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

抛物双曲方程初边值问题的解结构及其在数值分析中的应用

基本信息

批准号：
16540174
负责人：
KOBAYASI Kazuo
金额：
$ 1.6万
依托单位：
Waseda University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

项目摘要

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.

这项研究主要涉及标量保护法的熵解决方案以及非线性退化抛物线方程。我们获得的主要结果是以下内容：1。我们引入了一种具有局部Lipschitz连续通量功能的标量保护定律的重新规定耗散溶液的新概念，并证明该溶液等于Renilalan et ef benililan et allialan et al trused fuls。在（无限）L1溶液的研究中。我们还为仅在L1设置中的收缩放松系统构建了重新归一化的耗散解决方案。 2。我们研究了标量保护定律的两种类型的库奇问题的无界弱解，即是重新归一化的熵溶液和动力学溶液。事实证明，如果u是动力学溶液，那么它确实是一种重新归一化的熵解决方案。相反，我们证明，如果U是满足一定额外条件的重新归一化熵解决方案，则它将成为动力学解决方案。 3。我们研究具有初始和非均匀边界条件的非线性退化抛物线方程的比较原理。我们证明了任何（有限的）熵子和超溶液的比较定理。到目前为止，已经获得了L1的合同性和（无限）熵解决方案的唯一性，但比较结果是新的。在那里使用的方法是由于Kruzkov引起的所谓的加倍变量技术。我们的方法基于动力学配方和动力学技术。通过开发具有边界条件的退化抛物线方程的动力学技术，我们获得了比较特性。 4。我们将第3项中所述的上述结果扩展到无界熵解决方案的情况。我们考虑了L1框架中的问题，并证明了对重新归一化熵解决方案的比较定理和存在定理。