The paper presents a mathematical model for describing the behaviour of a crowd in an emergency evacuation situation. The model is based on the gradient flow approach for a suitable functional, according to the well established theory of minimizing movements, introduced in [E. De Giorgi, in: Res. Notes Appl. Math. 29, 81–98 (1993; Zbl 0851.35052)] and developed in the book [L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the space of probability measures. Basel: Birkhäuser (2008; Zbl 1145.35001)].
Given a room \(\Omega\) with an exit \(\Gamma_{\text{out}}\), the model, of Eulerian type, consists in a gradient flow evolution, in the space of probabilities \({\mathcal P}(\overline\Omega)\) endowed with the Wasserstein distance \(W_2\), from an initial given density \(\rho_0\). The functional governing the model is
\[ F(\rho)= \begin{cases} \int_\Omega D(x)\,d\rho\quad &\text{if }\rho\in K,\\ +\infty\quad &\text{otherwise}\end{cases} \]
being \(D(x)\) the distance of the point \(x\in\Omega\) from \(\Gamma_{\text{out}}\) and
\[ K= \{\rho\in{\mathcal P}(\overline\Omega),\;\rho= \rho_\Omega(x)\,dx+ \rho_{\text{out}},\;\rho_\Omega\leq 1,\;\text{supp}(\rho_{\text{out}}) \subset \Gamma_{\text{out}}\}. \]
It is shown that the discrete implicit Euler scheme converges to a limit density-velocity pair \((\rho,u)\) satisfying the continuity equation
\[ \partial_t\rho+ \text{div}(\rho u)= 0,\quad \rho(0,\cdot)= \rho_0, \]
and such that \(u(t,\cdot)= P_{C_{\rho(t,\cdot)}}(-\nabla D)\), where \(P_C\) denotes the \(L^2\)-projection on \(C\) and \(C_{\rho(t,\cdot)}\) is the convex cone of feasible velocities \(u\) with \(\text{div\,}u\geq 0\) on \(\{\rho(t,\cdot)= 1\}\).
Some examples illustrating the model, as well as some modeling issues and possible extensions, are included at the end of the paper.