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Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations. (English) Zbl 1215.35045
The paper deals with a well-posedness theory for weak measure solutions of the Cauchy problem for a family of nonlocal interaction equations. These equations are continuum models for interacting particle systems with attractive/repulsive pairwise interaction potentials. The authors develop an existence theory that enables one to go beyond the blow-up time in classical norms and allows for solutions to form atomic parts of the measure in finite time. The weak measure solutions are shown to be unique and exist globally in time. Moreover, in the case of sufficiently attractive potentials, the authors show the finite-time total collapse of the solution onto a single point for compactly supported initial measures. The presented approach is based on the theory of gradient flows in the space of probability measures endowed with the Wasserstein metric. In addition to classical tools, the authors exploit the stability of the flow with respect to the transportation distance to greatly simplify many problems by reducing them to questions about particle approximations.

MSC:
 35D30 Weak solutions to PDEs 35B44 Blow-up in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs 45K05 Integro-partial differential equations 49K20 Optimality conditions for problems involving partial differential equations
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