Transport equations with fractal noise - existence, uniqueness and regularity of the solution

TL;DR

This paper establishes existence, uniqueness, and regularity results for solutions to a class of transport equations with fractal noise, using semigroup theory and fixed point methods, with applications to stochastic PDEs driven by fractional Brownian motion.

Contribution

It introduces a framework for analyzing transport equations with fractal noise, including a novel product definition and norm estimates, extending to stochastic PDEs with fractional Brownian noise.

Findings

Proved existence and uniqueness of mild solutions.

Established regularity properties of solutions.

Applied results to stochastic PDEs driven by fractional Brownian noise.

Abstract

The main result of the present paper is a statement on existence, uniqueness and regularity for mild solutions to a parabolic transport diffusion type equation that involves a non-smooth coefficient. We investigate related Cauchy problems on bounded smooth domains with Dirichlet boundary conditions by means of semigroup theory and fixed point arguments. Main ingredients are the definition of a product of a function and a (not too irregular) distribution as well as a corresponding norm estimate. As an application, transport stochastic partial differential equations driven by fractional Brownian noises are considered in the pathwise sense.