The Microscopic Derivation and Well-Posedness of the Stochastic Keller–Segel Equation

Abstract

In this paper, we propose and study a stochastic aggregation–diffusion equation of the Keller–Segel (KS) type for modeling the chemotaxis in dimensions $$d=2,3$$ d = 2 , 3 . Unlike the classical deterministic KS system, which only allows for idiosyncratic noises, the stochastic KS equation is derived from an interacting particle system subject to both idiosyncratic and common noises. Both the unique existence of solutions to the stochastic KS equation and the mean-field limit result are addressed.

On global existence and blowup of solutions of stochastic Keller-Segel type equation

“…We further show in Theorem 3.2, by means of a priori estimates, that the local solution is in fact global, provided the initial mass is sufficiently small. Similar results were obtained in [15] for p = 4 only.…”

“…The works addressing stochastic Keller-Segel model include [10] and [15]. In [10], the authors establish the local existence of solutions for a large class of nonlinear SPDEs, including stochastic Keller-Segel type equation on torus.…”

“…In [10], the authors establish the local existence of solutions for a large class of nonlinear SPDEs, including stochastic Keller-Segel type equation on torus. The work [15] by Huang and Qiu addresses a Keller-Segel type stochastic aggregation-diffusion equation in the entire space, in which the interactions are determined by Bessel potentials. The authors obtain the SPDE as the large particle limit of a system of stochastic differential equations with common noise (in divergence form) and prove, using the contraction argument in spirit of [6], the local existence and uniqueness for weak solutions on the interval [0, T ], where T depends on the L 4 -norm of the initial value.…”

“…[2,4]) that global existence of solutions is impossible, since the solution has finite time blowup. The finite time blowup of the solutions are not considered in [15]. Hence, to the best of our knowledge, the question of a finite time blowup for stochastic Keller-Segel model, has not been addressed in the literature yet.…”

“…In this paper we deal with the stochastic Keller-Segel equation, with the stochastic perturbation of the following two types. The first choice of noise is of divergence type, which is a particular case of the noise in [15]:…”