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.

我们研究一类双曲-椭圆耦合系统的初值问题，包括辐射气体方程。对于最简单的辐射气体模型系统：1.给出了经典全局解的不存在和存在的充分条件。2.证明了Sobolev空间中光滑解的全局存在性和渐近衰减性。该解逼近了用粘性Brugers方程的自相似解定义的扩散波。3.证明了用无粘性Brugers方程的中心稀疏波定义的稀疏波的渐近稳定性。4 .我们证明了冲击剖面行波解的存在性。仅当冲击强度大于临界值时，这些冲击波才具有不连续性。此外，我们还证明了光滑激波的渐近稳定性。5.证明了黎曼问题弱解的总体存在性。弱解中包含的跳跃呈指数衰减，随着时间趋于无穷大，解逼近相应的平滑冲击波。 6.给出了熵函数的数学定义，并证明了熵函数的存在性与对称性的等价性系统的。接下来，我们制定稳定性条件，并在该条件下证明 Sobolev 空间中光滑解的全局存在性和渐近衰减性。此外，我们观察到该解是通过对应的双曲-抛物线耦合系统的解在时间上渐近近似的。这些结果适用于辐射气体方程。