The Keller–Segel equation on curved planes
Ákos Nagy
University of California, Santa Barbara
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York University, Friday, January 20, 2023.
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!
