ON WELL-POSEDNESS FOR STOCHASTIC BURGERSTYPE EQUATIONS

Abstract

In this thesis, different classes of generalized stochastic Burgers-type equations are studied. In particular, we focus on stochastic Burgers equation and its different types of generalizations. Here, we have mainly discussed three types of generalized equations: first equation considers the polynomial types nonlinearity in place of quadratic nonlinearity, second one is equipped with the fractional differential operator or mixed fractional differential operator in place of Laplacian operator and last one mainly uses of different types of stochastic noises. The main aspects of discussion is to show the existence and uniqueness of the solution to the these equations. The very first goal is to study of the existence of weak solutions of the one-dimensional generalized stochastic Burgers equation with polynomial nonlinearity perturbed by space time white noise with Dirichlet boundary conditions and a􀀀Hölder continuous coefficient in the noise term with a 2 1 2 ;1

. The existence result is established by solving an equivalent martingale problem. The second aim is to investigate the global existence and uniqueness of solutions to the one-dimensional generalized stochastic Burgers equation containing a nonlinearity of polynomial type and perturbed by cylindrical Volterra process having Dirichlet boundary conditions. In addition, we are also interested to prove that there exists an invariant measure for the same equation with the quadratic nonlinearity. As a third task, we investigate the existence and uniqueness of solutions to the fractional Burgers-type nonlinear stochastic partial differential equation driven by cylindrical fractional Brownian motion in Hölder spaces. The existence proof relies on a finite dimensional Galerkin approximation. Moreover, the rate of convergence of the Galerkin approximation as well as fully discretization of solution are also obtained. i ii Finally, we address a class of stochastic nonlinear partial differential equation of Burgerstype driven by pseudo differential operator (D+Da) where Da 􀀀(􀀀D) a2 with a 2 (0;2) and which is perturbed by the fractional Brownian sheet. The existence and uniqueness of an Lpvalued (local) solution is established for the initial boundary valued problem to this equation