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

(1)带脐点守恒律的粘性剖面的存在性：在带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超压缩激波，它们被称为非经典激波。在本研究中，我们研究了在Schaefer-Shearer分类的情形I和情形II中，构成激波的两种状态具有粘性剖面的可容许条件。我们证明了：如果Hugoniot曲线的基点不在中位数上，那么在Hugoniot曲线上构成压缩激波和过压缩激波的几乎所有状态都有激波剖面。在情形1中，如果基点位于中位数，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。在情形二中，我们得到了中值上超压激波存在的几乎充分必要条件。主要工具是将莫尔斯第一定理推广到非紧水平集。（ii）拉瓦尔喷管内的稳定流动：拉瓦尔喷管由会聚入口段、喉部和发散排气段组成，用于将亚音速流动加速为超音速流动。入口处的压力保持恒定，例如p_0，这是通过在入口处附加一个足够大的腔室来实现的。若使出口压力pj略低于p_0，则静止流动在收敛段加速，在发散段减速。随着p_j的不断减小，亚音速流最终加速到喉部声速；这叫做窒息。如果p_1进一步减小，则在发散段加速为超声速流动，在发散段出现驻波。最后，流动平稳，压速平稳下降，喉部声速平稳；这被称为理想喷嘴流量。在这项研究中，我们对不一定服从伽马定律的一般流动提供了上述现象的数学描述。此外，我们还研究了驻波喉部解的分岔和Hugoniot曲线的几何形状。少

(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