Aggregation equation

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 ($K\geq 0$ above) or segregate ($K\leq 0$) and whether the interaction is isotropic. Accordingly, different assumptions on $K$ lead to finite time blow up (aggregation) as well as to finite time extinction.

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