On Unique Ergodicity in Nonlinear Stochastic Partial Differential Equations

© 2016 Springer Science+Business Media New YorkWe illustrate how the notion of asymptotic coupling provides a flexible and intuitive framework for proving the uniqueness of invariant measures for a variety of stochastic partial differential equations whose deterministic counterpart possesses a finite number of determining modes. Examples exhibiting parabolic and hyperbolic structure are studied in detail. In the later situation we also present a simple framework for establishing the existence of invariant measures when the usual approach relying on the Krylov–Bogolyubov procedure and compactness fails

Global Existence and Regularity for the 3D Stochastic Primitive Equations of the Ocean and Atmosphere with Multiplicative White Noise

The Primitive Equations are a basic model in the study of large scale Oceanic and Atmospheric dynamics. These systems form the analytical core of the most advanced General Circulation Models. For this reason and due to their challenging nonlinear and anisotropic structure the Primitive Equations have recently received considerable attention from the mathematical community. In view of the complex multi-scale nature of the earth's climate system, many uncertainties appear that should be accounted for in the basic dynamical models of atmospheric and oceanic processes. In the climate community stochastic methods have come into extensive use in this connection. For this reason there has appeared a need to further develop the foundations of nonlinear stochastic partial differential equations in connection with the Primitive Equations and more generally. In this work we study a stochastic version of the Primitive Equations. We establish the global existence of strong, pathwise solutions for these equations in dimension 3 for the case of a nonlinear multiplicative noise. The proof makes use of anisotropic estimates, $L^{p}_{t}L^{q}_{x}$ estimates on the pressure and stopping time arguments.Comment: To appear in Nonlinearit

Existence and uniqueness for stochastic 2D Euler flows with bounded vorticity

The strong existence and the pathwise uniqueness of solutions with (Formula presented.) -vorticity of the 2D stochastic Euler equations are proved. The noise is multiplicative and it involves the first derivatives. A Lagrangian approach is implemented, where a stochastic flow solving a nonlinear flow equation is constructed. The stability under regularizations is also proved

Invariant measures for the stochastic one-dimensional compressible navier–stokes equations

We investigate the long-time behavior of solutions to a stochastically forced one-dimensional Navier–Stokes system, describing the motion of a compressible viscous fluid, in the case of linear pressure law. We prove existence of an invariant measure for the Markov process generated by strong solutions. We overcome the difficulties of working with non-Feller Markov semigroups on non-complete metric spaces by generalizing the classical Krylov–Bogoliubov method, and by providing suitable polynomial and exponential moment bounds on the solution, together with pathwise estimates

Asymptotics of the coleman-gurtin model

This paper is concerned with the integrodi erential equation arising in the Coleman-Gurtin's theory of heat conduction with hereditary memory, in presence of a nonlinearity ' of critical growth. Rephrasing the equation within the history space framework, we prove the existence of global and exponential attractors of optimal regularity and finite fractal dimension for the related solution semigroup, acting both on the basic weak-energy space and on a more regular phase space