Nathan Glatt-Holtz

Local martingale and pathwise solutions for an abstract fluids model

We establish the existence and uniqueness of both local martingale and local pathwise solutions of an abstract nonlinear stochastic evolution system. The primary application of this abstract framework is to infer the local existence of strong, pathwise solutions to the 3D primitive equations of the oceans and atmosphere forced by a nonlinear multiplicative white noise. Instead of developing our results specifically for the 3D primitive equations we choose to develop them in a slightly abstract framework which covers many related forms of these equations (atmosphere, oceans, coupled atmosphere-ocean, on the sphere, on the {\beta}-plane approximation etc and the incompressible Navier-Stokes equations). In applications, all of the details are given for the {\beta}-plane approximation of the oceans equations.

Local and global existence of smooth solutions for the stochastic Euler equations with multiplicative noise

We establish the local existence of pathwise solutions for the stochastic Euler equations in a three-dimensional bounded domain with slip boundary conditions and a very general nonlinear multiplicative noise. In the two-dimensional case we obtain the global existence of these solutions with additive or linear-multiplicative noise. Lastly, we show that, in the three dimensional case, the addition of linear multiplicative noise provides a regularizing effect; the global existence of solutions occurs with high probability if the initial data is sufficiently small, or if the noise coefficient is sufficiently large.

UNIQUE ERGODICITY FOR FRACTIONALLY DISSIPATED, STOCHASTICALLY FORCED 2D EULER EQUATIONS

We establish the existence and uniqueness of an ergodic invariant measure for 2D fractionally dissipated stochastic Euler equations on the periodic box for any power of the dissipation term.

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 earths 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,

Existence and Regularity of Invariant Measures for the Three Dimensional Stochastic Primitive Equations

L^{p}_{t}L^{q}_{x}

Cauchy convergence schemes for some nonlinear partial differential equations

estimates on the pressure and stopping time arguments.

Parameter Estimation for the Stochastically Perturbed Navier-Stokes Equations

We establish the continuity of the Markovian semigroup associated with strong solutions of the stochastic 3D Primitive Equations, and prove the existence of an invariant measure. The proof is based on new moment bounds for strong solutions. The invariant measure is supported on strong solutions and is furthermore shown to have higher regularity properties.

Invariant Measures for Passive Scalars in the Small Noise Inviscid Limit

Motivated by ongoing work in the theory of stochastic partial differential equations we develop direct methods to infer that the Galerkin approximations of certain nonlinear partial differential equations are Cauchy (and therefore convergent). We develop such a result for the Navier–Stokes equations in space dimensions two and three, and for the primitive equations in space dimension two. The analysis requires novel estimates for the nonlinear portion of these equations and delicate interpolation results concerning subspaces.

On Unique Ergodicity in Nonlinear Stochastic Partial Differential Equations

We consider a parameter estimation problem of determining the viscosity [nu] of a stochastically perturbed 2D Navier-Stokes system. We derive several different classes of estimators based on the first N Fourier modes of a single sample path observed on a finite time interval. We study the consistency and asymptotic normality of these estimators. Our analysis treats strong, pathwise solutions for both the periodic and bounded domain cases in the presence of an additive white (in time) noise.

Inviscid limits for a stochastically forced shell model of turbulent flow

AbstractWe consider a class of invariant measures for a passive scalar f driven by an incompressible velocity field