Cauchy Problem for Hyperbolic System of Conservation Laws

研究者研究了描述等熵相变的2 * 2守恒律系统的柯西问题。满足麦克斯韦等面积原理的两个常态构成一个可容许的定解；这些麦克斯韦状态的一个小扰动将是它们的初始数据。主要结果是：存在一个全局的随时间传播的相边界，它在满足Abeyaratne-Knowles动力学条件的意义上是允许的；当时间趋于无穷时，相边界外的态趋向于麦克斯韦态。研究了3 * 3系统模拟的等温相变，在这种情况下，速度和比容倾向于麦克斯韦状态，但熵密度可能倾向于非常数分布。（H.Yamahara）有一次研究者提出了一个猜想，即柯西问题被良好定的Gevrey类的指标是由主符号的最小多项式的零的多次决定的，而不是由主符号的最小多项式的零的多次决定的。如果特征根的多重数是常数，这是正确的。如果放弃这种恒定多样性的假设，情况实际上要复杂得多。研究人员给出了一个4 * 4双曲系统的例子，表明除了特征根的多重性外，主体的Jordan范式的简并性决定了适当的Gevrey指标。研究了微分-差分方程dx/=ax(t)+Bx(t-r), r > 0的渐近稳定性。研究了特征方程根的分布，得到了零解渐近稳定的一个充分必要条件。研究了方程dx/=ax(t-r)+Bx(t-nr), r > 0。对于n=2,3，研究者研究了零解的（a, b）集是渐近稳定的。给出了方程dx/=- α {1-*x*^2}R(θ)x（*t*）具有星形周期解的一个充分（本质上，必要）条件。少

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