This project continues mathematical research combining general operator-theoretic functional analysis with differential equations. Three overlapping areas will be treated in the work. The first concerns nonlinear partial differential equations of evolution which are of mixed type in the space variable. Initial work will focus on a concrete example, the Kompaneetz Equation of plasma physics. It can be written as an abstract Cauchy problem. The first task is to determine properties of the resolvent; in particular whether or not it has a dense range. The second area involves semigroups of operators, particularly those whose infinitesimal generators split into a linear and locally smooth nonlinear part. The object is to determine conditions for a globally defined semigroup of operators determining the solution for all time and to bound the difference of any two solutions as a function of time and the initial function norms. Finally, work will be done on spin polarization Thomas-Fermi theory involving a minimizing problem for pairs of functions. One main obstacle to finding extrema is the functional involving the two unknowns is not convex. By topological arguments, one can show that extrema exist. The next step will be to find a method which will produce the minima in a concrete fashion. The project combines abstract mathematical concepts with models of various phenomena in the physical world. Among the goals of the research is one of fitting models into the proper generalized structures and apply the existing theory. In this way one gains insight into the models and, at times, discovers hidden properties.
