On a Quasi - Linear Parabolic Equation Occurring in Aerodynamics

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 P. A. Lagerstrom and F. K. 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 NavierStokes equations for one-dimensional non-stationary flow of a compressible viscous fluid gives