## Stochastic PDEs beyond standard monotonicity: well posedness and regularity of solutions

##### Abstract

Nonlinear stochastic partial differential equations (SPDEs) are used to model wide variety of
phenomena in physics, engineering, finance and economics. In many such models the equations
exhibit super-linear growth. In general, equations with super-linear growth are ill-posed.
However if the growth satisfies some monotonicity-like conditions, then well-posedness can be
shown. This thesis focuses on SPDEs that satisfy monotonicity-like conditions and consists of
two main parts.
In part one, we have generalised the results using local-monotonicity condition by establishing
the existence and uniqueness of solution to nonlinear stochastic partial differential equations
(SPDEs) when the coefficients satisfy local monotonicity condition. This is done by identifying
appropriate coercivity condition which helps in obtaining the desired higher order moment
estimates without explicitly restricting the growth of the operators acting on the solution in
the stochastic integral terms. As a result, we can solve various semilinear and quasilinear
stochastic partial differential equations with locally monotone operators, where derivatives may
appear in the operator acting on the solution under the stochastic integral term. Examples
of such equations are stochastic reaction-diffusion equations, stochastic Burger equations and
stochastic p-Laplace equations where the diffusion operator need not necessarily be Lipschitz
continuous. Further, the operator appearing in bounded variation term is allowed to be the
sum of finitely many operators, each having different analytic and growth properties. As an application,
well-posedness of the stochastic anisotropic p-Laplace equation driven by Levy noise
has been shown.
In second part of this thesis, new regularity results for solution to semilinear SPDEs on
bounded domains are obtained. The semilinear term is continuous, monotone except around the
origin and is allowed to have polynomial growth of arbitrary high order. Typical examples are
the stochastic Allen-Cahn and Ginzburg-Landau equations. This is done by obtaining some Lp-
estimates which are subsequently employed in obtaining higher regularity of solutions. This is
motivated by ongoing work to obtain rate of convergence estimates for numerical approximations
to such equations.
Key