Reaction-diffusion equations with transport noise and critical superlinear diffusion: local well-posedness and positivity

TL;DR

This paper studies stochastic reaction-diffusion equations with transport noise, establishing local well-posedness, regularity, and positivity, and providing tools for analyzing complex systems with superlinear diffusion.

Contribution

It introduces new methods for handling transport noise and critical spaces, proving higher order regularity even with non-smooth initial data.

Findings

Established local well-posedness and positivity of solutions.

Developed $L^p(L^q)$-theory and maximal regularity estimates.

Provided criteria for blow-up and regularity improvements.

Abstract

In this paper we consider a class of stochastic reaction-diffusion equations. We provide local well-posedness, regularity, blow-up criteria and positivity of solutions. The key novelties of this work are related to the use transport noise, critical spaces and the proof of higher order regularity of solutions -- even in case of non-smooth initial data. Crucial tools are $L^p(L^q)$-theory, maximal regularity estimates and sharp blow-up criteria. We view the results of this paper as a general toolbox for establishing global well-posedness for a large class of reaction-diffusion systems of practical interest, of which many are completely open. In our follow-up work (Agresti and Veraar: Global existence ... ), the results of this paper are applied in the specific cases of the Lotka-Volterra equations and the Brusselator model.