Quasilinear rough evolution equations

We investigate the abstract Cauchy problem for a quasilinear parabolic equation in a Banach space of the form $d{u}_t-L_t(u_t)u_{t}dt=N_t(u_t)dt+F(u_t)\cdot d{\mathbf{X}}_t$, where $\mathbf{X}$ is a $\gamma$-Hölder rough path for $\gamma \in (1\mathrm{/}3,1\mathrm{/}2)$. We explore the mild formulation that combines functional analysis techniques and controlled rough paths theory which entail the local well-posedness of such equations. We apply our results to the stochastic Landau-Lifshitz-Gilbert and Shigesada-Kawasaki-Teramoto equation. In this framework we obtain a random dynamical system associated to the Landau-Lifshitz-Gilbert equation.