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

In: The Nonlinear Diffusion Equation

Author

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  • J. M. Burgers

    (University of Maryland, Institute for Fluid Dynamics and Applied Mathematics)

Abstract

We consider the nonlinear diffusion equation* (1.1) {u_t} + u{u_x} = v{u_{xx}}, $${u_t} + u{u_x} = v{u_{xx}},$$ 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.

Suggested Citation

  • J. M. Burgers, 1974. "The Hopf-Cole Solution of the Nonlinear Diffusion Equation and Its Geometrical Interpretation for the Case of Small Diffusivity," Springer Books, in: The Nonlinear Diffusion Equation, chapter 0, pages 9-20, Springer.
    • Handle: RePEc:spr:sprchp:978-94-010-1745-9_2
    DOI: 10.1007/978-94-010-1745-9_2
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