a system, a family of regularized stochastic systems is introduced. For each such regularized system, the existence of a unique strong solution (in the sense of stochastic analysis) is established by the method of martingale problems and pathwise uniqueness. The regularization is removed in the limit by obtaining a weakly convergent sequence from the family of regularized solutions, and identifying the limit as a solution of the three-dimensional stochastic Navier-Stokes equation with Markov switching.