Non-linear partial differential equations of kinetic type
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This thesis is concerned with the analytical study of non-linear partial differential equations (PDEs) of kinetic type which admit multiple stationary solutions. We consider a kinetic model which is given by a non-linear PDE in the sense of McKean and describes the time-evolution of the density ft = ft(x, v), (x, v) 2 T ⇥ R, of a collection of interacting particles moving in the one dimensional torus. We focus on tackling the main diculties which arise from the fact that the non-linear PDE has unbounded coecients, is non-elliptic and not in gradient form. In particular, we employ techniques to show well-posedness of the solution ft in a weighted Lp space. When the density ft does not depend on the spatial variable, we study the long time behavior of the spacehomogeneous PDE by using and comparing two different approaches: hypocoercivity theory and gradient flow theory.