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Study on the initial value problem for quasilinear hyperbolic-elliptic coupled systems

拟线性双曲椭圆耦合系统初值问题研究

基本信息

批准号：
07454029
负责人：
KAWASHIMA Shuichi
金额：
$ 4.35万
依托单位：
KYUSHU UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1995
资助国家：
日本
起止时间：
1995 至 1997
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-07454029/
关键词：
hyperbolic-elliptic system initial value problem blow-up of solution global solution asymptotic stability rarefaction wave shock wave Riemann problem 輻射気体漸近挙動進行波解拡散波

项目摘要

We study the initial value problem for a class of hyperbollic-elliptic coupled systems including the equation of a radating gas. For the simplest model system of a radiating gas :1.We give sufficient conditions for the non-existence and the existence of classical global-solutions.2.We prove the global existence and asymptotic decay of smooth solutions in Sobolev spaces. The solution approaches the diffusion wave which is defined in terms of the self-similar solution of the viscous Burgers equation.3.We prove the asymptotic stability of rarefaction waves which are defined in terms of the centered rarefaction wave of the inviscid Brugers equation.4.We show the existence of traveling wave solutions of shock profile. These shock waves have a discontinuity only when the shock strength is greater than a critical value. Moreover, we prove the asymptotic stability of smooth shock waves.5.We show the glogal existence of weak solutions to the Riemann problem. The jump contained in the weak solution decays exponentially, and the solution approaches the corresponding smooth shock wave as time tends to infinity.6.We give a mathematical definition of the entropy function and prove the equivalence of the existence of an entropy function and the symmetrization of the system. Next, we formulate the stability condition, and under that condition we show the global existence and asymptotic decay of smooth solutions in Sobolev spaces. Furthermore, we observe that the solution is time-asymptotically approximated by the solution to the eorresponding hyperbolic-parabolic coupled system. These results are applicable to the equation of a radiating gas.

研究了一类双曲-椭圆耦合方程组的初值问题。对于最简单的辐射气体模型系统：1.给出了经典整体解的存在性和不存在性的充分条件; 2.证明了Sobolev空间中光滑解的整体存在性和渐近衰减性。3.证明了由无粘Brugers方程的中心稀疏波定义的稀疏波的渐近稳定性; 4.证明了激波行波解的存在性.只有当激波强度大于临界值时，这些激波才具有不连续性。证明了光滑激波的渐近稳定性。5.证明了Riemann问题弱解的全局存在性。弱解中包含的跳跃是指数衰减的，并且随着时间趋于无穷大，解趋近于相应的光滑激波。6.给出了熵函数的数学定义，证明了熵函数的存在性与系统的对称性的等价性。其次，我们给出了稳定性条件，并在此条件下证明了Sobolev空间中光滑解的整体存在性和渐近衰减性。此外，我们观察到，该解决方案是时间渐近近似的相应的双曲抛物耦合系统的解决方案。这些结果适用于辐射气体的方程。