This project aims at establishing global regularity/asymptotic Stability results for certain geometric/physical nonlinear wave equations. Also, it is hoped that further insight into possible singularity formation is gained. The first goal is to generalize the PI's earlier result on gobal regularity of Wave Maps with small energy and smooth data from the three dimensional Minkowski space to the Lobachevskyi plane to enompass the case of arbitrary targets of bounded geometry, using partly the PI's method(this is jointly with D. Tataru). The next goal is to analyze the global stability of certain large-energy Wave Maps (geodesic, spherically symmetric, equivariant) under small-energy perturbations of the initial data (assumed smooth). It is expected that certain geometric properties of the target (geodesic convexity, negative curvature) should ensure that the Wave Maps behave like a free wave at infinity. A related goal is to establish the stability of stationary Wave Maps, expected based on physical intuition. Finally, the PI hopes to make progress on understanding possible singularity formation either for Wave Maps(when the target is a sphere) or related equations, such as Yang Mills. One toy problem would be to investigate the stability of self-similar blow-up solutions for Wave Maps, e.g., known in dimensions four and higher.
The significance of this work consists in hopefully paving the way for a better understanding of the extremely complex Einstein's equations(these are related to the Wave Maps discussed above) which govern the spacetime geometry of our universe, as well as other nonlinear wave eqations coming up in the description of fundamental natural phenomena, for example equations of elasticity or the Yang-Mills equations describing elementary particles. The subject is beautifully situated at the crossroads of several purely mathematical disciplines, such as Harmonic Analysis and Geometry, as well as more applied areas, such as Numerical Mathematics(computational simulations of partial differential equations), Physics and Cosmology. Indeed, one of the motivations for studying the qualitative behavior of breakdown of solutions, as occurs for example during gravitational collapse (Black Hole formation), is to lead to more accurate numerical modelling. The PI hopes that his research will not only be of interest to pure mathematicians but also researchers in these other disciplines.
