Quasilinear rough evolution equations

TL;DR

This paper develops a framework for analyzing quasilinear rough evolution equations driven by rough paths, establishing local well-posedness and applying it to stochastic PDEs like the Landau-Lifshitz-Gilbert equation.

Contribution

It introduces a novel combination of functional analysis and rough paths theory to handle quasilinear rough PDEs, extending existing methods to new classes of equations.

Findings

Proved local well-posedness for quasilinear rough PDEs.

Applied the theory to stochastic Landau-Lifshitz-Gilbert and Shigesada-Kawasaki-Teramoto equations.

Established a random dynamical system for the Landau-Lifshitz-Gilbert equation.

Abstract

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.