Global existence of martingale solutions to stochastic keller-segel system with degenerate diffusion

TL;DR

This work tackles the stochastic Patlak–Keller–Segel system with degenerate diffusion under multiplicative noise in bounded domains of dimension $d=2$ or $3$. It develops a variational, operator-based framework built on a Gelfand triple and an abstract SPDE with drift $A$ and diffusion $B$, then constructs a solution operator and uses a stochastic Schauder fixed point to prove the global existence of martingale solutions for nonnegative initial data in $H_{2}^{-1}(\mathcal{O})$ when the diffusion exponent satisfies $m\ge 3$. Key contributions include establishing the needed bounds for the chemotactic term via Newtonian/Bessel potentials, defining an appropriate Banach space setting, and deriving energy estimates that guarantee global existence in the weak sense. The results advance understanding of stochastic chemotaxis with degenerate diffusion, offering a robust existence theory where semigroup methods and direct variational approaches fail due to degeneracy and lack of coercivity.

Abstract

In this paper, we study the stochastic degenerate Keller-Segel system perturbed by linear multiplicative noise in a bounded domain $\mathcal{O}$. We establish the global existence of martingale solutions for this model with any nonnegative initial data in $H_{2}^{-1}(\mathcal{O})$. The main challenge in proving the existence of solutions arises from the degeneracy of the porous media diffusion and the lack of coercivity in the nonlinear chemotactic term. To overcome these difficulties, we construct a solution operator and apply the Schauder fixed point theorem within the variational framework.

Theorems & Definitions (17)

• Lemma 2.1: liu2015stochastic, Theorem 5.1.3
• Lemma 2.2: liu2015stochastic, Lemma 4.1.12
• Lemma 3.1
• proof
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Theorem 3.1
• proof
• Proposition 3.1