An Additive-Noise Approximation to Keller-Segel-Dean-Kawasaki Dynamics: Local Well-Posedness of Paracontrolled Solutions

TL;DR

This paper proves local well-posedness for a complex stochastic PDE modeling Keller-Segel dynamics with additive noise, employing paracontrolled calculus and renormalisation techniques to handle irregularities and heterogeneity.

Contribution

It introduces a novel additive-noise approximation to Keller-Segel hydrodynamics and establishes local well-posedness using advanced paracontrolled distribution methods.

Findings

Renormalisation diverges at most logarithmically due to symmetry

Solutions are well-posed locally in time despite irregular noise

Reduction in the number of diverging counterterms

Abstract

Using the method of paracontrolled distributions, we show the local well-posedness of an additive noise approximation to the fluctuating hydrodynamics of the Keller-Segel model on the two-dimensional torus. Our approximation is a non-linear, non-local, parabolic-elliptic stochastic PDE with an irregular, heterogeneous space-time noise. As a consequence of the irregularity and heterogeneity, solutions to this equation must be renormalised by a sequence of diverging fields. Using the symmetry of the elliptic Green's function, which appears in our non-local term, we establish that the renormalisation diverges at most logarithmically, an improvement over the linear divergence one would expect by power counting. Similar cancellations also serve to reduce the number of diverging counterterms.