This project is aimed at the analysis of hyperbolic systems of conservation laws. The focus is on: (1) vanishing viscosity limits to the multi-dimensional Euler equations of gas dynamics; and (2) new representation formulas for solutions of conservation laws. The first project aims at the characterization of limits of families of the vanishing viscosity, radial solutions to the multidimensional Euler equations with large, discontinuous data. The methods of compensated compactness will be adopted for systems of equations that do not possess invariant regions, using only the balance of total energy. In the second project the Principal Investigator will establish a new kinetic formulation for hyperbolic conservation laws. In this approach solutions of conservation laws are represented by probability measures on the phase space which, in turn, are represented as divergences of vector fields obtained as values of a contraction semigroup on suitable Hilbert spaces.
Hyperbolic systems of conservation laws are fundamental equations in physics. They model diverse phenomena in dynamics of gases, elasticity, and electromagnetism. Because of their importance in applications these equations have been extensively studied. However, there is no complete theory that allows one solving the equations for generic data and describing properties of the solutions. The project introduces several innovative analytical tools to approach these issues and creates a theoretical basis for solving a large variety of equations of this type. The research will potentially have impact on the areas of computational and applied mathematics where the hyperbolic systems are involved. Results of this research will be disseminated through presentations at national and international conferences, seminars and publications in scientific journals. Graduate student projects will be integrated into this research.