Asymptotic analysis of non linear diffusion partial differential equations and associated functional inequalities

Abstract : This work is dedicated to the study of the large time behaviour of some parabolic type partial differential equations. More specifically, we look into non linear diffusion equations that appear in a number of models arising in physics (e.g. the porous medium equation) or biology (e.g. the Patlak-Keller-Segel model for chemotaxis)Chapters I and II deal with an improved Sobolev inequality by means of its dual, the Hardy-Littlewood-Sobolev inequality, in the framework of the standard and fractional Laplacian, respectively. Chapter III is a review of the Onofri inequality,which acts as the Sobolev inequality for dimension two. New results are provided, and some of them are extended to Riemannian manifolds in Chapter IV. Finally, Chapter V deals with the stationary states of two parabolic models, used for thestudy of crowd motion and modeling in biologie (chemotaxis).
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Gaspard Jankowiak. Asymptotic analysis of non linear diffusion partial differential equations and associated functional inequalities. General Mathematics [math.GM]. Université Paris Dauphine - Paris IX, 2014. English. ⟨NNT : 2014PA090013⟩. ⟨tel-01067226⟩

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