Burgers J.M. The Nonlinear Diffusion Equation. Asymptotic Solutions and Statistical Problems

D.Reidel Publishing Company, Dordrecht, Holland, 1974. 185 pp.
The topic treated in this monograph is an of attempts to arrive at a statistical theory of turbulent fluid motion, modeled after the example of classical statistical mechanics as applied to the kinetic theory of gases. The attempts made clear that essential features of the spectrum of turbulence are due to the simultaneous presence of dissipative as well as nonlinear inertial terms in the equations of motion. It seemed attractive therefore to study the interaction between these terms in an extremely simplified equation, the one-dimensional nonlinear diffusion or heat flow equation.
The monograph contains the solution of the nonlinear equation, the mathematical theory of the auxiliary functions needed for the construction of the distribution function, the evaluation of a set of integrals of the distribution function, and the application of the results of these integrations.