This paper considers the Cauchy problem for a conservation law with a variable unilateral constraint, its motivation being, for instance, the modeling of a toll gate along a highway. This problem is solved by means of nonclassical shocks and its well posedness is proved. Then, the solutions so obtained are shown to coincide with the limits of the classical solutions to suitable conservation laws with discontinuous flux function that approximate the constrained problem.
We present a new class of macroscopic models for pedestrian flows. Each individual is assumed to move towards a fixed target, deviating from the best path according to the instantaneous crowd distribution. The resulting equation is a conservation law with a nonlocal flux. Each equation in this class generates a Lipschitz semigroup of solutions and is stable with respect to the functions and parameters defining it. Moreover, key qualitative properties such as the boundedness of the crowd density are proved. Specific models are presented and their qualitative properties are shown through numerical integrations. In particular, the present model accounts for the possibility of reducing the exit time from a room by carefully positioning obstacles that direct the crowd flow.
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