The Weak Vorticity Formulation of the 2-D Euler Equations and Concentration-Cancellation

The weak limit of a sequence of approximate solutions of the 2-D Euler equations will be a solution if the approximate vorticities concentrate only along a curve x(t) that is Holder-continuous with exponent A new proof is given of the theorem of DiPerna and Majda that weak limits of steady approximate solutions are solutions provided that the singularities of the inhomogeneous forcing term are sufficiently mild. An example shows that the weaker condition imposed here on the forcing term is sharp. A simplified formula for the kernel in Delort’s weak vorticity formulation of the two-dimensional Euler equations makes the properties of that kernel readily apparent, thereby simplying Delort’s proof of the existence of one-signed vortex sheets.