Cauchy Problem for Hyperbolic System of Conservation Laws
Project/Area Number 
09640233

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
解析学

Research Institution  Osaka ElectroCommunication University 
Principal Investigator 
ASAKURA Fumioki Faculty of Engineering, Osaka ElectroCommunication University Profesor, 工学部, 教授 (20140238)

CoInvestigator(Kenkyūbuntansha) 
SAKATA Sadahisa Faculty of Engineering, Osaka ElectroCommunication University Associate Profeso, 工学部, 助教授 (60175362)
YAMAHARA Hideo Faculty of Engineering, Osaka ElectroCommunication University Associate Profeso, 工学部, 助教授 (30103344)

Project Period (FY) 
1997 – 1998

Project Status 
Completed (Fiscal Year 1998)

Budget Amount *help 
¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 1998: ¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 1997: ¥1,600,000 (Direct Cost: ¥1,600,000)

Keywords  hyperbolic system / conservation laws / Initial value problem / asymptotic stability / phase boundary / wavefront tracking / Gevrey class / differentialdifference equation 
Research Abstract 
Large Time Stability of the Maxwell States (F.Asakura) The investigator studies the Cauchy problem for a 2 * 2system of conservation laws describing isentropic phase transitions. Two constant states satisfying the Maxwell equalarea 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 AbeyaratneKnowles 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 * 3system are also studied, In these cases, the velocity and the specific volume tend to the Maxwell states but the entropy density may tend to nonconstant distributions. AbeyaratneKnowles' driving traction is shown to be the difference of mechanical Gibbs function vspace2 ex Cauchy 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 wellposed, 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 * 4hyperbolic 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 differentialdifference equations (S.Sakata) The differentialdifference equation : dx/=ax(t)+Bx(tr), 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(tr)+Bx(tnr), 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 starshaped periodic solution. Less

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