Quasilinear rough evolution equations

We investigate the abstract Cauchy problem for a quasilinear parabolic equation in a Banach space of the form $du_t -L_t(u_t)u_t dt = N_t(u_t)dt + F(u_t)\cdot d\mathbf X_t$, where $\mathbf X$ is a $\gamma$-H\"older rough path for $\gamma\in(1/3,1/2)$. 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.