Local pathwise solutions and regularization by noises for the stochastic hyperbolic Keller-Segel equation

TL;DR

This paper studies the stochastic hyperbolic Keller-Segel equation, establishing local existence and uniqueness of solutions, and demonstrating noise-induced regularization effects that ensure global solutions under certain conditions.

Contribution

It provides the first local well-posedness results for the stochastic hyperbolic Keller-Segel equation and shows how multiplicative noise can induce regularization leading to global solutions.

Findings

Local existence and uniqueness of solutions in Sobolev spaces

Noise-induced regularization allows global solutions for large initial data

Global solutions for small initial data or large noise intensity

Abstract

In this paper, we investigate the Cauchy problem associated with the stochastic hyperbolic Keller-Segel (SHKS) equation featuring multiplicative noises on the torus $\mathbb{T}^d$. First, we establish the local existence and uniqueness of pathwise solutions to the SHKS equation within Sobolev spaces $H^s(\mathbb{T}^d)$ for $s>\frac{d}{2}+1$, under appropriate regularity conditions imposed on the nonlinear multiplicative noises. Subsequently, we explore two global results pertaining to noise-induced regularization: (1) The first result demonstrates that for polynomial-type nonlinear noises, when the noise intensity parameters meet specific threshold conditions, the SHKS equation possesses a unique pathwise solution for large initial data with probability one. This finding provides a partial answer to a question that has remained unresolved in the deterministic setting; (2) The second result reveals that, for small initial data, or equivalently when dealing with linear multiplicative noises with sufficiently large intensity (allowed to be negative), the SHKS equation admits a unique pathwise solution with high probability.