Cauchy Problem for Hyperbolic System of Conservation Laws

双曲守恒定律系统的柯西问题

基本信息

批准号：
09640233
负责人：
ASAKURA Fumioki
金额：
$ 1.92万
依托单位：
Osaka Electro-Communication University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640233/
关键词：
hyperbolic system conservation laws Initial value problem asymptotic stability phase boundary wave-front tracking Gevrey class differential-difference equation

项目摘要

Large Time Stability of the Maxwell States (F.Asakura)The investigator studies the Cauchy problem for a 2 * 2-system of conservation laws describing isentropic phase transitions. Two constant states satisfying the Maxwell equal-area principle constitute an admissible stationary solution ; a small perturbation of these Maxwell states will be their initial data. The main result is : there exists a global in time propagating phase boundary which is admissible in the sense that it satisfies the Abeyaratne-Knowles kinetic condition ; the states outside the phase boundary tend to the Maxwell states as time goes to infinity. Isothermal phase transitions modeled by a 3 * 3-system are also studied, In these cases, the velocity and the specific volume tend to the Maxwell states but the entropy density may tend to non-constant distributions. Abeyaratne-Knowles' driving traction is shown to be the difference of mechanical Gibbs function vspace2 exCauchy problem for nonstrictly hyperbolic systems i … More n Gevrey classes (H.Yamahara)Once the investigator gave a conjecture that the indices of Gevrey classes, in which the Cauchy problem is well-posed, are determined instead by the multipilcities of zeros of the minimal polynomial of the principal symbol. This is true provided that the multiplicities of the characteristic roots are constant.If one drops this assumption of constant multiplicities, the situation is in fact much more complicated. The investigator gave an example of 4 * 4-hyperbolic system which shows that, besides multiplicities of the characteristic roots, the degeneracy of the Jordan normal form of the principal part determine the appropriate Gevrey indices.Asymptotic stability for a linear system of differential-difference equations (S.Sakata)The differential-difference equation : dx/=ax(t)+Bx(t-r), r > 0 is studied. The investigator, studying the distribution of the roots of the characteristic equation, found a necessary and sufficient condition for the null solution to be asymptotically stable. The equation dx/=ax(t-r)+Bx(t-nr), r > 0 is also studied. For n=2,3, the investigator studied the set of (a, b) for the null solution to be asymptotically stable.A sufficient (substantially, necessary) condition is given for the system of equation dx/=-alpha{1-*x*^2}R(theta)x(*t*) to have a star-shaped periodic solution. Less

麦克斯韦国家（F.Asakura）的较大时间稳定性研究者研究了库奇的问题，该问题描述了2 * 2个保护法，描述了等等相位过渡。满足麦克斯韦等分地区原理的两个常数状态构成了可接受的固定解决方案；这些麦克斯韦状态的少量扰动将是其初始数据。主要的结果是：在繁殖相边界的全球范围内，这是可以接受的，因为它满足了Abeyaratne-Knowles动力学条件；随着时间的流逝，相边界以外的状态趋向于麦克斯韦状态。由3 * 3个系统建模的等温相跃迁也是研究的，在这些情况下，速度和特定体积趋向于麦克斯韦状态，但熵密度可能倾向于非稳定分布。 Abeyaratne-Knowles' driving traction is shown to be the difference of mechanical Gibbs function vspace2 exCauchy problem for nonstrictly hyperbolic systems i … More n Gevrey classes (H.Yamahara)Once the investigator gave a conjecture that the indices of Gevrey classes, in which the Cauchy problem is well-posed, are determined instead by the multipilcities of zeros of the minimal polynomial of the principal 象征。如果特征根的乘法是恒定的，则这是事实。如果一个人丢弃了恒定乘法的假设，则情况实际上更为复杂。研究者举例说明了4 * 4个高纤维系统的示例，该系统表明，除了特征根的多样性之外，主部分的约旦正常形式的退化还决定了适当的gevrey指标。平衡系统的线性稳定性差异方程（s.sakata）的线性稳定性（S.Sakata）差异方程式：dx/dx/dx/bx（dx/dx/bx）（d dx/bx）+bx（t）+bx（t）+bx（t）+bx（t）+bx（t）+bx（t）+bx（t）+bx（t）+bx（t）+bx（t）+bx（t）+bx（T）研究者研究了特征方程的根部分布，发现了无效溶液不对称稳定的必要条件。方程dx/= ax（t-r）+bx（t-nr），r> 0也是研究的。对于n = 2,3，研究人员研究了一组（a，b），以使零溶液不对称地稳定。方程dx/= - alpha {1-*x*x*x*^2} r（theta）r（theta）x（theta）x（*t*）的足够（基本上是必要的）条件，使得具有星状的周期性溶液。较少的