Analysis of nonlinear diffusion equations of second and fourth order [Elektronische Ressource] / Maria Pia Gualdani

Analysis of nonlineardi usion equations ofsecond and fourth orderDissertationzur Erlangung des GradesDoktorder Naturwissenschaftenam Fachbereich Physik, Mathematik und Informatikder Johannes Gutenberg-Universit atin MainzMaria Pia Gualdanigeboren in Montevarchi (Italien)Mainz, Juli 2005AbstractDue to the ongoing miniaturization of semiconductor devices, quantume ects play a more and more dominant role. Usually, quantum phenomenaare modeled by using kinetic equations, but sometimes a uid-dynamicaldescription presents several advantages; for example the better tractabilityfrom a numerical point of view and the assignation of boundary conditions.In the following work we study three uid-t ype nonlinear partial di eren tialequations of the second and fourth order; these models are related to themodeling of semiconductor devices. The rst part concerns the study of afully implicit semidiscretization in time and of the long-time asymptotics ofa Fokker-Planck equation of degenerate type. The second part is devoted tothe study of a quantum hydrodynamic model in one space dimension andthe asymptotic decay of the model is formally shown. In the last sectionof the work existence and long-time behaviour of a nonlinear fourth-orderparabolic equation (reduced quantum drift-di usion model) in one spacedimension are proved and some numerical examples are given.2ContentsChapter 1. Introductory Overview 41.1 Short summary of part I . . . . . . . . . . . . . . . .