Marek Capiński

Stochastic Navier-stokes equations with multiplicative noise

An abstract stochastic Navier-Stokes equation with multiplicative white noise is considered. 2-dimensional Navier-Stokes equations with noise depending on first order derivatives of the solution are covered by the abstract model. Existence and uniqueness of a solution is proved for small initial data, and the associated local stochastic flow is constructed

Nonstandard methods for stochastic fluid mechanics

Standard Preliminaries Nonstandard Preliminaries Weak Solutions of Navier-Stokes Equations Statistical Solutions of Navier-Stokes Equations Stochastic Navier-Stokes Equations Other Equations of Hydromechanics Euler Equation.

Stochastic Navier-Stokes equations

We construct a solution to stochastic Navier-Stokes equations in dimension n≤4 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.

A convergence result for stochastic partial differential equations

For regular approximations wnof the Brownian motion wthe solutions of the PDE converge to the unique solution of the stochastic partial differential equation . The assumptions on the operators A and B are not restrictive including for instance random,A, and admit physically meaningful applications.

On the existence of a solution to stochastic Navier—Stokes equations

We consider stochastic Navier{Stokes equations on a possibly unbounded domain O R d , where d is equal to 2 or 3. First we prove the existence of a martingale solution for the initial value being a probability measure on the space of square integrable R d-valued functions. Then we show the existence of a spatially homogeneous solution to the equation on the whole R d , driven by a spatially homogeneous Wiener random eld. 0. Introduction Let O be a possibly unbounded open subset of R d , where d is equal to 2 or 3. We assume that O is connected with the boundary @O of class C 2. Let us x a bounded time interval 0; T]. The paper is concerned with the existence of a martingal solution to the following system of stochastic Navier{Stokes equations (0.1)

Pathwise global attractors for stationary random dynamical systems

SummaryThe asymptotic behaviour of random dynamical systems in Polish spaces is considered. Under the assumption of existence of a random compact absorbing set, assumption supposed to hold path by path, a candidate pathwise attractorA(ω) is defined. The goal of the paper is to show that, in the case of stationary dynamical systems,A(ω) attracts bounded sets, is measurable with respect to the σ-algebra of invariant sets, and is independent of ω when the system is ergodic. An application to a general class of Navier-Stokes type equations perturbed by a multiplicative ergodic real noise is discussed in detail.

Local existence and uniqueness of strong solutions to 3-D stochastic Navier-Stokes equations

Abstract. For stochastic Navier-Stokes equations in a 3-dimensional bounded domain we first show that if the initial value is sufficiently regular, then martingale solutions are strong on a random time interval and we estimate its length. Then we prove the uniqueness of the strong solution in the class of all martingale solutions.

A Simple Proof of Existence of Weak and Statistical Solutions of Navier-Stokes Equations

The Galerkin approximation to the Navier–Stokes equations in dimension N, where N is an infinite non-standard natural number, is shown to have standard part that is a weak solution. This construction is uniform with respect to non-standard representation of the initial data, and provides easy existence proofs for statistical solutions.

Navier-Stokes equations with multiplicative noise

The authors use the methods of nonstandard analysis to give a solution to stochastic Navier-Stokes equations in dimension <or=4 with noise depending in a specific way on the first-order derivatives of the solution. Uniqueness holds for the two-dimensional case.

Attractors for three–dimensional Navier–Stokes equations

For the three‐dimensional Navier‐Stokes equations we propose two new approaches to the notion of an attractor. They involve multi‐valued semiflows constructed via the nonstandard framework used for solving the equations, where even in dimension three we have uniqueness of solution for the corresponding equation.