Stochastic Navier--Stokes Equations for Turbulent Flows

Abstract

This paper concerns the fluid dynamics modelled by the stochastic flow   η (t, x) = u (t, η (t, x)) + σ (t, η (t, x)) •Ẇ η(0, x) = x where the turbulent term is driven by the white noiseẆ. The motivation for this setting is to understand the motion of fluid parcels in turbulent and randomly forced fluid flows. Stochastic Euler equations for the undetermined components u(t, x) and σ(t, x) of the spatial velocity field is derived from the first principles. The resulting equations include as particular cases the deterministic and randomly forced counterparts of these equations. In the second part of the paper we prove the existence and uniqueness of a strong local solution of the stochastic Navier-Stokes equation in W 1 p R d , d > 1, p > d. In the 2D case, the existence and uniqueness of a global strong solution is shown. In the third part, we deal with the propagation of Wiener chaos by the stochastic Navier-Stokes equation and its relation to statistical moments of the solution.

Maximal $L^p$-Regularity for Stochastic Evolution Equations

“…For stochastic parabolic evolution equations, maximal L p -regularity results have been obtained previously by Krylov for second order problems on R d [44,46,47,48,49], by Kim for second order problems on bounded domains in R d [43], and by Mikulevicius and Rozovskii for Navier-Stokes equations [63]. A systematic theory of maximal L p -regularity for stochastic evolution equations, however, based on abstract operator-theoretic properties of the operators governing the equation, has yet to be developed.…”

“…Maximal L p -regularity techniques have been pivotal in much of the recent progress in the theory of parabolic evolution equations (see [2,22,25,54,76,86] and the references therein). Among other things, such techniques provide a systematic and powerful tool to study nonlinear and time-dependent parabolic problems.For stochastic parabolic evolution equations, maximal L p -regularity results have been obtained previously by Krylov for second order problems on R d [44,46,47,48,49], by Kim for second order problems on bounded domains in R d [43], and by Mikulevicius and Rozovskii for Navier-Stokes equations [63]. A systematic theory of maximal L p -regularity for stochastic evolution equations, however, based on abstract operator-theoretic properties of the operators governing the equation, has yet to be developed.…”

Weak Solutions of a Stochastic Model for Two-Dimensional Second Grade Fluids

“…Since then, stochastic partial differential equations and stochastic models of fluid dynamics have been the object of intense investigations which have generated several important results. We refer, for instance, to [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] . Similar investigations for Non-Newtonian fluids have almost not been undertaken except in very few work; we refer, for instance, to [23][24][25] for some computational studies of stochastic models of polymeric fluids.…”

Large Deviations for Stochastic Tamed 3D Navier-Stokes Equations

“…But the uniqueness is open, and one only knows the existence of a locally unique strong solution (cf. [16,27]). Moreover, without the term g N , the existence and ergodicity of invariant measures for equation (1) have already been studied by using Kolmogorov operators (cf.…”