Probabilistic Methods in Fluids

Preface.

International Research Institute in Mathematics and its Applications.

Workshop Participants.

We construct diffusion processes associated with the Navier-Stokes system in R³ and use them to prove the existence and uniquenes of local solution of the Cauchy problem for this system in some functional space.

A mean field result is proved for an abstract model, under a class of conditions on the rescaling of the energy. Propagation of chaos, variational characterization of the limit Gibbs density h and an equation for h are proved. The general results are applied to a model of 3D vortex filaments described by stochastic processes, including Brownian motion and Brownian Bridge, other semimartingales, and fractional Brownian Motion.

We construct Feller processes called Meixner-type processes by making the parameters of the characteristic exponent of a Meixner process state space dependent. Our main tool is the theory of pseudo-differential operators. A further aim of this paper is to popularize these methods among probabilists.

In this paper we study Itô integral in 2-smooth Banach spaces. Burkholder inequality is proved using Itô formula in certain subclass of such spaces. Relationship with an integral introduced recently by Mikulevicius and Rozovskii is discussed. Finally, Wong-Zakai type approximation for such integrals is proved.

Some results on the pathwise asymptotic behaviour of the weak solutions to a stochastic 2D-Navier-Stokes equation are established. In fact we prove some results concerning the asymptotic behaviour with general decay rate (exponential, sub and super-exponential).

The existence and uniqueness of solutions to a class of semilinear stochastic hyperbolic equations in ℛ^d are considered. First an energy inequality for a linear stochastic hyperbolic equation is established. Then it is proven that there exists a unique continuous local solution for the associated nonlinear equation in the Sobolev space H₁(ℛ^d) when the nonlinear terms are locally bounded and Lipschitz-continuous. Under an additional condition on the energy bound, the solution exists for all time. The results are shown to be applicable to stochastic wave equations with polynomial nonlinearities of degree m with m≤3 for d=3, and for any m≥1 for d=1 or 2.

We survey the use of Loeb space methods in stochastic fluid mechanics, with particular emphasis on recent results concerning the existence of attractors for the stochastic Navier–Stokes equations.

We study the two-dimensional Navier-Stokes equations with periodic boundary conditions perturbed by a space-time white noise. We prove that, for almost every initial data with respect to a measure supported by spaces with negative regularity, there exists a unique global solution in the strong probabilistic sense. The nonlinear term is defined thanks to techniques borrowed from Wick renormalisation and paraproducts in Besov spaces. Note however that no renormalisation is made here and the nonlinear term is not modified. This result was given in ⁹, here we give simplified proofs. Then we prove ergodicity of the Gaussian invariant measure.

A survey of results on invariant measures of the Lévy-Khinchine type for 2D Euler and stochastic Navier–Stokes equations is given. Uniqueness results of the corresponding Liouville respectively Kolmogorov flows are discussed. Stochastic dynamics associated with the invariant measures are also discussed (stochastic Stokes equation for the vorticity in the Gaussian case, Doob's independent Brownian motions process in the compound Poisson case).

Some recent notions and results, like invariant measures for the Navier-Stokes equations, random attractors, random invariant measures and vortex filaments are reviewed. Some conjectures about their relation are expressed.

This paper intends to review the main properties of the solutions of Burgers equation with random initial conditions of white noise type. These properties are closely related to those of the convex hull of a Brownian motion with parabolic drift. A special attention is given to the latter.

The flow of viscous incompressible fluid on an n-dimensional flat torus is presented as the expectation of a certain stochastic process on the group of diffeomorphisms of the torus. The above-mentioned process is governed by a stochastic analogue of the second Newton's law subjected to the mechanical constraint that garantees incompressibility. The diffusion term of the process is connected with viscosity coefficient of the fluid. The constraint is given in invariant geometric terms, the Newton's law is formulated in terms of Nelson's mean backward derivatives. The Navier-Stokes equation is derived as an Euler type equation in "algebra" of the group. The construction is translated into the finite-dimensional language of processes on the torus (as far as it is possible). Relations with some other stochastic approaches to viscous hydrodynamics is discussed.

