Mathematical analysis of vortex sheets

Abstract

We consider the motion of the interface separating two domains of the same fluid that moves with different velocities along the tangential direction of the interface. The evolution of the interface (the vortex sheet) is governed by the Birkhoff-Rott (BR) equations. We consider the question of the weakest possible assumptions such that the Birkhoff-Rott equation makes sense. This leads us to introduce chord-arc curves to this problem. We present three results. The first can be stated as the following: Assume that the Birkhoff-Rott equation has a solution in a weak sense and that the vortex strength is bounded away from 0 and ∞. Moreover, assume that the solution gives rise to a vortex sheet curve that is chord-arc. Then the curve is automatically smooth, in fact analytic, for fixed time. The second and third results demonstrate that the Birkhoff-Rott equation can be solved if and only if only half the initial data is given.

Euler equations for incompressible ideal fluids

“…The Kelvin-Helmholtz problem exhibits also some basic similar instabilities, but it is in some sense simpler. This is explained at the end of section 7, where it is also shown that some recent results of [44] , [71] and [72], on the regularity of the vortex sheet (interface), do contribute to the understanding of the instabilities of the original problem.…”

“…Up to now, the best (to the best of our knowledge) known result is due to S. Wu [71] [72]. The hypothesis C α loc (R t ; C 1+β loc (R λ )) is replaced by H 1 loc (R t × R λ ) .…”

On the Limit as the Density Ratio Tends to Zero for Two Perfect Incompressible Fluids Separated by a Surface of Discontinuity

“…The jump discontinuity in the tangential component of velocity across the material interface, which appears in the two-phase Euler model, is responsible for the ill-posedness of this system of PDE when surface tension effects are ignored (see [9] and [20]). On the other hand, in the presence of surface tension, the two-phase system is well-posed.…”

The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

“…[133,20,36], and it is known to be ill posed if one goes beyond the analytic framework [36,21]. As for the relation between the analyticity and the regularity of the solution to the Birkhoff-Rott equations, see also [88,146] and references therein. The Navier-Stokes equations in R n with vortex sheets as initial data are locally well-posed when n = 3 and globally well-posed when n = 2 since the velocity field in this class has enough local regularity to construct a unique solution.…”