Observability inequalities, unique continuation and applications to control problems for partial differential equations

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NSF Award Abstract - DMS-0072496Mathematical Sciences: Hyperbolic Systems of Conservation Laws - Viscous Conservation Laws - ApplicationsAbstract0072496 TrivisaThis research deals with the general areas of hyperbolic conservation laws and viscous conservation laws. The first set of questions concerns the well-posedness of solutions tohyperbolic systems of conservation laws with large initial data, the uniqueness and regularity of solutions, and the study of some hyperbolic systems of conservation laws in several space dimensions. The second set of questions lies in the area of viscous conservation laws and deals with the study of the compressible Navier-Stokes equations with applications in fluid dynamics and combustion theory. The main objective of this research project is to initiate a systematic investigation of the qualitative behavior of solutions to general multidimensional Navier-Stokes equations with large initial data.This is interdisciplinary research, lying on the interface between continuum physics and the theory of hyperbolic (and viscous) conservation laws. The investigator studies partial differential equations arising in continuum physics with the expectation that the underlying physical structure will direct the analysis, while in return the mathematical analysis of some nonlinear partial differential equations will further the understanding of continuum physics.