Large Deviations for Stochastic Evolution Equations with Small Multiplicative Noise

The Freidlin-Wentzell large deviation principle is established for the distributions of stochastic evolution equations with general monotone drift and small multiplicative noise. As examples, the main results are applied to derive the large deviation principle for different types of SPDE such as stochastic reaction-diffusion equations, stochastic porous media equations and fast diffusion equations, and the stochastic p-Laplace equation in Hilbert space. The weak convergence approach is employed in the proof to establish the Laplace principle, which is equivalent to the large deviation principle in our framework.Comment: 31 pages, published in Appl. Math. Opti

Large Deviations for the Stochastic Shell Model of Turbulence

In this work we first prove the existence and uniqueness of a strong solution to stochastic GOY model of turbulence with a small multiplicative noise. Then using the weak convergence approach, Laplace principle for so- lutions of the stochastic GOY model is established in certain Polish space. Thus a Wentzell-Freidlin type large deviation principle is established utilizing certain results by Varadhan and Bryc.Comment: 21 pages, submitted for publicatio

Strategic Systems of the Future (SSF)

This was a keynote lecture on two occasions, Naval Postgraduate School-National University of Singapore (TDSI)-Lawrence Livermore Annual Security Meeting in Monterey CA (May 22, 2012) and Defense Science Forum of Indian Defense Research Development Organization (DRDO) Headquarters (July 10, 2012

A simple proof of global solvability of 2-D Navier-Stokes equations in unbounded domains

In this paper we provide an elementary proof of the classical result of J.L. Lions and G. Prodi on the global unique solvability of two-dimensional Navier-Stokes equations that avoids compact embedding and strong convergence. The method applies to unbounded domains without special treatment. The essential idea is to utilize the local monotonicity of the sum of the Stokes operator and the inertia term. This method was first discovered in the context o

Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise

A Wentzell-Freidlin type large deviation principle is established for the two-dimensional Navier-Stokes equations perturbed by a multiplicative noise in both bounded and unbounded domains. The large deviation principle is equivalent to the Laplace principle in our function space setting. Hence, the weak convergence approach is employed to obtain the Laplace principle for solutions of stochastic Navier-Stokes equations. The existence and uniqueness of a strong solution to (a) stochastic Navier-Stokes equations with a small multiplicative noise, and (b) Navier-Stokes equations with an additional Lipschitz continuous drift term are proved for unbounded domains which may be of independent interest.Stochastic Navier-Stokes equations Large deviations Girsanov theorem