Mathematical Sciences: Nonlinear Partial Differential Equations and Statistical Physics

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9623220 Kiessling A basic strategy in statistical mechanics of nonhomogeneous systems is to construct an asymptotically exact nonlinear PDE problem with few variables which approximates the original linear problem with its overwhelmingly many variables. In the first part of the proposal we deal with two unsolved problems of this kind, namely the microcanonical ensemble for: point vortices; for classical gravitating hard particles. In both systems one has encountered technical problems due to nonconcavity of the entropy. We propose a new strategy that circumvents these problems by reinstalling concavity in a higher-dimensional parameter space from which the original problem obtains by some kind of projection. Progress in this area should significantly advance our understanding of nonlinear PDE without convexity and their relation to statistical mechanics. In the second part of the proposed project we deal directly with properties of systems of nonlinear PDEs that are firmly established in their relationship to statistical mechanics. Our main interest is in the symmetry properties of solutions. We recently constructed sharp isoperimetric estimates for two-dimensional elliptic PDE systems and compared them to a priori identities of Rellich-Pohozaev type, thus firstly obtaining conditions under which solutions are radially symmetric without requiring uniqueness or minimizing properties. We want to continue this research and optimize the conditions for two-dimensional elliptic systems such as Ginzburg-Landau and Bennett equations, extend the technique to higher dimensions and apply it to Thomas-Fermi models, finally extend the technique to parabolic transport equations such as Landau-Boltzmann equations, which in particular should yield a global existence result in time. %%% Long-lived vortices are an ubiquitous feature in turbulent flows, with hurricanes in the Earth's weather system and the great red spot in Jupiter's atmosphere being two promin ent examples. Vortices also occur as structural defects in superfluids, superconductors. Beside being of scientific interest, it is of pressing technical and meteorological importance to understand precisely the conditions under which such vortices do form. An integer part of the proposed project aims at making a significant contribution to this endeavor. The mathematical framework consists of certain systems of nonlinear partial differential equations which are deeply rooted in the scientific discipline of statis- tical mechanics. The techniques which shall be developed in the first part of the proposed project will allow us to extract the relevant differential equations from statistical mechanics under far more realistic conditions than treated so far. In the second part, we shall further develop a recent technique of us that yields qualitative and quantitative statements about the symmetry properties of the solutions to the nonlinear equations. In favorable cases this reduces the complexity of certain equations significantly. Beside vortices, we aim at applying our techniques to the problem of controlling more realistic plasma structures than previously treated, of the following categories: charged particle beams, which are of preeminent interest in various branches of technology development; stellar structures, which are of basic astrophysical interest. Finally, we see the possibility of an extension of our techniques to a dynamical set of transport equations for plasmas which have significance, in particular, for peaceful thermonuclear energy research. ***