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 which have a dipole singularity in the complex plane. These data are uniformly bounded in $H^1$ and lead to an earlier singularity formation. The presence of a small viscosity in the streamwise direction changes the behavior of the singularities.

References in corpus (1)

• Complex-space singularities of 2D Euler flow in Lagrangian coordinates

Cited by in corpus (15)

• Almost global existence for the Prandtl boundary layer equations
• Sobolev stability of Prandtl expansions for the steady Navier-Stokes equations
• Analysis of complex singularities in high-Reynolds-number Navier-Stokes solutions
• High Reynolds number Navier-Stokes solutions and boundary layer separation induced by a rectilinear vortex
• Sublayer of Prandtl boundary layers
• Energy dissipation caused by boundary layer instability at vanishing viscosity
• Insights from a pseudospectral study of a potentially singular solution of the three-dimensional axisymmetric incompressible Euler equation
• The van Dommelen and Shen singularity in the Prandtl equations
• On the local well-posedness of the Prandtl and the hydrostatic Euler equations with multiple monotonicity regions
• On the inviscid limit of the Navier-Stokes equations
• Viscous-Inviscid Interactions in a Boundary-Layer Flow Induced by a Vortex Array
• Separation of the two-dimensional unsteady Prandtl boundary layers under an adverse pressure gradient
• Singularities and unsteady separation for the inviscid two-dimensional Prandtl's system
• Real analytic local well-posedness for the Triple Deck
• High frequency analysis of the unsteady Interactive Boundary Layer model