Wellposedness of the Cauchy problems for hyperbolic systems with large data

大数据双曲系统柯西问题的适定性

基本信息

批准号：
16540153
负责人：
TANAKA Naoki
金额：
$ 2.3万
依托单位：
Shizuoka University (2006)Okayama University (2004-2005)
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

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540153/
关键词：
semigroup of Lipschitz operators evolution equation wellposedness in the sense of Hadamard subtangential condition dissipativity condition stability condition approximation theorem for semigroups of operators semigroup of operators semilin

项目摘要

1. Semigroups of locally Lipschitz operators are characterized by subtangential conditions and semilinear stability conditions in terms of a family of metric-like functionals. The result is applied to the mixed problem for the complex Ginzburg-Landau equation.2. It is interesting to compute solutions of Cauchy problems for partial differential equations numerically and to discuss the question of convergence which arises in that case. Such problems are treated by the theory of semigroups of operators. A convergence theorem and an approximation theorem for semigroups of Lipschitz operators are given. These theorems are applied to a semi-discrete approximation problem for a quasi-linear wave equation with damping and a finite difference method for a quasi-linear equation of Kirchhoff type.3. An approximation theorem is given for abstract quasi-linear evolution equations in the sense of Hadamard. The result is applied to an approximation problem for a degenerate Kirchhoff equation.4. The situation that the domains of differential operators are not dense in the underlying space arises when the mixed problems for certain partial differential equations in the space of continuous functions are studied. This leads to the study of the abstract Cauchy problems for quasi-linear evolution equations with non-densely defined operators. The result is applied to obtain the global wellposedness for quasi-linear wave equations of Kirchhoff type with acoustic boundary conditions and the local solvability of quasi-linear wave equations with Wentzell boundary conditions in the space of continuous functions.

1。局部Lipschitz运算符的半群的特征在于按照公制功能系列的统一条件和半线性稳定条件。结果适用于复杂的金茨堡 - 兰道方程的混合问题2。有趣的是，以数值计算部分微分方程的库奇问题解决方案，并讨论在这种情况下出现的收敛问题。操作员的半群理论对此问题进行了处理。给出了Lipschitz运算符的半群的收敛定理和近似定理。这些定理应用于具有阻尼的准线性波方程的半差异近似问题，以及用于Kirchhoff类型的准线性方程的有限差方法3。在哈玛德的意义上给出了抽象的准线性演化方程的近似定理。结果应用于退化的kirchhoff方程的近似问题4。当研究了连续函数空间中某些部分微分方程的混合问题时，在基础空间中不密集的域并不致密的情况。这导致研究了与未定义的运算符的准线性演化方程的抽象库奇问题。结果可用于获得具有声学边界条件的Kirchhoff类型的准线性波方程的全局良好性，并且在连续函数空间中具有Wentzell边界条件的准线性波方程的局部溶解性。