Modelling Aggregation-Diffusion Equations

Let us discuss the modelling for the so-called Aggregation-Diffusion Equations

\[\tag{ADE} \label{eq:ADE} \frac{\partial \rho}{\partial t} = \mathrm{div} \Big ( \rho \nabla ( U'(\rho) + V + {W * \rho } ) \Big).\]

In aggregation-diffusion equation we model distribution of a quantity, described in terms of its density $\rho$, i.e.

\[\textrm{amount of quantity in } A \textrm{ at time } t = \int_A \rho (t, x) d x.\]

Diffusion

We start with diffusion. A standard approach is through continuity equations of the form

\[\partial_t \rho = - \mathrm{div} \, \mathbf j\]

where \(\mathbf j\) is called flux, and describes the direction and ammount of flow of the quantity. For the heat equation we model the quantity moving from places of high concentration to places of low concentration, in particular opposite to the flux \(\mathbf j = - \nabla \rho\). This relation is called Darcy’s law. To describe the flow in some mediums, a non-linear version of Darcy’s law can be introduced: \(\mathbf j = - \nabla \rho^m\). This leads to the so-called Porous Medium Equation

\[\tag{PME} \label{eq:PME} \partial_t \rho = \Delta \rho^m\]

Notice that \(\Delta \rho^m = \mathrm{div} (\rho \nabla U'(\rho)).\) For a detailed modelling and many references we point to [1].

Aggregation and Confinement

Consider $N$ with positions \(X_i\) of masses \(a_i\) and the attracting/repulsive system

\[\frac{d X_i}{d t} = - \sum_{\substack{ j=1 \\ j \ne i}}^N { a_j \nabla W (X_i - X_j) } { - a_i \nabla V (X_i) } , \qquad i = 1, \cdots, N.\]

We say that $W$ is an aggregation potential if \(\nabla W (x) \cdot x \ge 0\), and \(V\) we say it is a confining potential if \(\nabla V(x) \cdot x \ge 0\).

Using Mean-Field theory, the distribution of these particles as \(N \to \infty\) satisfies the Aggregation Equation

\[\label{eq:AE} \tag{AE} \partial_t \rho = \mathrm{div} (\rho \nabla ( W * \rho + V ) )\]

For details on the Mean-Field limits see [2]. The solutions of this problem are often measures, and a suitable notion of weak solutions needs to be introduced.

Joining \eqref{eq:PME} and \eqref{eq:AE} we recover \eqref{eq:ADE}.

References

  1. Vázquez, Juan Luis. (2006) The Porous Medium Equation. Oxford University Press. Link
  2. Jabin, Pierre-Emmanuel and Wang, Zhenfu. (2017) Mean Field Limit for Stochastic Particle Systems , in Active Particles, Volume 1. Springer International Publishing. Link