Fokker-Planck and Mean-Field Equations

Research project P 14876 Fokker-Planck and Mean-Field Equations Peter MARKOWICH 27.11.2000 The transient behaviour of charged many-particle systems is in several physical applications (plasma, semiconductors) governed by (non)linear (phase space) Fokker Planck systems (in the sequel FP) or by (non)linear drift-diffusion systems in position space(in the sequel: DDP). Aside from existence- and uniqueness questions the concise description of the long-time behaviour of such systems is of distinctive importance for mathematical and physical purposes. Most recently the so-called "entropy method" has become the most powerful tool for a analytical description of FP`s and DDP`s long-time behaviour. A main goal of this method is to prove convergence (as "time t -+ oo") of a thermodynamical system to its equilibrium state and to determine the rate of convergence. Usually the time-dependent (mathematical, i.e. the negative physical) entropy E(t) of the system is considered. Typically E(.) is a decreasing function of time tending to its minimium value as t . However the entropy methods developed so far are applicable only in particular situations requiring, e.g., particular convex Sobolev inequalities, particular assumptions on confining potentials, a given electrostatic field, the existence of an equilibrium solution minimizing an energy functional etc. Also, the entropy (-entropy dissipation) technique is not yet well-developed (from a mathematical point of view) for quantum systems. It is the aim of the proposed project to develop entropy methods for situations where the existing results do not apply. In particular it is planned * to prove existence (and uniqueness) of equilibrium solutions of DDP minimizing the corresponding energy, * to investigate the free boundary problem arising in the limit "Debye length to 0" for these equilibrium solutions, * to develop an entropy method for a larger class of nonlinear DDP, * to prove existence, uniqueness and nonlinear stability of Schr6dinger-Poisson with general distributions, * To investigate spatially i nhomogeneous (phase space) quantum FP.