Uniqueness of solutions for Keller-Segel system of porous medium type coupled to fluid equations

TL;DR

This paper proves the uniqueness of weak solutions for a coupled Keller-Segel and fluid system in three dimensions, using duality and vanishing viscosity methods, with new estimates of Green functions.

Contribution

It introduces a novel approach to establish uniqueness for a porous medium Keller-Segel system coupled with fluid equations, including new Green function estimates.

Findings

Uniqueness of H"older continuous weak solutions in 3D

Development of Green function estimates for variable coefficient parabolic equations

Application of duality and vanishing viscosity methods

Abstract

We prove the uniqueness of H\"older continuous weak solutions via duality argument and vanishing viscosity method for the Keller-Segel system of porous medium type equations coupled to the Stokes system in dimensions three. An important step is the estimate of the Green function of parabolic equations with lower order terms of variable coefficients, which seems to be of independent interest.