This research addresses fundamental issues in the theory of hyperbolic systems of conservation laws, in one or more space dimensions. The project also includes the analysis of some new physical models of nonlinear wave propagation, and applications to the theory of differential games.
For systems of equations in two or three space dimensions, the P.I. will study basic problems concerning the existence and stability of solutions, in suitable functional spaces. A major goal will be the description of singularities determined by the focusing of nonlinear waves in radially symmetric solutions, and the understanding of oscillation effects, arising from transport phenomena with highly irregular velocity fields. The P.I. also plans to develop new mathematical techniques for the analysis of non-linear waves, described by a class of integro-differential equations. The theoretical results will be complemented by some rigorous studies on the performance of computational algorithms, for solutions containing shock waves.
Phenomena related to the propagation of waves can be found nearly everywhere in nature, and are of great importance in science and engineering. Non-linear effects are responsible for changes in the shape of the waves, and allow what is mathematically described as "singularity formation". In practice, this means that waves can break, such as sea waves rolling up along a beach or shock waves forming at the passage of a supersonic airplane. In two or three space dimensions, the singularities can be even more dramatic because of "focusing" effects, producing a high concentration of energy near a single point. This is exemplified by an optical lens, concentrating sun rays at a single spot. In the numerical computation of solutions, the loss of regularity due to wave breaking is a considerable source of difficulties, because it can greatly reduce the accuracy of computer algorithms.
This research project is concerned with mathematical models describing the propagation of nonlinear waves. Its main goal is to understand the formation and the evolution of singularities, in one or more space dimensions. Particular attention will be given at specific equations of physical significance, such as one describing waves in a liquid crystal. The research will also address some issues related to numerical computation. In particular, the P.I.~will study the effectiveness of discrete approximations, in case of solutions containing shock waves. Moreover, work will begin on the design of algorithms that can automatically recognize the location of shock fronts, with the eventual goal of producing more efficient computational codes.
