The investigator will study the Grad generalized differential equations which model the equilibrium and slowly varying evolution of thermonuclear confined plasmas in axisymmetric toroidal devices. Nonlinear and nonlocal terms enter these equations through derivatives of the increasing rearrangement of a function which models the poloidal flux of the magnetic field. The first and second derivatives of this flux correspond respectively to magnetic field and current profiles. It is important for physical reasons to determine whether these profiles can develop singularities at the magnetic axis of the torus. Professor Laurence will analyze the regularity of minimizers of a variational formulation of these equations. He will treat the case in which the polodial cross section is convex. He will introduce a set of approximating variational problems and examine the minimizer of the original variational problem as a limit. The m'th approximating problem is a free boundary problem with m free boundaries. He will try to extend regularity results for the approximating problems to the regularity of a weak solution to the original problem. The increasing rearrangement was first introduced by Hardy, Littlewood and Polya. Its properties are of interest in many areas of analysis, from harmonic analysis to partial differential equations and isoperimetric inequalities. Professor Laurence will examine how derivates of the rearrangement are affected by isolated critical points of the underlying function. He will also study convex symmetrization of functions over convex domains or over annuli A with convex boundaries, as well as certain isoperimetric inequalities related to level set analysis. The project falls into the general area of applied mathematics, with applications to plasma physics.
