A stochastic representation for backward incompressible Navier-Stokes equations

Abstract

Abstract. By reversing the time variable we derive a stochastic representation for backward incompressible Navier-Stokes equations in terms of stochastic Lagrangian paths, which is similar to Constantin and Iyer's forward formulations in [6]. Using this representation, a self-contained proof of local existence of solutions in Sobolev spaces are provided for incompressible Navier-Stokes equations in the whole space. In two dimensions or large viscosity, an alternative proof to the global existence is also given. Moreover, a large deviation estimate for stochastic particle trajectories is presented when the viscosity tends to zero.

Forward–backward stochastic differential systems associated to Navier–Stokes equations in the whole space

“…Hence, by the relation (7.9) or (7.11), we derive a probabilistic representation along the stochastic particle systems for the strong solutions of Navier-Stokes equations. In fact, the representation formula (7.11) is analogous to those of Constantin and Iyer [13] and Zhang [57] with the stochastic flow methods. Nevertheless, for the coefficients we only need G ∈ H m and especially, f ∈ L 2 (0, T ; H m−1 ) which fails to be continuous when m = 2, and these conditions are much weaker than those of [13,57] where G and f (t, ·) are spatially Lipschitz continuous and valued in C k+1,α and H k+2,q (R d ) (֒→ C k+1,α ) respectively, with some (k, α, q) ∈ N × (0, 1) × (d, ∞).…”

“…With the Lagrangian approach, Constantin and Iyer [13,14] derived a stochastic representation for the incompressible Navier-Stokes equations based on stochastic Lagrangian paths and gave a self-contained proof of the existence. Later, Zhang [57] considered a backward analogue and provided short elegant proofs for the classical existence results. In this section, we shall derive from our representation (see Theorem (3.1)) an analogous Lagrangian formula, through which we show the connections with the Lagrangian approach.…”

“…By truncating the time interval of the FBSDS, we approximate in Section 6 the Navier-Stokes equation by a class of FBSDSs and associated PDEs. In Section 7, from our relationship between FBSDS and the Navier-Stokes equation, we derive a probabilistic Lagrangian representation for the velocity field, which is shown to imply those of [13,57], and we also give a variational characterization of the the Navier-Stokes equation. Finally in Section 8 as an appendix, we prove Lemmas 2.2 and 4.1.…”

Stochastic Lagrangian Particle Approach to Fractal Navier-Stokes Equations

“…When ν(dy) = dy/|y| d+α with α ∈ (0, 2), L = −c α (−∆) α/2 is the usual fractional Laplacian operator by multiplying a constant. As a simplified model of equation (1.1), the following fractal Burgers equation has been studied by Biler, Funaki and Woyczynski [3] and Kiselev, Nazarov and Schterenberg [15], The aim of this paper is to study equation (1.1) by using a stochastic Lagrangian particle trajectories approach following [6,27]. More precisely, Constantin and Iyer [6] gave the following elegant stochastic representation for the regularity solution u of Navier-Stokes equation (corresponding to L = ν∆ in (1.1)):…”

2D backward stochastic Navier–Stokes equations with nonlinear forcing

“…As a motivation BSNSEs emerge in regard to inverse problems to determine the stochastic noise coefficients from the terminal velocity field as observed. In [6,24], a stochastic representation in terms of Lagrangian paths for the backward incompressible Navier-Stokes equations without forcing is shown and used to prove the local existence of solutions in weighted Sobolev spaces and the global existence results in two dimensions or with a large viscosity. In [16,17], the existence and uniqueness of adapted solutions are given to the backward stochastic Lorenz equations and to the backward stochastic Navier-Stokes equations (1.3) in a bounded domain with σ ≡ 0, ν < 0 and the external force f (t, y, z) ≡ f (t).…”