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：
11640219
负责人：
ASAKURA Fumioki
金额：
$ 2.18万
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：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640219/
关键词：
hyperbolic system conservation laws Initial value problem phase boundary entropy condition Riemann problem umbilic point asymptotic stability 不完全圧縮衝撃波軌道接続問題波面追跡法

项目摘要

1.Stability of the Maxwell States in Thermo-Elasticity : In the isothermal elasticity, the Maxwell states can be defined by the equal-area principle. We proved in this research that these Maxwell states are asymptotically stable in time. Moreover, the entropy function is expressed by means of the mechanical Gibbs function, even if the states are not stationary. In the polytropic thermo-elasticity, on the other hand, the Maxwell states are defined to constitute a phase boundary such that the entropy of the both sides coincides. We proved that there exists a unique transitional map in a neighborhood of a pair of Maxwell states together with the kinetic condition. However, we have shown that the Riemann problem has at least two solutions under certain condition. In this case, if we prescribe the increase or decrease of the temperature after the phase transition, we can single out a unique solution. The above study indicates in the polytropic elasticity, different from the isothermal elast … More icity, the Maxwell states must be unstable.2.Geometric Uniqueness Theorem in the Riemann Problem : We obtain a uniqueness theorem for the Riemann problem for general 2x2-system of conservation laws in a strictly hyperbolic domain whose boundary contains an isolated umbilic point. The condition for uniqueness is given by the following : for j=1 and 2, the gradient of the j-characteristic direction and the secant from the center of the j-Hugoniot curve to the point on the curve are confined to fixed disjoint sectors for j=1 or 2, respectively. This condition is a generalization of that obtained by T.-P.Liu in 70's. Moreover, in the process of our study, we gave a proof of the theorem declared by him but the details of its proof are not yet published.3.Admissible Discontinuous Solutions for Nonstrictly Hyperbolic Conservation Laws : We carried out a geometric study of the Hugoniot curves for conservation laws whose flux vector is a quadratic function of the state variables and has an isolated umbilic point. We found precise regions where the Lax entropy condition holds. In particular, for the Schaeffer-Shearer's class I, where the geometric structure is most complicated, we gave a mathematical proof of claims that had been postulated only by numerical studies. Here, it is essential that the Hugoniot curves are rational curves. Less

1.麦克斯韦状态在热弹性中的稳定性：在等温弹性中，麦克斯韦状态可以通过相等的原理来定义。我们在这项研究中证明了这些麦克斯韦状态在时间上不对称。此外，即使状态不静止，熵函数也通过机械吉布斯函数表示。另一方面，在多面热弹性弹性性中，麦克斯韦的状态被定义为构成一个相边界，使得两侧的熵一致。我们证明了在一对麦克斯韦州附近的独特过渡图以及动力学状况。但是，我们已经表明，在某些情况下，Riemann问题至少具有两种解决方案。在这种情况下，如果我们准备相变之后温度的升高或降低，则可以单击一个唯一的解决方案。上述研究表明在多面性弹性中，与等温弹性不同……更重要的是，麦克斯韦状态必须不稳定。2。Riemann问题中的几何唯一性定理：我们获得了Riemann问题的独特性理论，用于对一般2x2-Syperavery的唯一性，即严格的超级性超大型人物，其边界与umbiripal compitical umbilial umbilial comptical comptical comptim umbilial。唯一性的条件由以下内容给出：对于J = 1和2，J thacteristic方向的梯度和从J-Hugoniot曲线中心到曲线上的点的SECANT分别局限于j = 1或2的固定分离扇区。这种情况是T.-p.liu在70年代获得的条件。此外，在我们的研究过程中，我们给出了他声明的定理的证明，但尚未发表其证明的细节。3。可加入的不连续的不连续的解决方案：对荷叶顿曲线的几何学研究，用于保存曲线的几何研究，该曲线是综合效应的二次变量和彼此的指标。我们发现了宽松熵条件保持的精确区域。特别是，对于Schaeffer-Shearer的I类，几何结构最复杂，我们给出了仅数值研究发布的主张的数学证明。在这里，雨果曲线是理性曲线至关重要的。较少的