![]() Markov semigroups and the Harris-Meyn-Tweedie theory. Weighted L2 inequality for the scattering equation. Application to a general Fokker-Planck equation. Relative entropy for linear and positive PDE. A Krein-Rutman theorem for conservative operator. Self-adjoint operator with compact resolvent. Dynamical system, equilibrium and entropy methods. Poincaré inequality and long time asymptotic (with rate) in L2 Fisher information, log Sobolev inequality and long time convergence to the equilibrium (with rate) in L1. ![]() Rescaled (self-similar) variables and Fokker-Planck equation. Smoothing effect thanks to Nash argument. In a second part, we will mainly consider the long term asymptotic issue. Semigroup Hille-Yosida-Lumer-Phillips' existence theory. ![]() Duality argument and the well-posedness issue. Linear evolution equation, semigroup and generator. Uniqueness of solutions thanks to Gronwall argument and duality argument. Existence of solutions by the mean of the characterics method and renormalization theory of DiPerna-Lions. Existence of solutions for parabolic equations by the mean of the variational approach and the existence theorem of J.-L. In a first part, we will present several results about the well-posedness issue for evolution PDE.
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