This paper by Julian D. Cole discusses a quasi-linear parabolic equation that arises in aerodynamics, specifically in the study of shock waves and turbulence. The equation, given by: \[ \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = \nu \frac{\partial^2 u}{\partial x^2}, \] is similar to the heat equation but includes a nonlinear term \( u \frac{\partial u}{\partial x} \), which is crucial for understanding shock wave propagation and turbulence. The author explores the equation's properties, including its relationship to shock wave theory and turbulence theory, and provides a general solution for initial value problems. Key points include: 1. **Shock Wave Theory**: The equation can describe the flow through a shock wave in a viscous fluid. Solutions can be related to shock waves in several ways, and the nonlinear term steepens the wave fronts, while the viscous term prevents discontinuities. 2. **Turbulence Theory**: The equation is used as a mathematical model for turbulence, where the nonlinear and viscous terms are essential for understanding energy transfer and dissipation. 3. **General Properties**: The paper discusses the equation's behavior under different conditions, such as steady-state solutions and the influence of viscosity on wave propagation. 4. **Examples of Solutions**: Two examples are provided: one showing the approach to a steady state by a shock wave and another demonstrating the decay of an arbitrary periodic initial disturbance. 5. **Conclusion**: The main effects of the equation are the nonlinear steepening of velocity profiles and the diffusion of momentum and energy by viscosity. The solution's behavior depends on a characteristic Reynolds number \( R_0 \), and future work should focus on more precise approximations and higher-dimensional cases. The paper highlights the importance of the equation in understanding complex fluid dynamics phenomena and suggests further research directions.This paper by Julian D. Cole discusses a quasi-linear parabolic equation that arises in aerodynamics, specifically in the study of shock waves and turbulence. The equation, given by: \[ \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = \nu \frac{\partial^2 u}{\partial x^2}, \] is similar to the heat equation but includes a nonlinear term \( u \frac{\partial u}{\partial x} \), which is crucial for understanding shock wave propagation and turbulence. The author explores the equation's properties, including its relationship to shock wave theory and turbulence theory, and provides a general solution for initial value problems. Key points include: 1. **Shock Wave Theory**: The equation can describe the flow through a shock wave in a viscous fluid. Solutions can be related to shock waves in several ways, and the nonlinear term steepens the wave fronts, while the viscous term prevents discontinuities. 2. **Turbulence Theory**: The equation is used as a mathematical model for turbulence, where the nonlinear and viscous terms are essential for understanding energy transfer and dissipation. 3. **General Properties**: The paper discusses the equation's behavior under different conditions, such as steady-state solutions and the influence of viscosity on wave propagation. 4. **Examples of Solutions**: Two examples are provided: one showing the approach to a steady state by a shock wave and another demonstrating the decay of an arbitrary periodic initial disturbance. 5. **Conclusion**: The main effects of the equation are the nonlinear steepening of velocity profiles and the diffusion of momentum and energy by viscosity. The solution's behavior depends on a characteristic Reynolds number \( R_0 \), and future work should focus on more precise approximations and higher-dimensional cases. The paper highlights the importance of the equation in understanding complex fluid dynamics phenomena and suggests further research directions.