Non-Linear Evolution Equations Driven by Rough Paths

Abstract

We prove existence and uniqueness results for (mild) solutions to some non-linear parabolic evolution equations with a rough forcing term. Our method of proof relies on a careful exploitation of the interplay between the spatial and time regularity of the solution by capitialising some of Kato's ideas in semigroup theory. Classical Young integration theory is then shown to provide a means of interpreting the equation. As an application we consider the three dimensional Navier-Stokes system with a stochastic forcing term arising from a fractional Brownian motion with h > 1/2.