Existence of global stochastic flow and attractors for Navier–Stokes equations

For 2-D stochastic Navier-Stokes equations on the torus with multiplicative noise we construct a perfect cocycle and show the existence of global random compact attractors. The equations considered do not admit a pathwise method of solution. Received: 9 June 1998 / Revised version: 17 December 1998     

On the method of Da Prato and Debussche for the 3D stochastic Navier Stokes equations

3D stochastic Navier-Stokes equations with a suitable nondegenerate noise are considered. Following a method introduced by Da Prato and Debussche, it is proved that every Markov process associated to the equations has a Strong Feller like continuity property with respect to initial conditions. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Martingale and stationary solutions for stochastic Navier-Stokes equations

Summary We prove the existence of martingale solutions and of stationary solutions of stochastic Navier-Stokes equations under very general hypotheses on the diffusion term. The stationary martingale solutions yield the existence of invariant measures, when the transition semigroup is well defined. The results are obtained by a new method of compactness.     

Ergodicity of the 3D stochastic Navier-Stokes equations driven by mildly degenerate noise

We prove that any Markov solution to the 3D stochastic Navier-Stokes equations driven by a mildly degenerate noise (i.e. all but finitely many Fourier modes are forced) is uniquely ergodic. This follows by proving strong Feller regularity and irreducibility.     

Large deviations for the Boussinesq equations under random influences

A Boussinesq model for the Bénard convection under random influences is considered as a system of stochastic partial differential equations. This is a coupled system of stochastic Navier–Stokes equations and the transport equation for temperature. Large deviations are proved, using a weak convergence approach based on a variational representation of functionals of infinite-dimensional Brownian motion.     

A probabilistic interpretation and stochastic particle approximations of the 3-dimensional Navier-Stokes equations

We develop a probabilistic interpretation of local mild solutions of the three dimensional Navier-Stokes equation in the L^p spaces, when the initial vorticity field is integrable. This is done by associating a generalized nonlinear diffusion of the McKean-Vlasov type with the solution of the corresponding vortex equation. We then construct trajectorial (chaotic) stochastic particle approximations of this nonlinear process. These results provide the first complete proof of convergence of a stochastic vortex method for the Navier-Stokes equation in three dimensions, and rectify the algorithm conjectured by Esposito and Pulvirenti in 1989. Our techniques rely on a fine regularity study of the vortex equation in the supercritical L^p spaces, and on an extension of the classic McKean-Vlasov model, which incorporates the derivative of the stochastic flow of the nonlinear process to explain the vortex stretching phenomenon proper to dimension three. Supported by Fondecyt Project 1040689 and Nucleus Millennium Information and Randomness ICM P01-005.     

Stochastic Navier-Stokes equations

We construct a solution to stochastic Navier-Stokes equations in dimension n4 with the feedback in both the external forces and a general infinite-dimensional noise. The solution is unique and adapted to the Brownian filtration in the 2-dimensional case with periodic boundary conditions or, when there is no feedback in the noise, for the Dirichlet boundary condition. The paper uses the methods of nonstandard analysis.The research of this author was supported by an SERC Grant.     

Stochastic conservation laws: Weak-in-time formulation and strong entropy condition

This article is an attempt to complement some recent developments on conservation laws with stochastic forcing. In a pioneering development, Feng and Nualart [8] have developed the entropy solution theory for such problems and the presence of stochastic forcing necessitates introduction of strong entropy condition. However, the authors' formulation of entropy inequalities are weak-in-space but strong-in-time. In the absence of a priori path continuity for the solutions, we take a critical outlook towards this formulation and offer an entropy formulation which is weak-in-time and weak-in-space.     

On the stochastic Benjamin-Ono equation

We discuss the Cauchy problem for the stochastic Benjamin-Ono equation in the function class Hs(R), s>3/2. When there is a zero-order dissipation, we also establish the existence of an invariant measure with support in H²(R). Many authors have discussed the Cauchy problem for the deterministic Benjamin-Ono equation. But our results are new for the stochastic Benjamin-Ono equation. Our goal is to extend known results for the deterministic equation to the stochastic equation.     

Regularity of the sample paths of a class of second-order spde's

We study the sample path regularity of the solutions of a class of spde's which are second order in time and that includes the stochastic wave equation. Non-integer powers of the spatial Laplacian are allowed. The driving noise is white in time and spatially homogeneous. Continuing with the work initiated in Dalang and Mueller (Electron. J. Probab. 8 (2003) 1), we prove that the solutions belong to a fractional L²-Sobolev space. We also prove Hölder continuity in time and therefore, we obtain joint Hölder continuity in the time and space variables. Our conclusions rely on a precise analysis of the properties of the stochastic integral used in the rigourous formulation of the spde, as introduced by Dalang and Mueller. For spatial covariances given by Riesz kernels, we show that our results are optimal.     

Existence and uniqueness of solutions to the backward 2D stochastic Navier–Stokes equations

The backward two-dimensional stochastic Navier–Stokes equations (BSNSEs, for short) with suitable perturbations are studied in this paper, over bounded domains for incompressible fluid flow. A priori estimates for adapted solutions of the BSNSEs are obtained which reveal a pathwise L^∞(H)$L^{\infty }\left(H\right)$ bound on the solutions. The existence and uniqueness of solutions are proved by using a monotonicity argument for bounded terminal data. The continuity of the adapted solutions with respect to the terminal data is also established.     

