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和非线性退化抛物方程的熵解问题。我们得到的主要结果如下：1。引入了具有局部Lipschitz连续通量函数的标量守恒律重正化耗散解的新概念，并证明了该解等价于Benilan等人在（无界）L1解研究中引入的重正化熵解。我们还构造了一个L1条件下收缩松弛系统的重正规化耗散解。2. 研究了标量守恒律柯西问题的两类无界弱解，即重正规化熵解和动力学解。证明了如果u是一个动力学解，那么它确实是一个重整熵解。反过来，我们证明了如果u是一个满足附加条件的重整化熵解，那么它就是一个动力学解。3. 研究了具有初始边界条件和非齐次边界条件的非线性退化抛物方程的比较原理。证明了任意（有界）熵的子解和超解的比较定理。目前为止已经得到了（无界）熵解的L1收缩性和唯一性，但比较结果是新的。使用的方法是所谓的加倍变量技术，由于克鲁兹科夫。我们的方法是基于动力学公式和动力学技术。通过发展带边界条件的退化抛物方程的动力学技术，得到了一个比较性质。4. 我们将上述第3项的结果推广到无界熵解的情况。我们在L1框架下考虑这个问题，证明了重正化熵解的比较定理和存在性定理。

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.