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.     

Stochastic hydrodynamic-type evolution equations driven by Lévy noise in 3D unbounded domains—Abstract framework and applications

The existence of martingale solutions of the hydrodynamic-type equations in 3D possibly unbounded domains is proved. The construction of the solution is based on the Faedo–Galerkin approximation. To overcome the difficulty related to the lack of the compactness of Sobolev embeddings in the case of unbounded domain we use certain Fréchet space. Besides, we use compactness and tightness criteria in some nonmetrizable spaces and a version of the Skorohod theorem in non-metric spaces. The general framework is applied to the stochastic Navier–Stokes, magneto-hydrodynamic (MHD) and the Boussinesq equations.     

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.     

Martingale solutions and Markov selection of stochastic 3D Navier-Stokes equations with jump

In this paper, we study the existence of martingale solutions of stochastic 3D Navier-Stokes equations with jump, and following Flandoli and Romito (2008) [7] and Goldys et al. (2009) [8], we prove the existence of Markov selections for the martingale solutions.     

Martingale solutions for the three-dimensional stochastic nonhomogeneous incompressible Navier–Stokes equations driven by Lévy processes

In this paper, the three-dimensional stochastic nonhomogeneous incompressible Navier–Stokes equations driven by Lévy processes consisting of the Brownian motion, the compensated Poisson random measure and the Poisson random measure are considered in a bounded domain. We obtain the existence of martingale solutions. The construction of the solution is based on the classical Galerkin approximation method, the stopping times, the stochastic compactness method and the Jakubowski–Skorokhod theorem.     

Uniqueness and absolute continuity of weak solutions for parabolic SPDE's

The paper presents necessary and sufficient conditions for the absolute continuity of measures generated by infinite-dimensional martingale problems. This result is applied to the study of the existence and uniqueness of weak solutions to nonlinear parabolic SPDE's. The paper also addresses the problem of stochastic integration with respect to a martingale in a quasi-complete locally convex topological vector space.This work was partially supported by NSF Grant # DMS-9002997 and ONR Grant # N00014-91-J-1526.     

2D backward stochastic Navier-Stokes equations with nonlinear forcing

The paper is concerned with the existence and uniqueness of a strong solution to a two-dimensional backward stochastic Navier-Stokes equation with nonlinear forcing, driven by a Brownian motion. We use the spectral approximation and the truncation and variational techniques. The methodology features an interactive analysis on the basis of the regularity of the deterministic Navier-Stokes dynamics and the stochastic properties of the Itô-type diffusion processes.     

Backward stochastic differential equations with Azéma's martingale

The aim of this paper is to study backward stochastic differential equations (BSDE) driven by Azéma's martingale and the associated deterministic functional equations. More precisely, we introduce BSDE's vs. Azéma's martingale in a general frame, then we prove that the existence of a solution to a Markovian BSDE implies the existence of a solution to a deterministic functional equation of a new type. Uniqueness for the functional equation is proved in a particular case. Then we discuss BSDE's vs. an asymmetric martingale: half Brownian motion/half Azéma's martingale, which leads to an asymmetric deterministic functional equation.     

Ergodicity for the 3D stochastic Navier–Stokes equations perturbed by Lévy noise

In this work we construct a Markov family of martingale solutions for 3D stochastic Navier–Stokes equations (SNSE) perturbed by Lévy noise with periodic boundary conditions. Using the Kolmogorov equations of integrodifferential type associated with the SNSE perturbed by Lévy noise, we construct a transition semigroup and establish the existence of a unique invariant measure. We also show that it is ergodic and strongly mixing.     

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     

Pullback attractors for a non-autonomous generalized 2D parabolic system in an unbounded domain

The existence of a pullback attractor is proven for a non-autonomous generalized 2D parabolic system in an unbounded domain. The asymptotic compactness of the solution operator is obtained by the uniform estimates on the tails of solutions.     

Backward problems for stochastic differential equations on the Sierpinski gasket

In this paper, we study the non-linear backward problems (with deterministic or stochastic durations) of stochastic differential equations on the Sierpinski gasket. We prove the existence and uniqueness of solutions of backward stochastic differential equations driven by Brownian martingale (defined in Section 2) on the Sierpinski gasket constructed by S. Goldstein and S. Kusuoka. The exponential integrability of quadratic processes for martingale additive functionals is obtained, and as an application, a Feynman–Kac representation formula for weak solutions of semi-linear parabolic PDEs on the gasket is also established.     

Spatially homogeneous solutions of 3D stochastic Navier–Stokes equations and local energy inequality

We study the three-dimensional stochastic Navier–Stokes equations with additive white noise, in the context of spatially homogeneous solutions in R³${\mathbb{R}}^3$, i.e. solutions with a law invariant under space translations. We prove the existence of such a solution, with the additional property of being suitable in the sense of Caffarelli, Kohn and Nirenberg: it satisfies a localized version of the energy inequality.     

Non-monotone stochastic generalized porous media equations

By using the Nash inequality and a monotonicity approximation argument, existence and uniqueness of strong solutions are proved for a class of non-monotone stochastic generalized porous media equations. Moreover, we prove for a large class of stochastic PDE that the solutions stay in the smaller L²-space provided the initial value does, so that some recent results in the literature are considerably strengthened.     

Stochastic generalized porous media and fast diffusion equations

We present a generalization of Krylov-Rozovskii's result on the existence and uniqueness of solutions to monotone stochastic differential equations. As an application, the stochastic generalized porous media and fast diffusion equations are studied for σ-finite reference measures, where the drift term is given by a negative definite operator acting on a time-dependent function, which belongs to a large class of functions comparable with the so-called N-functions in the theory of Orlicz spaces.     

Zur existenz periodischer losungen von pseudodifferential gleichungen mit nacheilendem argument

In this paper an existence theorem for periodic solutions of pseudo-differential equations with delayed argument is proved by using an algebra of pseudo-differential operators. The existence of periodic solutions of the heat equation with delayed argument is included.     

The analysis of a finite element method for the Navier–Stokes equations with compressibility

Summary. A finite element formulation is developed for the two dimensional nonlinear time dependent compressible Navier–Stokes equations on a bounded domain. The existence and uniqueness of the solution to the numerical formulation is proved. An error estimate for the numerical solution is obtained. Received September 9, 1997 / Revised version received August 12, 1999 / Published online July 12, 2000     

Martingale weak solutions of the stochastic Landau–Lifshitz–Bloch equation

The present paper studies the stochastic Landau–Lifshitz–Bloch equation which is recommended as the only valid model at temperature around the Curie temperature and is especially important for the simulation of heat-assisted magnetic recording. We study the stochastic Landau–Lifshitz–Bloch equation in the case that the temperature is raised higher than the Curie temperature. The global existence of martingale weak solutions is proved by using a new argument and regularity properties of the weak solutions are discussed.     

Probabilistic gradient approximation for a viscous scalar conservation law in space dimension d≥2

In this paper, we are interested in the solution of a viscous scalar conservation law. We remark that its first order spatial derivatives solve a system of partial differential equations presenting a nonlocal nonlinearity. We associate a nonlinear martingale problem with this system. After proving existence and uniqueness for the martingale problem, we obtain a propagation of chaos result for a system of interacting diffusion processes. We deduce that it is possible to approximate the solution of the viscous scalar conservation law thanks to the interacting diffusions