Aggregation equation

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The aggregation equation consists in the scalar equation

\[\begin{array}{rll} u_t+\text{div}(uv) & = 0 & \text{ in } \mathbb{R}^d \times \mathbb{R}_+\\ u(x,0) & = u_0\\ v (x,t) & = -\nabla (K*u(.,t))(x) & \end{array} \]

where the kernel $K$ satisfies several properties but always being such that $\Delta K \in L^1_{\text{loc}}(\mathbb{R}^d)$. This equation arises in many models in biology, where $u_0$ represents the density of some population that is self-interacting through the vector field $v$, in this context, $K$ determines many properties of the interaction between different "agents" within the population. For instance the sign of $K$ determines whether there is a tendency to aggregate or segregate (depending on whether $-x \cdot \nabla K(x)$ is always negative or always positive, respectively), or whether the interaction is isotropic, or whether it enjoys a homogeneous scaling (when say $K$ is homogeneous). Accordingly, different assumptions on kernel lead to finite time blow up or extinction. In the biological literature a common kernel is $K(x) = |x|$.

  • Note that these equations can be seen as a Wasserstein Gradient Flow (only possibly for some choices of $K$)
  • Note the similarities and differences when the energy generating these dynamics is written in a Nonlocal Phase Field model

Note the close connection with the nonlocal porous media equation, the key difference being that $\Delta K$ is not locally integrable in that case.

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