Cauchy Problem for Hyperbolic System of Conservation Laws

双曲守恒定律系统的柯西问题

• 批准号: 09640233
• 负责人: ASAKURA Fumioki
• 金额: $ 1.92万
• 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: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640233/
• 关键词: hyperbolic system; conservation laws; Initial value problem; asymptotic stability; phase boundary; wave-front tracking; Gevrey class; differential-difference equation

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

麦克斯韦态的大时间稳定性(F.Asakura)研究者研究描述等熵相变的2*2守恒律系的柯西问题。满足麦克斯韦等面积原理的两个恒定态构成了一个可容许的定常解；这些麦克斯韦态的微小扰动将是它们的初始数据。主要结果是：在满足Abearatne-Knowles动力学条件的情况下，存在一个整体的时间传播相界面，它是可容许的；随着时间的推移，相界面外的态趋于Maxwell态。文中还研究了由3×3系统模拟的等温相变，在这种情况下，速度和比体积趋于麦克斯韦态，而熵密度可能趋于非恒定分布。证明了非严格双曲型方程组…的Abearatne-Knowles的驱动力是力学Gibbs函数vspace2exCauchy问题的不同之处更多的n个Gevrey类(H.Yamahara)一旦研究者猜想，Gevrey类的指数，其中柯西问题是适定的，而不是由主符号的最小多项式的零点的乘积决定的。如果特征根的重数是恒定的，这是正确的。如果一个人放弃这个恒定重数的假设，情况实际上要复杂得多。给出了一个4*4-双曲型系统的例子，表明除了特征根的重数外，主体的Jordan标准型的退化决定了适当的Gevrey指标.研究了线性微分差分方程组(S.Sakata)的渐近稳定性：dx/=ax(T)+Bx(t-r)，r&gt；0.研究者研究了特征方程的根的分布，找到了零解渐近稳定的充要条件。研究了方程dx/=ax(t-r)+Bx(t-nr)，r&gt；0。当n=2，3时，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(Theta)x(*t*)存在星形周期解的充分(实质上是必要的)条件.较少

项目成果

F.Asakura: "Global solutions with a single transonic shock wave for quasilinear hyperbolic systems" Methods and Applications of Analysis. 4(1). 33-52 (1997)

F.Asakura：“拟线性双曲系统的单一跨音速冲击波的全局解决方案”分析方法和应用。

F.Asakura: "The Glimm Lax theory via wave-frant tracking" Science Bulletin of Josai University. Special Issue 5. 131-142 (1998)

F.Asakura：“通过波域跟踪的 Glimm Lax 理论”城西大学科学通报。

S.Sakata: "Asymptotic stability for a linear system of differential-difference equations" Funkcialaj Ekvacioj. 41(3). 435-449 (1998)

S.Sakata：“微分差分方程线性系统的渐近稳定性”Funkcialaj Ekvacioj。

F.Asakura: "Large Time Stability of the Maxwell States" Methods and Applications of Analysis. 6 掲載予定. (1999)

F.Asakura：“麦克斯韦态的大时间稳定性”分析方法和应用 6（1999 年）。

H.Yamahara: "An example of the Cauchy problem in Gevray classes" Proceeding of the International Symposium in Honor of Prof.Vaillant on His 65th Birthday Ehime Univ.60-62 (1998)

H.Yamahara：“Gevray 类中柯西问题的一个例子”纪念 Vaillant 教授 65 岁生日的国际研讨会论文集 Ehime Univ.60-62 (1998)