On singular parabolic pde's and their role in unsteady boundary layers
by
William R.C. Phillips
Swinburne University of Technology

Second order partial differential equations are classified as elliptic, hyperbolic or parabolic, according to the spectrum of eigenvalues of the coefficient functions and this information dictates the boundary / initial conditions necessary to ensure a unique stable solution. For example, in a well posed parabolic pde, at least one eigenvalue is zero with the remainder of the same sign while in a singular parabolic pde one eigenvalue is zero with the remainder of mixed sign. Thus, while an initial condition at one instant in time and boundary conditions at two locations in space for all time (open hyperspace) are usually adequate to render a stable solution in a well posed parabolic pde, a further boundary condition is necessary in the singular parabolic situation resulting in boundary conditions on the closed hyperspace, as with an elliptic pde, rendering the calculation more difficult. Boundary layer flows in fluid dynamics are typically described by parabolic pde's and although most lie in the well posed category there are exceptions. Examples in the singular parabolic class include some oil spreading flows, where the unsteady boundary layer can have two beginnings, one at the leading edge and another at the trailing edge; another is the boundary layer behind a shock wave as it moves over a flat plate. This talk will outline the story behind singular parabolic partial differential equations, and how to identify and classify them using unsteady boundary layers as examples.

Date received: December 13, 2009