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Partial differential equations of Classical Physics

Final Report Summary - PDECP (Partial differential equations of Classical Physics)

The project was chiefly aimed at understanding the phenomena of shock formation and development. Compressible fluids is the most familiar context in which shocks occur, showing up as the spontaneous generation first of infinite gradients and then of discontinuities in the fluid velocity and the thermodynamic variables, such as pressure and temperature, describing the local state of the fluid. The laws of evolution in this context are the compressible Euler equations. Another context in which shocks occur is electromagnetic wave propagation in nonlinear dielectrics, where shocks similarly arise as the spontaneous formation of infinite gradients in the electromagnetic field. Here the evolution is governed by Maxwell's equations, with a nonlinear relation between the electric field and the electric displacement.
In both cases we have nonlinear 1st order systems of hyperbolic partial differential equations with the characteristics depending on the unknowns, and shock formation is signaled by the degeneration of foliations by characteristic hypersurfaces. Methods which allow the complete analysis of the shock formation problem were introduced by the principal investigator and developed further in his monograph with S. Miao on the classical Euler equations. These methods capitalize on the fact that the equations satisfied by variations through solutions of a solution of the original nonlinear system depend only on the Lorentzian geometry defined on the underlying spacetime manifold by the characteristic cone field. An essential role is played by the study of the relation between this Lorentzian geometry, as manifested in the geometry of foliations by characteristic hypersurfaces, and the Galilean structure of the underlying physical spacetime (Euclidean spatial geometry). These methods, in combination with the short pulse method which was introduced by the principal investigator in his study of the formation of black holes in general relativity, have led S. Miao and P. Yu to the solution of a scalar model problem for electromagnetic shock formation. The same point of view inspires the work of the principal investigator with D.R. Perez on the formation of shocks in the propagation of electromagnetic plane waves
in a nonlinear crystal dielectric. The further development of shocks, with the generation and evolution of discontinuities, had been formulated by the principal investigator as a free boundary problem with initial conditions on a characteristic hypersurface which are singular on its past boundary surface. This problem was solved, in the framework of relativistic fluid mechanics, in the spherically symmetric case, in joint work of the principal investigator with Andre Lisibach. The research work on the general case by the principal investigator was completed in November 2015, and a research monograph of about 500 pages in length is currently being written up which is intended for publication in the EMS Monographs in Mathematics series of the EMS Publishing House.

In a different line of research, the principal investigator and I. Kaelin have studied the mechanics of crystalline solids with a continuous distribution of dislocations. In particular they have solved the problem of the equilibrium configurations in free space of solids containing uniformly distributed dislocations of the simplest types, determining the resulting internal stress
fields. This involves the solution of geometric nonlinear systems of elliptic partial differential equations.