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David Kaspar : Scalar conservation laws with Markov initial data

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The inviscid Burgers' equation has the remarkable property that its dynamics preserve the class of spectrally negative L\'{e}vy initial data, as observed by Carraro and Duchon (statistical solutions) and Bertoin (entropy solutions). Further, the evolution of the L\'{e}vy measure admits a mean-field description, given by the Smoluchowski coagulation equation with additive kernel. In this talk we discuss ongoing efforts to generalize this result to scalar conservation laws, a special case where this is done, and a connection with integrable systems. Includes work with F. Rezakhanlou.

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