Delayed blow-up and enhanced diffusion by transport noise for systems of reaction-diffusion equations

TL;DR

This paper demonstrates that transport noise can delay blow-up and enhance diffusion in reaction-diffusion systems, leading to global solutions under certain conditions, by extending regularization techniques in stochastic PDEs.

Contribution

It introduces a novel regularization effect of multiplicative transport noise on reaction-diffusion systems, connecting recent advances in stochastic PDE regularization and $L^p(L^q)$-methods.

Findings

Transport noise delays blow-up of solutions.

Enhanced diffusion effect observed with sufficient noise.

Global existence achieved for exponentially decreasing mass systems.

Abstract

This paper is concerned with the problem of regularization by noise of systems of reaction-diffusion equations with mass control. It is known that $\textit{strong}$ solutions to such systems of PDEs may blow-up in finite time. Moreover, for many systems of practical interest, establishing whether the blow-up occurs or not is an open question. Here we prove that a suitable multiplicative noise of transport type has a regularizing effect. More precisely, for sufficiently noise intensity and spectrum, the blow-up of strong solutions is delayed and an enhanced diffusion effect is also established. Global existence is shown for the case of exponentially decreasing mass. The proofs combine and extend recent developments in regularization by noise and in the $L^p(L^q)$-approach to stochastic PDEs, highlighting new connections between the two areas.