Quantifying and minimizing collision intensity in stationary crowd flow
We develop a continuum model to quantify kinematic collision risk in stationary crowd flow. From the density-weighted inward relative flux, we derive a collision functional based on the symmetric velocity gradient. For incompressible flows with constant density, this functional is convex and positively homogeneous of degree one in a combination of second derivatives of the stream function. We minimize it subject to prescribed boundary flux. On bounded simply connected domains with Lipschitz boundary, under a finite-energy lifting assumption, we prove existence of minimizers in a relaxed class that allows this derivative combination to be a vector-valued Radon measure. We show that disks are the only domains with a nontrivial zero-trace kernel, corresponding to rigid rotation about the center. Using the compact embedding for functions of bounded deformation, we prove that relaxed energy sublevels are compact without normalization on every other admissible domain, and after removing one rotational component on disks. We identify the relaxed functional with the strong $W^{1,1}$ lower semicontinuous envelope of the functional on the prescribed-trace class with integrable controlled derivatives. Then, we derive optimality conditions via convex duality. We construct explicit radial flows on annular sectors and prove that they minimize the energy over the full admissible classes, including nonradial competitors. On the square, distinct minimizers with the same prescribed boundary flux have different local collision intensities. The square--disk comparison shows that the saddle flow loses optimality when the domain changes, even after normalization.