# Semilinear and quasilinear stochastic evolution equations in Banach spaces

Hornung, Luca

Abstract (englisch):
In this thesis, we investigate the Cauchy problem for the quasilinear stochastic evolution equation
\begin{equation*}
\begin{cases}
in a Banach space $X.$
In the first part of the thesis, we concentrate on the parabolic situation, i.e. we assume that ${-}A(u(t))$ is for every $t$ a generator of an analytic semigroup and that $A(u(t))$ has a bounded $H^{\infty}$-calculus. Under a local Lipschitz assumption on $u\mapsto A(u)$ we prove existence and uniqueness of a local strong solution up to a maximal stopping time that can be characterised by a blow-up alternative. We apply our local well-posedness result to a second order parabolic partial differential equation on $\mathbb{R} ^d$, to a generalised Navier-Stokes equation describing non-Newtonian fluids and to a convection-diffusion equation on a bounded domain with Dirichlet, Neumann or mixed boudary conditions. In the last situation, we can even show that the solution exists globally.