Cauchy Problem for Hyperbolic System of Conservation Laws

点击翻译按钮获取中文摘要

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