Quasilinear rough evolution equations
Abstract
We investigate the abstract Cauchy problem for a quasilinear parabolic equation in a Banach space of the form \( dut -Lt(ut)ut dt = Nt(ut)dt + F(ut)· d Xt \), where \( X\) is a \( γ\)-H\"older rough path for \( γ∈(1/3,1/2)\). 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.
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