On diperna‐majda concentration sets for two‐dimensional incompressible flow

TLDR: In this paper, the existence of weak solutions to the incompressible Euler equations for vortex sheet initial data is an open problem, and two approaches to this problem are the smoothing of vortex sheets by approximated identifies (which provides a sequence of smooth initial conditions, converging to vortex sheet starting conditions, for which classical solutions exist) and viscous smoothing.

Abstract: : The existence of weak solutions to the incompressible Euler equations for vortex sheet initial data is an open problem. Two approaches to this problem are the smoothing of vortex sheets by approximated identifies (which provides a sequence of smooth initial conditions, converging to vortex sheet initial conditions, for which classical solutions exist) and viscous smoothing of vortex sheets. These ideas are among the principal motivations for the recent series of papers in which the nature of the limiting behavior of sequences of solutions of Euler's equations, are examined. The concept of generalized Young measure-valued solution of the Euler equations is introduced. In two dimensions, approximate solution sequences always have subsequences which converge, in an appropriate sense, to non-oscillatory generalized Young measure-valued solutions of Euler's equations. Infinitesimal vortices of zero circulation were described in and named phantom vortices. The vortices in our example are somewhat different in that they have two length scales. The two length scales are important, for one cannot pack phantom vortices on successively finer lattices, as above, keeping the vorticity absolutely summable, without at the same time obtaining strong convergence in L-sq. Reprints.