二维压力梯度系统黎曼问题的整体解
Global Solutions of a Two-Dimensional Riemann Problem for the Pressure Gradient System
摘要：
本文研究了用压力梯度系统模拟的二维（$2$-D）黎曼问题，该压力梯度系统是一个$2$-D双曲型守恒律组。黎曼初始数据由四个扇形区域中的四个恒定状态组成，使得在相邻状态之间产生两个激波和两个涡面。对于一个混合复合型非线性系统，这个黎曼问题可以化为一个自相似坐标系下的边值问题，其渐近边界条件为黎曼初值，并有两个由黎曼初值确定的音速圆.这些解保持了四个恒定状态和四个平面波在外音速圈之外。这两个激波保持平面直到它们在两个不同的点处与外音速圆相遇，然后产生一个衍射激波，该衍射激波预计将连接这两个点，其确切位置是{\it apriori}未知的，被认为是一个自由边界。然后，$2$-D黎曼问题可以转化为一个自由边界问题，其中跨音速绕射激波是连接这两点的单相自由边界，而外音速圆的另一部分则构成问题的固定边界的一部分。我们建立了自由边界问题解的整体存在性，以及$C^{0,1}$--跨越两点的衍射冲击和跨越外声波边界的解的规律性，这是最佳的。在自相似坐标系中，衍射激波不能与内音速圆相交是本文的一个重要发现。结果，这个$2$-D黎曼问题得到了全局解，它的解包含两个涡面和一个连接由黎曼数据产生的两个原始激波的全局$2$-D激波。
英文摘要：
We are concerned with a two-dimensional ($2$-D) Riemann problem for compressible flows modeled by the pressure gradient system that is a $2$-D hyperbolic system of conservation laws. The Riemann initial data consist of four constant states in four sectorial regions such that two shock waves and two vortex sheets are generated between the adjacent states. This Riemann problem can be reduced to a boundary value problem in the self-similar coordinates with the Riemann initial data as its asymptotic boundary data, along with two sonic circles determined by the Riemann initial data, for a nonlinear system of mixed-composite type. The solutions keep the four constant states and four planar waves outside the outer sonic circle. The two shocks keep planar until they meet the outer sonic circle at two different points and then generate a diffracted shock to be expected to connect these two points, whose exact location is {\it apriori} unknown which is regarded as a free boundary. Then the $2$-D Riemann problem can be reformulated as a free boundary problem, in which the diffracted transonic shock is the one-phase free boundary to connect the two points, while the other part of the outer sonic circle forms the part of the fixed boundary of the problem. We establish the global existence of a solution of the free boundary problem, as well as the $C^{0,1}$--regularity of both the diffracted shock across the two points and the solution across the outer sonic boundary which is optimal. One of the key observations here is that the diffracted transonic shock can not intersect with the inner sonic circle in the self-similar coordinates. As a result, this $2$-D Riemann problem is solved globally, whose solution contains two vortex sheets and one global $2$-D shock connecting the two original shocks generated by the Riemann data.
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