Global existence and blow-up for a stochastic transport equation with non-local velocity

TL;DR

This paper studies a stochastic non-local transport equation, establishing local existence, uniqueness, blow-up criteria, and analyzing long-term behavior under different noise types, including conditions for global existence and finite-time blow-up.

Contribution

It provides a comprehensive analysis of the stochastic transport equation, including conditions for global solutions and finite-time blow-up, with new insights into noise effects on solution behavior.

Findings

Global solutions exist almost surely under certain noise conditions.

Finite-time blow-up occurs with positive probability in linear noise cases.

Introduces the concept of stability of exiting times and its limitations.

Abstract

In this paper we investigate a non-linear and non-local one dimensional transport equation under random perturbations on the real line. We first establish a local-in-time theory, i.e., existence, uniqueness and blow-up criterion for pathwise solutions in Sobolev spaces $H^{s}$ with $s>3$. Thereafter, we give a complete picture of the long time behavior of the solutions based on the type of noise we consider. On one hand, we identify a family of noises such that blow-up can be prevented with probability $1$, guaranteeing the existence and uniqueness of global solutions almost surely. On the other hand, in the particular linear noise case, we show that singularities occur in finite time with positive probability, and we derive lower bounds of these probabilities. To conclude, we introduce the notion of stability of exiting times and show that one cannot improve the stability of the exiting time and simultaneously improve the continuity of the dependence on initial data.