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

本文主要研究标量守恒律方程和非线性退化抛物方程的熵解。主要结果如下：1.本文对具有局部Lipschitz连续流函数的标量守恒律引入了重整化耗散解的新概念，并证明了此解与Benilan等人在研究（无界）L1解时引入的重整化熵解是等价的.我们还构造了一个重整化耗散解的压缩松弛系统在仅仅一个L1设置。2.研究了标量守恒律Cauchy问题的两类无界弱解，即重整化熵解和动力学解。证明了如果u是一个动力学解，那么它确实是一个重整化熵解。相反，我们证明了，如果u是一个重整化熵解，满足一定的附加条件，那么它成为一个动力学解。3.研究了具有初始和非齐次边界条件的非线性退化抛物型方程的比较原理。我们证明了一个比较定理的任何（有界）熵的子和上的解决方案。到目前为止，已经获得了（无界）熵解的L1收缩性和唯一性，但比较结果是新的。使用的方法有所谓的加倍变量技术由于Kruzkov。我们的方法是基于动力学配方和动力学技术。通过发展退化抛物型方程边值问题的动力学技巧，得到了一个比较性质。4.我们将第3项中的结果推广到熵无界解的情形。在L1框架下研究了该问题，证明了重整化熵解的比较定理和存在定理。

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.