Global Existence And Stability Of A Multidimensional Supersonic Conic Shock Wave:Isothermal Case

This thesis is the continuation of the works [15] and [17]. In this thesis, when the flow is governed by both special relativistic and non-relativistic compressible Euler systems, we establish the global existence and stability of a three-dimensional superson-ic conic shock wave, formed when a perturbed steady supersonic isothermal flow goes across an infinitely long circular cone with a sharp angle. The flow is assumed to be isen-tropic and ir-rotational and therefore can be described by a 3-D steady potential equation in special relativistic case and/or non-relativistic case, which is quasi-linear hyperbolic equation with the supersonic direction acting as its time-direction. The main ingredient of the thesis is to find an appropriate multiplier and improved Hardy-type inequality to deal with Rankine-Hugoniot conditions on the conic shock surface and the slip bound-ary condition on the surface of the cone. By the establishment of the uniform weighted energy estimate for the related nonlinear problem, we show that a multi-dimensional su-personic conic shock attached to the vertex of the cone globally exists and is stable when the Mach number of the incoming supersonic flow is sufficiently large. Moreover, the multidimensional conic shock resulted by the non-relativistic limit of the relativistic po-tential equation is shown to be coincided with the non-relativistic case. As in [17], the assumption is removed that the vertex angle should be suitably small when considering the related problem for the polytropic gas in [15].