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Existence and Stability of Non-classical Weak Solutions to Hyperbolic Conservation Laws

双曲守恒定律非经典弱解的存在性和稳定性

基本信息

批准号：
15540221
负责人：
ASAKURA Fumioki
金额：
$ 1.98万
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：
2003
资助国家：
日本
起止时间：
2003 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540221/
关键词：
conservation laws shock waves gas dynamics entropy condition overcompressive shock wave undercompressive shock wave Viscous shock profile 双曲系保存則 Riemann問題粘性進行波漸近安定性零解

项目摘要

(i)Existence of Viscous Profiles for Conservation Laws with an Umbilic Point :In hyperbolic conservation laws with an umbilic point, we have not only compressive (classical) shock waves but also undercomressive shock waves and overcompressive shock waves that are called non-classical shock waves. In this investigation, we have studied the admissibility condition that two states composing a shock wave have viscous profiles in Case I and II of the Schaefer-Shearer's classification. We have proved that : if the base-point of the Hugoniot curve is not located on the median, then almost all states on the Hugoniot curve composing compressive and overcompressive shock waves have shock profiles. In Case I, if the base point is located on the median, then there exist sometimes under compressives shock waves having viscous profiles ; in this case, there are compressive shock waves with no viscous profile. We have succeeded in obtaining a necessary and sufficient condition for such non-existence. … More In case II, we have obtained a almost necessary and sufficient condition for the existence of overcompressive shock waves on a median. Main tool is a generalization of the first theorem of Morse to non-compact level sets.(ii)Steady Flows in the Laval Nozzle :The Laval nozzle consists of a converging entry section, a throat and a diverging exhaust section, and used to accelerate subsonic flow into supersonic flow. The pressure at the entrance is kept constant, say p_0, which is realized by attaching a sufficiently large chamber at the entrance. If the pressure pj at the exit is made slightly lower than p_0, the flow at rest accelerate in the converging section and decelerate in the diverging section. As p_j reduces more and more, finally, the subsonic flow accelerates into the sonic speed at the throat; this is called the choking. If p_1 reduces still more, the flow accelerates into supersonic flow in the diverging section and a standing shock wave appears there. Finally, the flow is smooth with steadily decreasing pressure and increasing speed, and sonic at the throat ; this is called the ideal nozzle flow. In this investigation, we provide mathematical descriptions of the above phenomena for general flows which do not necessarily obey the gamma law. Moreover, we study the bifurcation of the solution at the throat and the geometry of the Hugoniot curve for the standing shock waves. Less

（i）具有脐点的守恒律的粘性分布的存在性：在具有脐点的双曲守恒律中，我们不仅有压缩（经典）激波，而且有欠压缩激波和过压缩激波，它们被称为非经典激波。本文研究了在Schaefer-Shearer分类的第一和第二种情况下，组成激波的两个状态具有粘性剖面的容许条件。我们已经证明：如果Hugoniot曲线的基点不在中值上，那么在Hugoniot曲线上几乎所有由压缩和过压缩激波组成的状态都有激波剖面。在第一种情况下，如果基点位于中线上，则有时存在具有粘性剖面的压缩激波;在这种情况下，存在没有粘性剖面的压缩激波。我们已经成功地获得了这种不存在的充分必要条件。 ...更多信息在第二种情况下，我们得到了中值上存在超压缩激波的一个几乎充分必要条件。主要工具是将莫尔斯第一定理推广到非紧水平集。（ii）拉瓦尔喷管中的定常流：拉瓦尔喷管由收敛入口段、喉部和扩张排气段组成，用于将亚音速流加速成超音速流。入口处的压力保持恒定，例如p_0，这通过在入口处附接足够大的腔室来实现。如果使出口处的压力pj略低于p_0，则静止的流动在收敛段加速，而在发散段减速。随着p_j越来越小，最后亚音速流在喉部加速到音速，这就叫做阻塞。如果p_1进一步减小，在扩张段流动加速成超音速流动，并在那里出现驻激波。最后，流动是平稳的，压力稳定下降，速度增加，喉部有音速;这就是理想喷管流动。在这项研究中，我们提供了上述现象的数学描述一般流动不一定服从伽玛定律。此外，我们还研究了驻激波在喉道处解的分叉和Hugoniot曲线的几何形状。少