Global solutions for random vorticity equations perturbed by gradient dependent noise, in two and three dimensions
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Journal of Evolution Equations
Global solutions for random vorticity equations perturbed by gradient dependent noise, in two and three dimensions Ionu¸t Munteanu and Michael Röckner
Abstract. The aim of this work is to prove an existence and uniqueness result of Kato–Fujita type for the Navier–Stokes equations, in vorticity form, in 2D and 3D, perturbed by a gradient-type multiplicative Gaussian noise (for sufficiently small initial vorticity). These equations are considered in order to model hydrodynamic turbulence. The approach was motivated by a recent result by Barbu and Röckner (J Differ Equ 263:5395–5411, 2017) that treats the stochastic 3D Navier–Stokes equations, in vorticity form, perturbed by linear multiplicative Gaussian noise. More precisely, the equation is transformed to a random nonlinear parabolic equation, as in Barbu and Röckner (2017), but the transformation is different and adapted to our gradient-type noise. Then, global unique existence results are proved for the transformed equation, while for the original stochastic Navier–Stokes equations, existence of a solution adapted to the Brownian filtration is obtained up to some stopping time.
1. Introduction One of the most important features concerning the Navier–Stokes equation is its relation to the phenomenon of hydrodynamic turbulence that is often assumed to be caused by random background movements. That is why a randomly forced Navier– Stokes equation may be considered to model this. In this direction, we recall the pioneering work of Bensoussan and Temam [4] concerning the analytical study of a Navier–Stokes equation driven by a white noise type random force; followed later by numerous developments and extensions by many authors (see [7,8,10,13] and the references therein). We emphasize the approach in [13,14] that involves gradientdependent noise in order to model turbulence. In this light, we consider the following Navier–Stokes equation in dimension d = 2, 3, perturbed by gradient-dependent noise ⎧ N ⎪ ⎪ ⎪ ⎪ Ai (X )dβi (t) + ∇π dt on (0, ∞) × Rd , dX − X dt + (X · ∇)X dt = ⎨ i=1 (1.1) d, ⎪ ∇ · X = 0 on (0, ∞) × R ⎪ ⎪ ⎪ d ⎩ X (0) = x in L p (Rd , Mathematics Subject Classification: 60H15, 35Q30, 76F20, 76N10 Keywords: Stochastic Navier–Stokes equation, Turbulence, Vorticity, Biot–Savart operator, Gradient-type noise.
I. Munteanu and M. Röckner
J. Evol. Equ.
N where x : × Rd → Rd is a random variable; π denotes the pressure; {βi }i=1 is a system of independent Brownian motions on a probability space (, F, P) with normal filtration (Ft )t≥0 , x is F0 -adapted, and Ai are certain operators, linear in the gradient of the solution, specified below. Our aim in this paper is to study (1.1) by writing it in vorticity form (i.e., apply the curl operator to it) and by transforming it into the following random partial differential equation
dy = y(t) + −1 (t)[K ((t)y(t)) · ∇]((t)y(t)), t > 0; y(0) = U0 = curl x. dt (1.2) where (t) solves (3.5) below and K is the Biot–Savart operator. We analyze (1.2) N for a.e. fixed ω. In par
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