A structure of solutions of initial-boundary value problems for parabolic-hyperbolic equations and applications to numerical analises

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This research is mainly concerned with entropy solutions for scalar conservation laws as well as nonlinear degenerate parabolic equations. The main results which we obtained are the followings : 1. We introduce a new notion of renormalized dissipative solutions for scalar conservation laws with locally Lipschitz continuous flux-functions and prove that this solution is equivalent to renormalized entropy solutions which are introduced by Benilan et al. in the study of (unbounded) L1 solutions. We also construct a renormalized dissipative solution for contractive relaxation systems in merely an L1 setting. 2. We study two types unbounded weak solutions of the Cauchy problem for scalar conservation laws, that is a renormalized entropy solution and a kinetic solution. It is proved that if u is a kinetic solution, then it is indeed a renormalized entropy solution. Conversely, we prove that if u is a renormalized entropy solution which satisfies a certain additional condition, then it becomes a kinetic solution. 3. We study the comparison principle for nonlinear degenerate parabolic equations with initial and non-homogeneous boundary conditions. We prove a comparison theorem for any (bounded) entropy sub-and super-solution. The L1 contractivity and therefore uniqueness of (unbounded) entropy solutions has been obtained so far, but a comparison result is new. The method used there is the so called doubling variable technique due to Kruzkov. Our method is based upon the kinetic formulation and the kinetic technique. By developing the kinetic technique for degenerate parabolic equations with boundary conditions, we obtain a comparison property. 4. We extend the above result stated in the item 3 to the case of unbounded entropy solutions. We consider the problem in an L1 framework and prove a comparison theorem and existence theorem for renormalized entropy solutions.