Mathematics Colloquia and Seminars

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The question is how to obtain hydrodynamic equations (Euler, Navier-Stokes) from Liouville-type equations of Hamiltonian mechanics (classical or quantum). The original idea was due to Ch. Morrey (1956) who introduced a concept of a hydrodynamic limit and was able to formally derive an Euler equation from the classical Liouville equations (more precisely, from the corresponding BBGKY hierarchy). However, Morrey had to make some assumptions about the long-term behavior of the motion, and this included a statement on ergodicity, in the sense that all `reasonable' first integrals are functions of the energy, linear momentum and the number of particles. Since then the idea of hydrodynamic limit became very popular in the literature and has been successfully applied to a variety of models of (mostly stochastic) dynamics relating them to non-linear equations. However, in the original problem there was no substantial progress until the work by S. Olla, S.R.S. Varadhan and T. Yau (1992) where Morrey's assumptions were replaced by introducing a small noise into the Hamiltonian (which effectively kills other integrals of motion), and a classical Euler equation was correctly derived. The talk will be on a quantum model for which the hydrodynamic limit can be rigorously performed. The resulting Euler-type equation is similar to the one that arises for the classical counterpart of the same model. This suggests that perhaps classical and quantum hydrodynamic equations must look sismilar if they are written for local densities of `canonical' conserved quantities (the density of mass, linear momentum and energy). The work is in collaboration with C.Boldrighini (University of Rome-1) and A. Pellegrinotti (University of Rome-3).