Singularity formation for Prandtl’s equations

Abstract

We consider Prandtl’s equations for the impulsively started disk and follow the process of the formation of the singularity in the complex plane using the singularity tracking method. We classify Van Dommelen and Shen’s singularity as a cubic root singularity. We introduce a class of initial data, uniformly bounded in H1, which have a dipole singularity in the complex plane. These data lead to a solution blow-up whose time can be made arbitrarily short within the class. This is a numerical evidence of the ill-posedness of the Prandtl equations in H1. The presence of a small viscosity in the streamwise direction changes the behavior of the singularities. They stabilize at a distance from the real axis which depends on the amount of the viscosity. We show that the Van Dommelen and Shen singularity and the singularity predicted by E and Engquist in [W. E, B. Engquist, Blowup of the Solutions to the Unsteady Prandtl’s Equations, Comm. Pure Appl. Math. 50 (1997) 1287–1293.] have different complex structures.