Well-posedness and approximation of Cauchy problems for hyperbolic systems

双曲系统柯西问题的适定性和逼近

基本信息

批准号：
14540175
负责人：
TANAKA Naoki
金额：
$ 2.18万
依托单位：
OKAYAMA UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540175/
关键词：
abstract quasi-linear equation Hadamard well-posedness stability conservation law regularized semigroup generator Lipschitz semigroup integrated semigroup semigroup of Lipschite operators infinitesimal generators generation theorem c

项目摘要

1. Cauchy problems for hyperbolic systems of conservation laws : A class of weak solutions is introduced for the systems of nonlinear conservation laws for which the shock and rarefaction curves coincide and an analogue of the classical theorem Kruzhkov is given for such systems.2. Semigroups of locally Lipschitz operators : The continuous infinitesimal generators of semigroups of locally Lipschitz operators are characterized by the dissipative conditions defined by means of metric-like functionals and the so-called subtangential conditions.3. Integrated semigroups : (1) A class of perturbing operators is introduced for locally Lipschitz continuous integrated semigroups, and some perturbation theorems are given for such integrated semigroups. (2) Nonlinear perturbations of integrated semigroups are treated from the viewpoint of nonlinear semigroup theory and a characterization of nonlinear semigroups is discussed.4. Evolution operators generated by non-densely defined operators : It is … More shown that an evolution operator is generated by a family of closed linear operators whose common domain is not necessarily dense in the underlying Banach space, under the stability condition from the viewpoint of finite approximations.5. Abstract Cauchy problems for quasi-linear evolution equations in the sense of Hadamard : The notion of well-posedness in the sense of Hadamard is introduced for abstract quasi-linear evolution equations of degenerate type. A new type of stability condition is also introduced from the viewpoint of finite difference approximations. The obtained theorem is applied to the local solvability of a degenerate Kirchhoff equation with nonlinear perturbation.6. Abstract quasilinear equations of second order with Wentzell boundary conditions : A general framework is introduced to treat abstract quasilinear equations of-second order with Wentzell boundary conditions. The result is applied to a wave equation for a second order quasilinear differential operator in the continuous function space with Wentzell boundary condition. Less

1.双曲守恒定律系统的柯西问题：介绍了激波曲线和稀疏曲线重合的非线性守恒定律系统的一类弱解，并给出了此类系统的经典定理Kruzhkov的类比。 2．局部 Lipschitz 算子：局部 Lipschitz 算子半群的连续无穷小生成元的特征在于通过类度量泛函和所谓的耗散条件定义3. 积分半群： (1) 引入一类局部Lipschitz连续积分半群的微扰算子，并给出此类积分半群的一些微扰定理。 (2) 从以下角度处理积分半群的非线性微扰。讨论了非线性半群理论和非线性半群的表征。 4. 由非稠密定义算子生成的演化算子：表明演化算子是由族生成的从有限近似的角度来看，在稳定条件下，公共域不一定是稠密的闭线性算子。Hadamard 意义上的拟线性演化方程的抽象 Cauchy 问题：well- 的概念。为简并型的抽象拟线性演化方程引入了Hadamard意义上的适定性，并从有限差分近似的角度引入了一种新的稳定性条件，并将所得定理应用于该方程。具有非线性扰动的简并基尔霍夫方程的局部可解性。6．具有Wentzell 边界条件的抽象二阶拟线性方程：引入了一个通用框架来处理具有Wentzell 边界条件的抽象二阶拟线性方程，并将结果应用于波。具有 Wentzell 边界条件的连续函数空间中的二阶拟线性微分算子方程。