Non-Linear Evolution Equations Driven by Rough Paths

TL;DR

This paper establishes existence and uniqueness of solutions for certain non-linear parabolic equations driven by rough paths, using semigroup theory and Young integration, with applications to stochastic Navier-Stokes equations.

Contribution

It introduces a novel approach combining semigroup theory and Young integration to handle rough forcing in non-linear PDEs, including stochastic Navier-Stokes.

Findings

Proved existence and uniqueness of solutions for rough PDEs.

Applied the framework to stochastic Navier-Stokes equations with fractional Brownian motion.

Demonstrated the effectiveness of Young integration in interpreting rough forcing terms.

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.