The weak vorticity formulation of the 2-D Euler equations and concentration- cancellation.

*(English)*Zbl 0822.35111Summary: 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 HĂ¶lder-continuous with exponent \(1/2\).

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 simplifying Delort’s proof of the existence of one-signed vortex sheets.

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 simplifying Delort’s proof of the existence of one-signed vortex sheets.

##### MSC:

35Q35 | PDEs in connection with fluid mechanics |

76B47 | Vortex flows for incompressible inviscid fluids |

35Q30 | Navier-Stokes equations |

35A35 | Theoretical approximation in context of PDEs |