Convection-Diffusion Equations with Random Initial Conditions

TL;DR

This paper studies a generalized convection-diffusion equation with random initial conditions, establishing existence, regularity, and long-term behavior of solutions within a new nonlinear framework.

Contribution

It introduces a novel nonlinear approach to analyze convection-diffusion equations with stochastic initial data, proving key properties of solutions.

Findings

Existence of solutions under random initial conditions

Regularity results for the solutions

Analysis of long-term behavior of solutions

Abstract

We consider an evolution equation generalising the viscous Burgers equation supplemented by an initial condition which is a homogeneous random field. We develop a non-linear framework enabling us to show the existence and regularity of solutions as well as their long time behaviour.