Well-posedness and approximation of Cauchy problems for hyperbolic systems

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，并给出了该系统的经典定理Kruzhkov的类比。 2．局部Lipschitz算子半群：局部Lipschitz算子半群的连续无穷小生成元的特征在于由类度量泛函和所谓的次切线条件定义的耗散条件。 3．积分半群： (1) 为局部Lipschitz连续积分半群引入一类微扰算子，并给出了此类积分半群的一些微扰定理。 (2)从非线性半群理论的角度处理积分半群的非线性扰动，讨论了非线性半群的表征。 4．由非稠密定义算子生成的演化算子：从有限近似的角度来看，在稳定条件下，演化算子是由一系列闭合线性算子生成的，其公共域在底层 Banach 空间中不一定是稠密的。 5．摘要 Hadamard 意义上的拟线性演化方程的柯西问题：针对简并型的抽象拟线性演化方程引入了 Hadamard 意义上的适定性概念。还从有限差分近似的角度引入了一种新型的稳定性条件。将得到的定理应用于具有非线性摄动的简并Kirchhoff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 边界条件的二阶抽象拟线性方程。将结果应用于具有 Wentzell 边界条件的连续函数空间中二阶拟线性微分算子的波动方程。较少的

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