Cauchy Problem for Hyperbolic System of Conservation Laws

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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