Problems in Hyperbolic Field Theories

PI: Abdolreza Tahvildar-Zadeh, Rutgers University

DMS-0301207

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Problems in Hyperbolic Field Theories

This is a three-year proposal for studying some of the hyperbolic systems of partial differential equations arising in physical theories that are derivable from a Lagrangian, focusing on questions of long-time existence and asymptotic behavior of classical solutions, mapping properties of the linear operators involved, and nonlinear dynamical stability of static and stationary solutions. Specific problems proposed are (1) Obtaining space-time Strichartz estimates for solutions of the linear wave and Schroedinger equations in presence of potentials with critical (i.e. inverse-square) decay at infinity and/or with local singularities. (2) Proving stability of vortex-like wave maps from the Minkowski space into the sphere, utilizing the above estimates. (3) Obtaining a sharp dispersive estimate for solutions of the anisotropic Maxwell equations of crystal optics, one that encodes the direction-dependence of the decay. (4) Proving global existence of small-amplitude waves for the Euler-Maxwell system describing the dynamics of plasma modeled by an electron fluid moving in a constant ion background. (5) Using the wave map formulation of symmetry-reduced Einstein equations of general relativity to obtain results on the future and past asymptotic behaviors of Gowdy metrics, on the existence of constant mean curvature hypersurfaces in symmetric spacetimes with twist, on the oscillatory approach to the initial singularity in these spacetimes, and on the global existence for the Einstein-Vlasov system in cylindrical symmetry.

Field Theory is the most enduring paradigm of classical as well as modern physics. Electromagnetics, fluid and solid mechanics, weak and strong interactions of elementary particles, and Einstein's theory of gravitation are all describable in the framework of a field theory. One of the most important physical phenomena to be understood in this framework is the phenomenon of waves, their creation, propagation, interaction, and dispersion. Some examples are electromagnetic waves, material waves, and gravitational waves. Each of the problems proposed here has a direct consequence in the understanding of a specific aspect of the wave phenomenon. With the dawn of a new century, as advances in technology force scientists to address the inherently nonlinear behavior of nature in more detail than ever before, mathematical analysts are in a position to take up the challenge of doing research in those areas of physical mathematics that have long been neglected by others. This fundamental research involves going beyond numerical simulations and approximate equations, and addressing the hard problems that lie at the core of the subject, i.e. in the theory of nonlinear partial differential equations. Understanding nonlinear waves is an important step in this direction. This is very much a collaborative effort, and in particular collaborations with members of mathematical communities in

other parts of the world where a tradition of caring about physical problems is well-maintained, provides us with an opportunity to play a role in preventing the erosion of the leading status of the US in these key areas of mathematical sciences.