Cauchy Problem for Hyperbolic System of Conservation Laws

1.热弹性中麦克斯韦状态的稳定性：在等温弹性中，麦克斯韦状态可以用等面积原理来定义。在本研究中，我们证明了这些麦克斯韦状态随时间渐近稳定。此外，熵函数用力学吉布斯函数表示，即使状态不是平稳的。另一方面，在多向热弹性中，麦克斯韦状态被定义为构成一个相边界，使得两边的熵重合。我们证明了在一对麦克斯韦态的邻域中存在唯一的过渡映射，并给出了动力学条件。然而，我们已经证明了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，我们可以挑出一个唯一的解。以上研究表明，在多向弹性中，与等温弹性不同，麦克斯韦状态必须是不稳定的。黎曼问题中的几何唯一性定理：在边界包含孤立脐点的严格双曲域上，我们得到了一般2x2-守恒律系黎曼问题的唯一性定理。唯一性的条件为：当j=1和2时，j- hugoniot曲线的j特征方向的梯度和j=1和2时，j- hugoniot曲线中心到曲线上点的割线分别局限于不相交的固定扇区。这个条件是t。p。得到的条件的推广。70年代的刘。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发表。非严格双曲型守恒定律的可容许不连续解：我们对守恒定律的Hugoniot曲线进行了几何研究，其通量矢量是状态变量的二次函数，并且具有孤立的脐点。我们找到了满足Lax熵条件的精确区域。特别地，对于几何结构最复杂的Schaeffer-Shearer's I类，我们给出了一个数学证明，证明了那些只在数值研究中被假设的主张。在这里，Hugoniot曲线必须是有理曲线。少

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