Cauchy Problem for Hyperbolic System of Conservation Laws

双曲守恒定律系统的柯西问题

• 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

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.热弹性中麦克斯韦态的稳定性：在等温弹性中，麦克斯韦态可以通过等面积原理来定义。我们在这项研究中证明了这些麦克斯韦态在时间上是渐近稳定的。此外，即使状态不是平稳的，熵函数也可以通过机械吉布斯函数来表达。另一方面，在多变热弹性中，麦克斯韦态被定义为构成相界，使得两侧的熵一致。我们证明了一对麦克斯韦态邻域中存在唯一的过渡图以及动力学条件。然而，我们已经证明黎曼问题在一定条件下至少有两个解。在这种情况下，如果我们规定相变后温度的升高或降低，我们可以找出唯一的解决方案。上述研究表明，在多变弹性中，与等温弹性冰态不同，麦克斯韦态必定是不稳定的。2.黎曼问题中的几何唯一性定理：我们得到了边界包含孤立脐点的严格双曲域中一般2x2守恒定律系统的黎曼问题的唯一性定理。唯一性的条件由以下给出：对于j=1和2，j-特征方向的梯度和从j-Hugoniot曲线的中心到曲线上的点的割线分别被限制在对于j=1或2的固定不相交扇区。这个条件是T.-P.Liu在70年代得到的条件的推广。此外，在我们的研究过程中，我们给出了他所声明的定理的证明，但其证明的细节尚未公布。3.非严格双曲守恒定律的可接受的不连续解：我们对守恒定律的Hugoniot曲线进行了几何研究，该守恒定律的通量向量是状态变量的二次函数，并且具有孤立的脐点。我们找到了宽松熵条件成立的精确区域。特别是，对于几何结构最复杂的 Schaeffer-Shearer I 类，我们给出了仅通过数值研究假设的主张的数学证明。在此，休格尼奥曲线必须是有理曲线。较少的

项目成果

H.Yamahara,: "Cauchy problem for hyperbolic systems in Gevrey class"Armales de la Faculte des Science de Toulouse. 9・1. 147-160 (2000)

H. Yamahara，：“Gevrey 级双曲系统的柯西问题”Armales de la Faculte des Science de Toulouse 9・1 (2000)。

F.Asakura: "Large Time Stability of the Maxwell States"Methods and Applications of Analysis. Vol.6, No.4. 477-594 (1999)

F.Asakura：“麦克斯韦态的大时间稳定性”分析方法和应用。

T.Mandai: "The method of Frobenius to fuchsian partial differential equations"J.of Math.Soc.Japan. (未定). (2000)

T.Mandai：“Frobenius 求解 fuchsian 偏微分方程的方法”J.of Math.Soc.Japan（待定）。

T.Mandai: "The Method of Frobenius to Fuchsian Partial Differential Equations"J.Math.Soc.Japan.. 52. 645-672 (2000)

T.Mandai：“Frobenius 求解 Fuchsian 偏微分方程的方法”J.Math.Soc.Japan.. 52. 645-672 (2000)

T.Mandai: "The Method of Frobenius to Fuchsian Partial Differential Equations"J.Math.Soc.Japan. 52. 645-672 (2000)

T.Mandai：“Frobenius 求解 Fuchsian 偏微分方程的方法”J.Math.Soc.Japan。