The Hopf-Cole Solution of the Nonlinear Diffusion Equation and Its Geometrical Interpretation for the Case of Small Diffusivity

Abstract

We consider the nonlinear diffusion equation*

$${u_t} + u{u_x} = v{u_{xx}},$$

((1.1))

where v is a positive constant. This equation has been chosen as a simplified form of the Navier—Stokes equations, and we may consider u as a quantity of the nature of a velocity, having the dimensions LT ⁻¹ with reference to the scales of length and time; while v plays the part of a diffusivity or kinematic viscosity, with dimensions L ² T ⁻¹. Half the square of u can be considered as a ‘kinetic energy’ per unit length of the x-scale, with dimensions L ² T ⁻²; and v(u _x )² will be a ‘dissipation of energy’ per unit length and in unit time, with dimensions L ² T ⁻³. These dimensional considerations are not of primary importance, but they will turn up from time to time. ‘Mass’ or ‘density’ does not occur.