The Keller–Segel equation on curved planes

Ákos Nagy

University of California, Santa Barbara

Part of this work is joint with Michael Sigal (Toronto)

This talk is partly based on arXiv:2107.12279

York University, Friday, January 20, 2023.

this presentation can be viewed at

The Keller–Segel equation

  • A little biology: chemotaxis is the movement of an organisms (typically bacteria) in the presence of a (chemical) substance with which the organism "interacts".

  • A mathematical model for chemotaxis was given by Keller and Segel in the 1970s:
    • Unknown densities: $\varrho, c : \mathbb{R}_{\mathrm{space}}^n \times \mathbb{R}_{\mathrm{time}} \rightarrow \mathbb{R}_+$
    • Both satisfy continuity equations: $\tfrac{\partial f}{\partial t} + \mathrm{div} (J_f) = 0$, where $J_f$ is the flux and $\sigma_f$ is the source of the quantity $f$.
    • Flux and source for the bacteria: $J_\varrho = \underbrace{- \mathrm{grad} (\varrho)}_{\mbox{diffusion}} + \underbrace{\varrho \: \mathrm{grad} (c)}_{\mbox{chemotaxis}}$ and $\sigma_\varrho = 0$.
    • Flux for the chemical: $J_c = \underbrace{- \mathrm{grad} (c)}_{\mbox{diffusion}}$ and $\sigma_c = \varrho$.

  • This yields the (parabolic) Keller–Segel equations: \begin{align} \left( \partial_t + \Delta \right) \varrho &= - \mathrm{div} \left( \varrho \: \mathrm{grad} (c) \right), \\ \left( \partial_t + \Delta \right) c &= \varrho. \end{align}
  • Example: Dictyostelium discoideum (amoeba) and cyclic-AMP (chemical).

The fast-diffusion Keller–Segel equation on the plane

  • \begin{equation} \left( \partial_t + \Delta \right) \varrho = - \mathrm{div} \left( \varrho \: \mathrm{grad} (c) \right) \quad \& \quad \left( \partial_t + \Delta \right) c = \varrho. \end{equation}

  • Too hard! Reasonable simplification: the chemical diffuses much faster than the bacteria do:
    Replace the second equation with $\Delta c = \varrho$.

  • In order to make sense of $\Delta c = \varrho$, we define \begin{equation} c_\varrho (x, t) := - \frac{1}{2 \pi} \int\limits_{\mathbb{R}^2} \ln \left( |x - y| \right) \varrho (y, t) \mathrm{d} y_1 \mathrm{d} y_2. \end{equation}

  • The fast-diffusion Keller–Segel equation on the plane: \begin{equation} \left( \partial_t + \Delta \right) \varrho = - \mathrm{div} \left( \varrho \: \mathrm{grad} \left( c_\varrho \right) \right). \end{equation}

  • The above equation is still not easy, it is a parabolic, nonlinear, and nonlocal PDE, but it is tamer in many ways, while still relevant for the applications.

  • Side note: This equation also models self-gravitating Brownian dust.

What has been known?

  • \begin{equation} \left( \partial_t + \Delta \right) \varrho = - \mathrm{div} \left( \varrho \: \mathrm{grad} \left( c_\varrho \right) \right) \end{equation}
  • Stationary (time-independent) solutions: \begin{equation} \varrho (x) = \tfrac{8 \lambda^2}{\left( \lambda^2 + |x - x_0|^2 \right)^2}, \quad \lambda \in \mathbb{R}_+ \: \& \: x_0 \in \mathbb{R}^2. \end{equation} Note: Mass = $m := \int \varrho (x) \mathrm{d}^2 x = 8 \pi$.

  • Time-dependent case: the mass is conserved $\oplus$ Virial theorem: \begin{equation} W (t) := \int\limits_{\mathbb{R}^2} |x|^2 \varrho (x, t) \mathrm{d}^2 x \quad \Rightarrow \quad \tfrac{\mathrm{d} W}{\mathrm{d} t} = \left( 8 \pi - m \right) \tfrac{m}{2 \pi}. \end{equation}

  • Small mass = long-time existence, large mass = blow up in finite time, and critical ($m = 8 \pi$) mass = ???

  • Maheux–Pierfelice: Keller–Segel on the hyperbolic plane. (The only study with curvature.)

Adding curvature

  • We want to study the (fast-diffusion) Keller–Segel equation on planes with curvature (or even topology).

  • Motivation: more realistic, interesting study case of a nonlinear and nonlocal geometric PDE, also why wouldn't we, it sounds fun!

  • Question: how does the equation change? Our choice: minimally.

