Department of Mathematics- Seminar on PDE - From large deviations around porous media, to PDEs
with irregular coefficients, to gradient flow structures
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We consider the large deviations of the rescaled zero-range process about its
hydrodynamic limit, the porous medium equation. This leads to the analysis of the
skeleton equation, an energy-critical, degenerate parabolic-hyperbolic PDE with irregular
drift. In this talk, we present a robust well-posedness theory for such PDEs based on
concepts of renormalized solutions, the equation's kinetic form, and commutator
estimates. The relationship of these large deviations principles to a formal gradient flow
interpretation of the porous medium equation will be demonstrated by deducing an
entropy dissipation equality from the large deviations and reversibility.