By T. Husain, S.M. Khaleelulla
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There are lots of bits and items of folklore in arithmetic which are handed down from consultant to pupil, or from collaborator to collaborator, yet that are too fuzzy and non-rigorous to be mentioned within the formal literature. commonly, it used to be an issue of success and site as to who discovered such folklore arithmetic.
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Extra resources for Barrelledness in Topological and Ordered Vector Spaces
E. in the probability space. All the rest of the details are left to the interested reader. Acknowledgements. We acknowledge support from the project MTM2008-06349C03-03 DGI-MICINN (Spain), 2009-SGR-345 from AGAUR-Generalitat de Catalunya and IPAM–UCLA where part of this work was done. References  L. A. Ambrosio, N. Gigli, and G. Savaré, Gradient flows in metric spaces and in the space of probability measures. Lectures Math. ETH Zurich, Birkhäuser, Basel, 2005.  J. D. Benamou and Y. Brenier, A computational fluid mechanics solution to the MongeKantorovich mass transfer problem.
Originally, this model was written in Uniqueness of bounded solutions to aggregation equations by optimal transport methods 5 two dimensions with linear diffusion, see , ,  for a state of the art in two dimensions and  in larger dimensions. Therefore, in the rest we will refer to as “PKS equation without diffusion” and the “PKS equation”, respectively. In the case without diffusion, it is known that bounded solutions will exist locally in time and that smooth fast-decaying solutions cannot exist globally.
They contain several mathematical biology models proposed in macroscopic descriptions of swarming and chemotaxis for the evolution of mass densities of individuals or cells. Uniqueness is shown for bounded nonnegative mass-preserving weak solutions without diffusion. In diffusive cases, we use a coupling method , , and thus we need a stochastic representation of the solution to hold. In summary, our results show, modulo certain technical hypotheses, that nonnegative mass-preserving solutions remain unique as long as their L1 -norm is controlled in time.