  • Question: how does the geometry change? Our choice: conformally:
    $|v|_{\mathrm{new}, x} = e^{\varphi (x)} \sqrt{v_1^2 + v_2^2}$
    , where $\varphi : \mathbb{R}^2 \rightarrow \mathbb{R}$ is smooth and compactly supported. Gauss curvature $= e^{- 2 \varphi} \Delta \varphi$.

  • The fast-diffusion Keller–Segel equation on the plane: \begin{equation} \left( \partial_t + e^{- 2 \varphi} \Delta \right) \varrho = - e^{- 2 \varphi} \mathrm{div} \left( \varrho \: \mathrm{grad} \left( c_{\varrho, \varphi} \right) \right) \quad \& \quad c_{\varrho, \varphi} (x, t) := - \frac{1}{2 \pi} \int\limits_{\mathbb{R}^2} \ln \left( |x - y| \right) \varrho (y, t) e^{2 \varphi (x)} \mathrm{d} y_1 \mathrm{d} y_2. \end{equation}
  • Maheux–Pierfelice (2020): Keller–Segel on the hyperbolic plane. (They used the same generalization.)

Main results: Stationary case

  • \begin{equation} \Delta \varrho = - \mathrm{div} \left( \varrho \: \mathrm{grad} \left( c_{\varrho, \varphi} \right) \right) \quad (\star) \end{equation}
  • Theorem 1 (N, 2022): Under very mild hypotheses, $(\star)$ is equivalent to the "no-flux equation": \begin{equation} J_{\varrho, \varphi} = - \mathrm{grad} (\varrho) + \varrho \: \mathrm{grad} \left( c_{\varrho, \varphi} \right) = 0 \quad \Leftrightarrow \quad \underbrace{\mathrm{grad} (\varrho)}_{\mbox{diffusion}} = \underbrace{\varrho \: \mathrm{grad} \left( c_{\varrho, \varphi} \right)}_{\mbox{chemotaxis}} \quad \Leftrightarrow \quad \ln \left( \varrho \right) - c_{\varrho, \varphi} = \mathrm{const.} \end{equation} Proof: "Agmon's trick" (integration by parts).

  • Theorem 2 (N, 2022): Under the same hypotheses, any solution of $(\star)$, satisfies $m = 8 \pi$ and $\varrho \approx r^{- 4}$.
    Proof: Stationary, curved Virial theorem.

  • Both theorems are novel already in the noncurved ($\varphi \equiv 0$) case.

Main results: Stationary case

  • Theorem 3 (N, 2022): There are arbitrarily small $\varphi$ so that conformally curved metric carries no stationary solutions.

  • Proof: Duality to the "hard" Kazdan–Warner equation on the sphere: \begin{equation} \Delta_{S^2} u = h_\varphi e^{2 u} - 1. \end{equation}

  • Kazdan and Warner studied the above equation to construct prescribe Gauss curvatures on the sphere.

  • Solvability condition: $\int\limits_{S^2} g_{S^2} \left( \mathrm{grad} \left( u_1 \right), \mathrm{grad} \left( h_\varphi \right) \right) e^{2 u} \mathrm{d} A_{S^2} = 0$, where $u_1$ is an (arbitrary) spherical harmonic of degree 1.

  • Corollary: a radial, decreasing of increasing $\varphi$ cannot satisfy the above condition.

Main results: Time-dependent case
Joint with I. M. Sigal; work in progress.

  • \begin{equation} \left( \partial_t + e^{- 2 \varphi} \Delta \right) \varrho = - e^{- 2 \varphi} \mathrm{div} \left( \varrho \: \mathrm{grad} \left( c_{\varrho, \varphi} \right) \right) \end{equation}
  • Short-time existence: Fixed point methods still work, just complicated by the presence of curvature; good heat kernel estimates are needed.

  • Long-time existence: Modified Virial theorem: \begin{equation} W (t) := \int\limits_{\mathbb{R}^2} f(x) \varrho (x) e^{2 \varphi (x)} \mathrm{d}^2 x \quad \Rightarrow \quad \dot{W} (t) \approx \left( 8 \pi - m \right) \tfrac{m}{2 \pi}. \end{equation}

  • Current state: for each $\varphi$, there is $\epsilon > 0$ and $f$, such that $\left| \dot{W} (t) - \left( 8 \pi - m \right) \tfrac{m}{2 \pi} \right| < \epsilon$.
    Proof: One can write down a PDE for $f$ and then prove that a solution, with the desired properties, exists.

  • Corollary: Small enough and large enough masses "behave as they should".

Future plans

  • Prove that $m = 8 \pi$ is still critical. Bonus points for the "meaning" for $8 \pi$.

  • Asymptotics: large time approximations for small masses and blow-up profiles for large masses.

  • More general geometries.

  • More general topologies!

  • Study of the full Keller–Segel system.

Thank you for your attention!