"Smoluchowski-Poisson dynamics" is just a tentative provisional name I give to the following transport equation:$$\partial_t m+\nabla_x\cdot(um)=0$$where $u(x,t)\in\mathbb R^n$ ($x\in\mathbb R^n$, $t\ge0$) is the gradient of a solution $f$ to the Poisson equation$$u=\nabla f,\ \ \Delta_x f=c'm$$i.e. (more precisely)$$u(x)=c\int_{\mathbb R^n} \frac{(y-x)\ m(dy)}{|y-x|^n}$$($c$ or $c'$ is a physical constant). The case I'm interested in is when the sign of $c$ is such that the velocity $u$, by which the (nonnegative) measure $m$ is transported, is attractive, resulting in atoms being formed and coalescing in finite time (to a final single atom if $m$ is finite).
My question is twofold: 1°) Does this equation have a name? 2°) Is there a theory of existence and uniqueness of solutions for initial conditions of the form $m(.,0)=\sum_1^N m_i\delta_{a_i}$ ? (One has to properly interpret the product $um$, as $u$ is singular. This amounts to replacing the original $u$ with $u(x)=c\int_{\mathbb R^n\setminus \{x\}} \frac{(y-x)\ m(dy)}{|y-x|^n}$).
Miscellaneous comments: 1.This is one of a family of models of chemotaxis in physics and biology, maybe the simplest. 2. For $n=1$ it is related to Burgers equation through a (rather obvious) change of variables. 3. There seems to exist interesting statistical solutions of this equation (not published yet if I'm not mistaken), some of which I believe should provide a paradigmatic example of spontaneous stochasticity (ill-posed Cauchy problem with a random solution), an important phenomenon in the physics of turbulence.