O. Barndorff-Nielsen and S. Levendorskiǐ used some classical pseudo-differential operators to construct Markov processes in order to model some situations in finance (and turbulence). In our note we describe various symbol classes consisting of non-classical but smooth symbols which lead to Markov processes and obey a symbolic calculus. In particular it is pointed out that in many cases it is possible to make parameters of the characteristic exponent of a Lévy process state-space dependent to get a corresponding Markov process generated by a psuedo-differential operator.

We analyze the properties of a stochastic differential equation (SDE) arising in the modeling of polymeric fluids. More precisely, we focus on the so-called FENE (Finite Extensible Nonlinear Elastic) model, for which the drift term in the SDE is singular.

We give a survey on results about the growth of the diameter of the image of a bounded subset of R^d under the action of a stochastic flow. We provide a new proof of the fact that, under reasonable assumptions, the diameter of this image set will almost surely grow at most linearly in time, and we establish an explicit upper bound for the linear growth rate which is both simpler and better than previous bounds. Our main tool is the Garsia–Rodemich–Rumsey Lemma.

We give a one dimensional analysis of the solution v^μ of the stochastic Burgers equation in d dimensions, with viscosity μ²~0, as obtained by Davies, Truman and Zhao. Our analysis shows how the graph of a simple action functional in one space variable can be used to decompose the caustics into hot and cool parts. The inviscid limiting Burgers velocity field has a jump discontinuity across a cool part but is continuous as you cross the hot part. Our analysis also enables us to get a hold on the intermittence of stochastic turbulence in terms of the recurrence of a one dimensional stochastic process ζ simply related to the reduced action. Some detailed examples are discussed.

We establish a version of the strong law of large numbers (SLLN) for mixing-type Markov chains and apply it to a class of random dynamical systems with additive noise. The result obtained implies the SLLN for solutions of the 2D Navier–Stokes system and the complex Ginzburg–Landau equation perturbed by a non-degenerate random force.

The aim of this paper is to present various mathematical models for wells. Special attention is paid to a non-standard description using nonlocal boundary conditions (BCs). We also develop numerical algorithms to handle nonlocal BCs. The choice of the appropriate model depends, of course, on concrete situation.

In this paper a representation of the Fourier transform of solutions to the Navier-Stokes Equations are obtained in terms of a stochastic recursion generated by a branching random walk. The notion of majorizing kernel is introduced and used to study regularity and existence of solutions of the Navier-Stokes equations. Similar representation of solutions to other equations are also discussed and its corresponding multiplicative recursion in the physical space are presented. This is joint work with R. Bhattacharya, L. Chen, S. Dobson, R. Guenther, C. Orum and E. Waymire.

The purpose of this paper is to investigate the Cauchy problem for the following stochastic Burgers equation

In this note, we prove a comparison result for the solutions of backward stochastic differential equations with two reflecting barrier processes. The result is then applied to obtain an existence result for solutions of a backward SDE with reflecting barrier processes under weak assumptions on the coefficients.

In this paper we study the WKB-Langer asymptotic expansion of the eigenfunctions of a Schrödinger operator . Applying these asymptotic formulae we prove that the exact L² eigenfunction Ψ_E(N,ℏ) (and its derivative ℏΨ′_E(N,ℏ)) of the Schrödinger operator with a well-shaped analytic potential are approximated up to arbitrary order ℏ^m by the semi-classical WKB-Langer approximate eigenfunction Ψ_Em(N,ℏ),m (and its derivative ℏΨ′_Em(N,ℏ),m) respectively in L², i.e. ‖Ψ_E(N,ℏ)-Ψ_Em(N,ℏ),m‖_L²=O(ℏ^m+1),‖ℏΨ′_E(N,ℏ)-ℏΨ′_Em(N,ℏ),m‖_L²=O(ℏ^m+1) uniformly for any N. Here E_m(N,ℏ) approximates E(N,ℏ) up to m-th order (in ℏ) and satisfies the m-th order quantization condition. There are applications of this limit to Burgers equations, turbulence and the large scale structure of the universe.