WELL-POSEDNESS AND ASYMPTOTIC ANALYSIS OF A CLASS OF STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

Abstract

The literature has devoted a great deal of attention to the analysis of stochastic partial differential equations (SPDEs) since the turn of the century. The models/general frame work we consider in this thesis have various applications in the fluid dynamics models, namely, describing the interaction between reaction mechanisms, convection effects, hy drodynamics models, etc. Aside from applications, the primary objective of phenomena in f luid dynamics models is to comprehend how random perturbations to an equation might affect its properties. Recent years have seen a shift away from the confines of classical f luid mechanics in studying of fluids and their turbulent behavior. Because of the sub ject’s many significant mathematical challenges and scientific/engineering applications, researchers from various fields of mathematics, such as nonlinear partial differential equa tions, functional analysis, harmonic analysis, stochastic analysis, ergodic theory, large deviations theory, and control filtering theory, are working together to advance the field. This thesis is the outcome of the following three projects: 1. We consider a stochastic generalized Burger-Huxley (SGBH) equation under differ ent types of noises. Existence and uniqueness of solutions and invariant measures, large deviation principle (LDP), uniform large deviation principle (UDLP), and ab solute continuity of the law of the solution are discussed to understand the behavior of various properties of solutions. 2. We consider 2D-stochastic Navier-Stokes equations (SNSE) in the vorticity form driven by infinite-dimensional noise (additive) and finite-dimensional multiplicative noise. Well-posedness of 2D-SNSE in the vorticity form driven by finite-dimensional noise has been explored. Furthermore, ULDP for 2D-SNSE in the vorticity form driven by two types of noises have established. i ii 3. Well-posedness of a class of SPDEs driven by L´evy noise has been established. Furthermore, an LDP for the laws of the solutions to a class of SPDEs driven by L´evy noise has been demonstrated. This class covers a large number of fluid dynamics models. The main goal of the thesis is to explore the well-posedness of different types of SPDEs and to analyze the large time behavior of the solutions. The random forcing may differ; that is, we consider both additive and multiplicative noises (with bounded or linear growth coefficients), which can be white in time and colored in space or space-time white noise. Moreover, the thesis deals with the following questions arising in stochastic analysis: • What about the existence and uniqueness of solutions to SGBH equation, 2D-SNSE in vorticity form, and a class of SPDEs perturbed by random forcing? • What about the absolute continuity of the law of the solution to SGBH equation with respect to the Lebesgue measure and the existence of the density? • What about the LDP (Wentzell-Friedlin type) as well as ULDP for the laws of the solutions to SGBH equation and 2D-SNSE in vorticity form? • What about the LDP (Wentzell-Friedlin type) for the law of the solutions to a class of SPDEs with fully monotone coefficients driven by L´evy noise? • WhatabouttheLDP(Donsker-Varadhantype) for the occupation measure of SGBH equation, which expresses the exact rate of exponential convergence? Our procedure to tackle this question includes the knowledges about the irreducibility and strong Feller properties of the associated Markovian semigroup. • Whatabout the existence and uniqueness of invariant measures of solution to SGBH equations and their ergodic behavior?