Entropy solutions for nonlinear degenerate parabolic equations and hyperbolic systems of conservation laws

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The theory of the Cauchy problem for one-dimensinal n×n systems of conservation laws has evolved from a set of works by A. Bressan et al via the front tracking method. The argument developed there is, however, rather complex. In this research we restricted ourself to the case of one-dimensional 2×2 systems of conservation laws and succeeded to make clear the geometrical structure of shock curves for the Riemann problem (with initial data of step functions). Using the structure theory established here we gave a simple argument in the front trucking method and proved the existence of entropy solutions to general Cauchy problems. Besides, we obtained the comparison and existence of entropy solutions to nonlinear anisotropic degenerate parabolic equations with initial-boundary conditions.