Delayed blow-up by transport noise

TL;DR

This paper demonstrates that transport noise can delay finite-time blow-up in certain nonlinear PDEs, leading to prolonged solution existence with high probability, especially in models like Keller-Segel and Fisher-KPP.

Contribution

It introduces a novel scaling limit showing how transport noise delays blow-up in nonlinear PDEs, extending the understanding of stochastic effects on PDE solutions.

Findings

Blow-up is delayed by transport noise under certain conditions.

Long-time existence is achieved for large initial data with high probability.

Results are applied to Keller-Segel, Fisher-KPP, and Kuramoto-Sivashinsky equations.

Abstract

For some deterministic nonlinear PDEs on the torus whose solutions may blow up in finite time, we show that, under suitable conditions on the nonlinear term, the blow-up is delayed by multiplicative noise of transport type in a certain scaling limit. The main result is applied to the 3D Keller-Segel, 3D Fisher-KPP and 2D Kuramoto-Sivashinsky equations, yielding long-time existence for large initial data with high probability.