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

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(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