Description

We are interested in the fluctuating hydrodynamics of the Keller–Segel model of chemotaxis, the so-called Keller–Segel–Dean–Kawasaki (KSDK) equation:

\begin{equation*} \begin{cases} (\partial_{t}-\Delta)\rho &= -\nabla\cdot(\rho\nabla c)-\nabla\cdot(\sqrt{\rho}\boldsymbol{\xi}),\\ -\Delta c &= \rho, \end{cases} \end{equation*}

where $\boldsymbol{\xi}$ denotes a vector-valued space-time white noise.

This equation is a mesoscopic model for the dynamics of a finite population in which particles disperse a chemical $c$, to which other particles react by moving up the chemical gradient. Due to the non-linear noise, we do not expect KSDK to be well-posed¹.

Instead, we consider an additive-noise approximation on the two-dimensional torus $\mathbb{T}^{2}=\mathbb{R}^{2}/\mathbb{Z}^{2}$ and establish its well-posedness with the theory of paracontrolled calculus².

Let $\sigma\in C_{T}\mathcal{H}^{2}$ be a space-time inhomogeneity, $\delta>0$ be the correlation length and $\boldsymbol{\xi}^{\delta}$ be a mollified space-time white noise. We show that there exists a sequence of deterministic functions $(f^{\delta})_{\delta>0}$ such that the solutions $\rho^{\delta}$ to \begin{equation*} \begin{cases} (\partial_{t}-\Delta)\rho &= \nabla\cdot(\rho\nabla\Phi_{\rho}-f^{\delta})+\nabla\cdot(\sigma\boldsymbol{\xi}^{\delta}),\\ -\Delta \Phi_{\rho} &= \rho-\langle1,\rho\rangle_{L^{2}(\mathbb{T}^{2})}, \end{cases} \end{equation*} converge as $\delta\to0$ to a unique limit $\rho$.

The sequence of counter-terms $(f^{\delta})_{\delta>0}$ diverges at most logarithmically as $\delta\to0$, whereas naïve power counting would suggest a divergence of order $\delta^{-1}$. This improvement is due to non-trivial symmetries in the Fourier multiplier $\Phi$. Furthermore by the same argument, $(f^{\delta})_{\delta>0}$ is identically equal to $0$ if $\sigma\equiv1$, which shows that inhomogeneities may induce divergences that need to be renormalised by fields rather than constants.