Research objectives and content
The objectives of this project are to undertake research in nonlinear evolutions equations related to elastodynamics and the analysis of microstructure. Modern approaches relate nonlinear dynamics of to minimisation of certain functionals to which direct methods from the calculus of variations are applied. Young measure solutions are constructed by weak conver- gence methods. I propose the investigation of existence of Young measure solutions for problems in nonlinear elastodynamics (especially systems of mixed-type exhibiting co-existence of phases) and the regularity properties of measure-valued solutions. At the end of the fellowship I expect to have acquired a better understanding of nonlinear continuum theories and familiarity with modern mathematical techniques. These problems share features with physical models of modern materials such as martensitic solids or shape memory materials. Thus their mathematical understanding should benefit industrial developement. Furthermore, a problem from gauge theo- ries is also proposed, namely the Ginzburg-Landau superconducting gradi- ent flow in a bounded domain.
Key Words: Young measures, nonlinear elasticity, weak convergence, calculus of variations, systems of conservation laws.