ON A QUASI-LINEAR PARABOLIC EQUATION OCCURRING IN AERODYNAMICS
ON A QUASI-LINEAR PARABOLIC EQUATION OCCURRING IN AERODYNAMICS
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DOI：
10.1090/qam/42889
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发表时间：
1951-01-01
影响因子：
0.8
通讯作者：
COLE, JD
中科院分区：
文献类型：
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作者：
COLE, JD
Tt+ uTx-* a？其中u（x，t）在某个域上，v是一个参数。一阶导数在t中的出现和二阶导数在x中的出现清楚地表明方程是抛物型的，类似于热方程，而有趣的附加特征是非线性项u du/dx的出现。因此，该方程的结构大致类似于纳维尔-斯托克斯方程，实际上出现在空气动力学的两个独立问题中。在真实的流体中弱非定常激波的近似理论中，出现了一个与（1）简单相关的方程。这在参考文献1（pp. 146-154），其中给出了（1）的一般解。在J. Burgers的湍流模型理论（参考文献2）中也给出了该方程，他在文中指出了模型理论和激波之间的关系。历史上，方程（1）首次出现在H.贝特曼（参考文献3）在1915年提到它值得研究，并给出了一个特殊的解决方案。当量(1)它本身具有一定的数学意义，并可能在随机过程理论中有应用。本文的目的是研究（1）的一般性质，并联系各种应用。我要感谢PA Lagerstrom教授和FK Chuang的有益合作。2.（1）与冲击波理论的关系。Eq的解决方案(1)可以近似地描述粘性流体中通过冲击波的流动。它们可以通过几种方式与冲击波相关。在参考文献1中，基于Navier-Stokes方程的可压缩粘性流体一维非定常流动的近似给出了
Tt+ uTx-* a?<(1) where u—u (x, t) in some domain and v is a parameter. The occurrence of the first derivative in t and the second in x clearly indicates the equation is parabolic, similar to the heat equation, while the interesting additional feature is the occurrence of the non-linear term u du/dx. The equation thus shows a structure roughly similar to that of the Navier-Stokes equations and has actually appeared in two separate problems in aerodynamics. An equation simply related to (1) appears in the approximate theory of a weak non-stationary shock wave in a real fluid. This is discussed in Ref. 1 (pp. 146-154) where a general solution of (1) is given. The equation is also given in J. Burgers' theory of a model of turbulence (Ref. 2) where he notes the relationship between the model theory and the shock wave. Historically, the equation (1) first appears in a paper by H. Bateman (Ref. 3) in 1915 when he mentioned it as worthy of study and gave a special solution. Eq.(1) is of some mathematical interest in itself and may have applications in the theory of stochastic processes. The aim of this paper is to study the general properties of (1) and relate the various applications. I wish to thank Professor PA Lagerstrom and FK Chuang for helpful collaboration. 2. Relationship of (1) to Shock Wave Theory. The solutions to Eq.(1) can approximately describe the flow through a shock wave in a viscous fluid. They can be related to the shock wave in several ways. In Ref. 1 an approximation based on the Navier-Stokes equations for one-dimensional non-stationary flow of a compressible viscous fluid gives