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.     

Unique solvability of equations of motion for ferrofluids

We study the differential system governing the flow of an incompressible ferrofluid under the action of a magnetic field. The system is a combination of the Navier-Stokes equations, the angular momentum equation, the magnetization equation and the magnetostatic equations. No regularizing term is added to the magnetization equation. We prove the local-in-time existence of the unique strong solution to the system posed in a bounded domain of R³ and equipped with initial and boundary conditions.     

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.     

Stochastic partial differential equations in M-type 2 Banach spaces

We study abstract stochastic evolution equations in M-type 2 Banach spaces. Applications to stochastic partial differential equations inL ^p spaces withp2 are given. For example, solutions of such equations are Hölder continuous in the space variables.The author is an Alexander von Humboldt Stiftung fellow     

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.     

Invariant measures for semilinear SPDE’s with local Lipschitz drift coefficients and applications

We study existence and a priori estimates of invariant measures μ for SPDE with local Lipschitz drift coefficients. Furthermore, we discuss the corresponding parabolic Cauchy-problem in L ¹(μ). Particular emphasis will be put on stochastic reaction diffusion equations.      

Nonlinear diffusion waves for hyperbolic p-system with nonlinear damping

This paper is concerned with the p-system of hyperbolic conservation laws with nonlinear damping. When the constant states are small, the solutions of the Cauchy problem for the damped p-system globally exist and converge to their corresponding nonlinear diffusion waves, which are the solutions of the corresponding nonlinear parabolic equation given by the Darcy's law. The optimal convergence rates are also obtained. In order to overcome the difficulty caused by the nonlinear damping, a couple of correction functions have been technically constructed. The approach adopted is the elementary energy method together with the technique of approximating Green function. On the other hand, when the constant states are large, the solutions of the Cauchy problem for the p-system will blow up at a finite time.     

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.     

A free boundary problem for the Navier–Stokes equations with measure data

The existence of a weak solution of an initial boundary-value problem for the plane nonstationary Navier–Stokes equations with Radon measure data on the free boundary, is established. The problem may be considered as a model of the blood flow around the heart valves. An inverse problem is studied, it allows us to find the boundary forces acting on the valve from the observed values of the velocity of the fluid in a fixed subregion.     

Exponential ergodicity and regularity for equations with Lévy noise

We prove exponential convergence to the invariant measure, in the total variation norm, for solutions of SDEs driven by α-stable noises in finite and in infinite dimensions. Two approaches are used. The first one is based on Liapunov’s function approach by Harris, and the second on Doeblin’s coupling argument in [8]. Irreducibility and uniform strong Feller property play an essential role in both approaches. We concentrate on two classes of Markov processes: solutions of finite dimensional equations, introduced in [27], with Hölder continuous drift and a general, non-degenerate, symmetric α-stable noise, and infinite dimensional parabolic systems, introduced in [29], with Lipschitz drift and cylindrical α-stable noise. We show that if the nonlinearity is bounded, then the processes are exponential mixing. This improves, in particular, an earlier result established in [28], with a different method.     

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     

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.     

Weighted estimates for stationary Navier-Stokes equations

We show Morrey-type estimates for the weak solution of the periodic Navier-Stokes equations in dimensionN, 5 <N < 10. ForN < 8, we prove the existence of a maximum solution.     

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     

Substochastic semigroups for transport equations with conservative boundary conditions

We consider the free streaming operator associated with conservative boundary conditions. It is known that this operator (with its usual domain) admits an extension A which generates a C₀-semigroup in L¹. With techniques borrowed from the additive perturbation theory of substochastic semigroups, we describe precisely its domain and provide necessary and sufficient conditions ensuring to be stochastic. We apply these results to examples from kinetic theory.     

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.     

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.     

Invariant measures for the linear stochastic Cauchy problem and R-boundedness of the resolvent

We study the asymptotic behaviour of solutions of the stochastic abstract Cauchy problem $$ \left\{ {\begin{array}{*{20}l} {dU\left( t \right) = AU\left( t \right)dt + BdW_H \left( t \right),\quad t \geqslant 0,} \hfill\ {U\left( 0 \right) = 0,} \hfill\ \end{array}} \right. $$ where A is the generator of a C₀-semigroup on a Banach space E, W_H is a cylindrical Brownian motion over a separable Hilbert space H, and $$ B \in \user1{\mathscr L}\left( {H,E} \right) $$ is a bounded operator. Assuming the existence of a solution U, we prove that a unique invariant measure exists if the resolvent R(λ, A) is R-bounded in the right half-plane {Reλ > 0}, and that conversely the existence of an invariant measure implies the R-boundedness of R(λ, A)B in every half-plane properly contained in {Re λ > 0}. We study various abscissae related to the above problem and show, among other things, that the abscissa of R-boundedness of the resolvent of A coincides with the abscissa corresponding to the existence of invariant measures for all γ -radonifying operators B provided the latter abscissa is finite. For Hilbert spaces E this result reduces to the Gearhart-Herbst-Prüss theorem. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

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.