Stochastic wave equation of pure jumps: Existence,uniqueness and invariant measures

In this paper, we investigate a class of nonlinear damped stochastic hyperbolic equations with jumps. The jump component considered here is described as a Poisson point process. This paper is divided into two parts. The first part deals with existence and uniqueness of global weak and strong solutions to this type of equations, based on the energy approach. The second part devotes to the existence and support of invariant measures corresponding to the weak solution semi-group, based on Markov property of the solution.     

Large deviations and approximations for slow-fast stochastic reaction-diffusion equations

A large deviation principle is derived for a class of stochastic reaction-diffusion partial differential equations with slow-fast components. The result shows that the rate function is exactly that of the averaged equation plus the fluctuating deviation which is a stochastic partial differential equation with small Gaussian perturbation. This result also confirms the effectiveness of the approximation of the averaged equation plus the fluctuating deviation to the slow-fast stochastic partial differential equations.     

A new approach to stochastic evolution equations with adapted drift

In this paper we develop a new approach to stochastic evolution equations with an unbounded drift A which is dependent on time and the underlying probability space in an adapted way. It is well-known that the semigroup approach to equations with random drift leads to adaptedness problems for the stochastic convolution term. In this paper we give a new representation formula for the stochastic convolution which avoids integration of non-adapted processes. Here we mainly consider the parabolic setting. We establish connections with other solution concepts such as weak solutions. The usual parabolic regularity properties are derived and we show that the new approach can be applied in the study of semilinear problems with random drift. At the end of the paper the results are illustrated with two examples of stochastic heat equations with random drift.     

Periodic-like solutions of variable coefficient and Wick-type stochastic NLS equations

Variable coefficient and Wick-type stochastic nonlinear Schrödinger (NLS) equations are investigated. By using white noise analysis, Hermite transform and extended F-expansion method, we obtain a number of Wick versions of periodic-like wave solutions and periodic wave solutions expressed by various Jacobi elliptic functions for Wick-type stochastic and variable coefficient NLS equations, respectively. In the limit cases, the soliton-like wave solutions are showed as well. Since Wick versions of functions are usually difficult to evaluate, we get some nonWick versions of the solutions for Wick-type stochastic NLS equations in special cases.     

Global existence of solutions to the compressible Navier-Stokes equation around parallel flows

The initial boundary value problem for the compressible Navier-Stokes equation is considered in an infinite layer of Rn. It is proved that if n?3, then strong solutions to the compressible Navier-Stokes equation around parallel flows exist globally in time for sufficiently small initial perturbations, provided that the Reynolds and Mach numbers are sufficiently small. The proof is given by a variant of the Matsumura-Nishida energy method based on a decomposition of solutions associated with a spectral property of the linearized operator.     

A PDE approach to large deviations in Hilbert spaces

We introduce a PDE approach to the large deviation principle for Hilbert space valued diffusions. It can be applied to a large class of solutions of abstract stochastic evolution equations with small noise intensities and is adaptable to some special equations, for instance to the 2D stochastic Navier–Stokes equations. Our approach uses a lot of ideas from (and in significant part follows) the program recently developed by Feng and Kurtz [J. Feng, T. Kurtz, Large Deviations for Stochastic Processes, in: Mathematical Surveys and Monographs, vol. 131, American Mathematical Society, Providence, RI, 2006]. Moreover we present easy proofs of exponential moment estimates for solutions of stochastic PDE.     

Invariant measure for the stochastic Ginzburg Landau equation

The existence of martingale solutions and stationary solutions of stochastic Ginzburg-Landau equations under general hypothesizes on the dimension, the non linear term and the added noise is investigated. With a few more assumptions, it is established that the transition semi-group is well defined and that the stationary martingale solution yields the existence of an invariant measure. Moreover this invariant measure is shown to be unique.     

Stochastic partial differential equations in Hölder spaces

Summary We prove the existence and regularity of solutions to stochastic partial differential equations of parabolic Itô type in Hölder spaces under the usual sublinear growth and local Lipschitz conditions. Some examples are given to which our main theorems apply.The work of the first author was supported in part by the NSF grant DMS-91-01360     

The scaling limit for a stochastic PDE and the separation of phases

Summary We investigate the problem of singular perturbation for a reaction-diffusion equation with additive noise (or a stochastic partial differential equation of Ginzburg-Landau type) under the situation that the reaction term is determined by a potential with double-wells of equal depth. As the parameter (the temperature of the system) tends to 0, the solution converges to one of the two stable phases and consequently the phase separation is formed in the limit. We derive a stochastic differential equation which describes the random movement of the phase separation point. The proof consists of two main steps. We show that the solution stays near a manifoldM of minimal energy configurations based on a Lyapunov type argument. Then, the limit equation is identified by introducing a nice coordinate system in a neighborhood ofM .Research partially supported by Japan Society for the Promotion of Science