二维定常可压缩Euler流通过直楔的高超音速极限
Hypersonic limit of two-dimensional steady compressible Euler flows passing a straight wedge
摘要：
本文提出了一个二维定常非等熵可压缩欧拉绕流高超声速极限问题。结果表明，当多方气体的绝热指数$γ$减小到$1$时，即将到来的均匀超声速流动的马赫数增大到无穷大。我们提出了一种形式的欧拉方程，这是有效的，如果未知量是措施，并构造了一个测量解决方案包含Dirac措施支持的表面上的楔形。证明了在楔体前方含有激波的可压缩Euler方程的解序列作为$γ\to1$，作为测度模糊地收敛到我们构造的测度解。这就证明了在二维直楔形物情况下高超音速气流通过障碍物的牛顿理论是正确的。这一结果也说明了在研究双曲型守恒律组的边值问题时考虑一般测度解的必要性。
英文摘要：
We formulated a problem on hypersonic limit of two-dimensional steady non-isentropic compressible Euler flows passing a straight wedge. It turns out that Mach number of the upcoming uniform supersonic flow increases to infinite may be taken as the adiabatic exponent $γ$ of the polytropic gas decreases to $1$. We proposed a form of the Euler equations which is valid if the unknowns are measures and constructed a measure solution contains Dirac measures supported on the surface of the wedge. It is proved that as $γ\to1$, the sequence of solutions of the compressible Euler equations that containing a shock ahead of the wedge converge vaguely as measures to the measure solution we constructed. This justified the Newton theory of hypersonic flow passing obstacles in the case of two-dimensional straight wedges. The result also demonstrates the necessity of considering general measure solutions in the studies of boundary-value problems of systems of hyperbolic conservation laws.
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