Determinstic and stochastic 2d Navier-Stokes equations with anisotropic viscosity

TL;DR

This paper studies deterministic and stochastic 2D Navier-Stokes equations with anisotropic viscosity, establishing well-posedness, existence, and uniqueness of solutions in specific function spaces.

Contribution

It proves global well-posedness for the deterministic case and existence plus uniqueness of solutions for the stochastic case with anisotropic viscosity.

Findings

Global well-posedness in anisotropic Sobolev space

Existence of martingale solutions in stochastic case

Pathwise uniqueness and strong solutions

Abstract

In this paper, we investigate both deterministic and stochastic 2D Navier Stokes equations with anisotropic viscosity. For the deterministic case, we prove the global well-posedness of the system with initial data in the anisotropic Sobolev space $\tilde{H}^{0,1}.$ For the stochastic case, we obtain existence of the martingale solutions and pathwise uniqueness of the solutions, which imply existence of the probabilistically strong solution to this system by the Yamada-Watanabe Theorem.