Deterministic and stochastic 2D Navier-Stokes equations with anisotropic viscosity

Martingale Solutions of the Stochastic 2D Primitive Equations with Anisotropic Viscosity

“…Especially, for the stochastic Navier-Stokes equations, by adding a term of Brinkman-Forchheimer type, Bessaih and Millet [1] established the existence and uniqueness of global weak solutions (in the PDE sense) in the whole space R 3 . Liang, Zhang and Zhu [39] investigated the existence of the martingale solutions and pathwise uniqueness of the solutions in a given anisotropic Sobolev space on R 2 or on the two dimensional torus T 2 . Comparing with the Navier-Stokes equations, it is worth to point out that the system of primitive equations is generally harder to deal with, the nonlinear term w∂ z v is a more difficult version in contrast to the nonlinearity of the Navier-Stokes equations since w = w(v) given by (1.4) involves a first order derivative.…”

“…As in [39], we can obtain that on each space appearing in the definition of X there exists a countable set of continuous real-valued functions separating points. Then all the conditions of the above Skorokhod theorem are satisfied.…”