Regularity of stagnation-point form solutions of the two-dimensional Euler equations

Abstract

A class of semi-bounded solutions of the two-dimensional incompressible Euler equations, satisfying either periodic or Dirichlet boundary conditions, is examined. For smooth initial data, new blowup criteria in terms of the initial concavity profile is presented and the effects that the boundary conditions have on the global regularity of solutions is discussed. In particular, by deriving a formula for a general solution along Lagrangian trajectories, we describe how periodicity can prevent blow-up. This is in opposition to Dirichlet boundary conditions which, as we will show, allow for the formation of singularities in finite time. Lastly, regularity of solutions arising from non-smooth initial data is briefly discussed.