Well-posedness and approximation of Cauchy problems for hyperbolic systems

1.双曲型守恒律方程组的Cauchy问题：对于激波曲线和稀疏曲线重合的非线性守恒律方程组，引入了一类弱解，并给出了经典Kruzhkov定理的类似结果.局部Lipschitz算子半群：局部Lipschitz算子半群的连续无穷小生成元由度量泛函定义的耗散条件和所谓的下切条件刻画.积分半群：（1）对局部Lipschitz连续积分半群引入了一类扰动算子，并给出了这类积分半群的若干扰动定理。(2)从非线性半群理论的角度讨论了积分半群的非线性扰动，并给出了非线性半群的一个刻画.由非密集定义的算子生成的演化算子： 关于我们 证明了在有限逼近的稳定性条件下，演化算子是由一族闭线性算子生成的，这些算子的公共区域在基本的Banach空间中不一定是稠密的. Hadamard意义下的拟线性发展方程的抽象Cauchy问题：对退化型抽象拟线性发展方程引入Hadamard意义下的适定性概念。从有限差分近似的观点出发，提出了一种新的稳定性条件。将所得定理应用于一类具有非线性扰动的退化Kirchhoff方程的局部可解性.带Wentzell边界条件的抽象二阶拟线性方程：给出了处理带Wentzell边界条件的抽象二阶拟线性方程的一般框架。将所得结果应用于连续函数空间中具有Wentzell边界条件的二阶拟线性微分算子的波动方程。少

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