Blow-up for a Stochastic Model of Chemotaxis Driven by Conservative Noise on ${\mathbb{R}}^2$

Abstract

We establish criteria on the chemotactic sensitivity $\chi$ for the non-existence of global weak solutions (i.e. \textit{blow-up} in finite time) to a stochastic Keller--Segel model with spatially inhomogeneous, conservative noise on ${\mathbb{R}}^2$. We show that if $\chi$ is sufficiently large then \emph{blow-up} occurs with probability $1$. In this regime our criterion agrees with that of a deterministic Keller--Segel model with increased viscosity. However, for $\chi$ in an intermediate regime, determined by the variance of the initial data and the spatial correlation of the noise, we show that \emph{blow-up} occurs with positive probability.

Example of conservative noise edited and issue with uniqueness of weak solutions addressed; 25 pages

References in corpus (3)

• High mode transport noise improves vorticity blow-up control in 3D Navier-Stokes equations
• Blow-up for the stochastic nonlinear Schrodinger equation with multiplicative noise
• Time--space white noise eliminates global solutions in reaction diffusion equations