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。凯奇（Cauchy）的保护法问题：为非线性保护法的系统引入了一类弱解决方案，为此，震动和稀疏曲线重合的是对这种系统的经典理论Kruzhkov的类似物。2。本地Lipschitz运算符的半群：局部Lipschitz运算符的半群的连续无限发电机的特征是由公制的功能和所谓的亚tangectional条件定义的耗散条件。3。集成的半群：（1）为局部Lipschitz持续集成的半群，引入了一类扰动操作员，并为这种集成的半群提供了一些扰动定理。（2）从非线性半群理论的角度来处理综合半群的非线性扰动，并讨论了非线性半群的特征。4。由非定义的运算符生成的进化运算符：更多地表明，在稳定条件下，从有限的近似角度来看，在稳定条件下，封闭的线性企业家族在基础BANACH空间中不一定在基础的Banach空间中生成进化运算符。5。在Hadamard的意义上，抽象的cauchy问题是：在Hadamard意义上的良好符合意义的概念是为了抽象的准级 - 线性进化方程。从有限差近似的角度来看，还引入了一种新型的稳定条件。所获得的理论应用于具有非线性扰动的退化基尔chhoff方程的局部溶解度。6。带有温泽尔边界条件的二阶的抽象准线性方程：引入一个通用框架来处理与温泽尔边界条件的第二阶阶的抽象准线性方程。在连续函数空间中，将结果应用于二阶析汇率差异操作员的波方程。较少的