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Moreover, we also consider the behavior in the presence of nonlinear diffusion terms, the most interesting case being the one of small diffusion coefficients. Via the implicit function theorem we give a quite general proof of a rather natural assertion for such models, namely that there exist stationary solutions that have the form of a local peak around the center of mass. Our approach even yields the order of the size of the support in terms of the diffusion coefficients.

All these results are obtained via a reformulation of the equations considered using the Wasserstein metric for probability measures, and are carried out in the case of a single spatial dimension.

We also introduce an efficient numerical simulation approach based on an appropriate hybrid discontinuous Galerkin method, which in particular allows flexible treatment of complicated geometries. Extensive numerical studies also provide a better understanding of the strengths and shortcomings of the herding model, in particular we examine trapping effects of crowds behind non-convex obstacles.

Our computational approaches are based on alternating direction methods of multipliers (ADMM) and appropriate splittings with augmented Lagrangians. Those are tested for a model scenario related to dynamic positron emission tomography (PET), which is a functional imaging technique in nuclear medicine.

The results of this paper provide insight into the potential impact of regularization with the $\ell^{1,\infty}$-norm for local sparsity in appropriate settings. However, it also indicates several shortcomings, possibly related to the non-tightness of the functional as a relaxation of the $\ell^{0,\infty}$-norm.

In this paper we study balanced growth path solutions of a Boltzmann mean field game model proposed by Lucas and Moll [

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