On stochastic modified 3D Navier–Stokes equations with anisotropic viscosity

Abstract

Navier-Stokes equations in the whole space R 3 subject to an anisotropic viscosity and a random perturbation of multiplicative type is described. By adding a term of Brinkman-Forchheimer type to the model, existence and uniqueness of global weak solutions in the PDE sense are proved. These are strong solutions in the probability sense. The Brinkman-Forchheirmer term provides some extra regularity in the space L 2α+2 (R 3 ), with α > 1. As a consequence, the nonlinear term has better properties which allow to prove uniqueness. The proof of existence is performed through a control method. A Large Deviations Principle is given and proven at the end of the paper. ∂ t u − ν ∆u + u · ∇u + ∇p + a u 2α u = f, divu = 0, u| t=0 = u 0 , 2000 Mathematics Subject Classification. Primary 60H15, 60F10; Secondary 76D06, 76M35.

Strong $L^2$ convergence of time numerical schemes for the stochastic two-dimensional Navier–Stokes equations

“…where the last upper estimate is a consequence of the inequality 4α ∈ [4,6]. This completes the proof of (4.6).…”

“…In dimension 3 the Gagliardo-Nirenberg inequality implies that for p ∈ [2,6], H 1 ⊂ L p ; more precisely…”

“…The stochastic case has been investigated as well by several authors among which F. Flandoli, M. Röckner and M. Romito; see for example [18] for an account of remaining open problems. The anisotropic 3D case with a stochastic perturbation has been studied in [20] for rotating fluids, and in [6] for a Brinkman Forchheimer modification.…”

Large deviation principle for stochastic convective Brinkman–Forchheimer equations perturbed by pure jump noise

“…The works [5,8,39,40,49,55], etc discuss the global solvability and asymptotic behavior of solutions to the stochastic counterpart of the system (1.1) and related models in the whole space or on a torus. Whereas on bounded domains, the existence and uniqueness of strong solutions and asymptotic behavior of solutions to the stochastic counterpart of the system (1.1) and related models are discussed in the works [28,37,38,57], etc.…”

Local and global strong solution to the stochastic 3-D incompressible anisotropic Navier-Stokes equations

“…For the anisotropic case, only H. Bessaih and A. Mille [3] considered the stochastic modified 3-D anisotropic Navier-stokes equations, which add a Brinkman-Forchheimer type term a|u| 2α u (a > 0, α > 1). They proved the existence and uniqueness of global weak solutions in R 3 and gave a large deviations principle.…”