Some mathematical studies on general systems of hyperbolic conservation laws

Abstract

This thesis is concerned with the mathematical studies on the general system of conservation laws. The research in this area has been one of the most important and challenging fields in mathematics not only because of many unsolved mathematical problems, but also because of its rich physical background and practical applications. In Chapter 1 we review some basic results on hyperbolic conservation laws. Our works are all based on or motivated by the classical results in this Chapter. In the first part of Chapter 2 we introduce a new nonlinear functional which improves the one given in [35] and it can be viewed as a better attempt for the generalized entropy functional for general equations. In the second part of Chapter 2 we give a new measure about the rarefaction waves, and a sharp decay estimate of the new measure is established for the cubic nonlinear system of conservation laws. In the third part of Chapter 2 we study some decay estimates in nonlinear hyperbolic system of conservation laws. By introducing a proper Glimm functional, we obtain some useful decay estimates which are proved helpful in obtaining decay rates of the admissible solutions to nonlinear hyperbolic conservation laws as t→∞. Keywords: conservation laws, wave tracing method, Glimm scheme, Glimm functional, decay rates