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 ½. 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.

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

In [Nonlinearity, 11 (1998), pp. 1625--1636], Clopeau, Mikelic, and Robert studied the inviscid limit of the two-dimensional incompressible Navier--Stokes equations in a bounded domain subject to Navier friction--type boundary conditions. They proved that the inviscid limit satisfies the incompressible Euler equations, and their result ultimately includes flows generated by bounded initial vorticities. Our purpose in this article is to adapt and, to some extent, simplify their argument in order to include pth power integrable initial vorticities, with p > 2.

On the inviscid limit for two-dimensional incompressible flow with navier friction condition

This paper deals with the derivation and analysis of a compressible Euler-type equation with singular commutator, which is derived from a hyperbolic limit of the kinetic description to the Cucker–Smale model of interacting individuals.

Euler-type equations and commutators in singular and hyperbolic limits of kinetic Cucker–Smale models

The main purpose of this work is to establish the existence of a weak solution to the incompressible 2D Euler equations with initial vorticity consisting of a Radon measure with distinguished sign in H

 − 1, compactly supported in the closed right half-plane, superimposed on its odd reflection in the left half-plane. We make use of a new a priori estimate to control the interaction between positive and negative vorticity at the symmetry axis. We prove that a weak limit of a sequence of approximations obtained by either regularizing the initial data or by using the vanishing viscosity method is a weak solution of the incompressible 2D Euler equations. We also establish the equivalence at the level of weak solutions between mirror symmetric flows in the full plane and flows in the half-plane. Finally, we extend our existence result to odd L

 1 perturbations, without distinguished sign, of our original initial vorticity.

Existence of Vortex Sheets with Reflection Symmetry in Two Space Dimensions

We present a sharp local condition for the lack of concentrations in (and hence the L 2 convergence of) sequences of approximate solutions to the incompressible Euler equations. We apply this characterization to greatly simplify known existence results for 2D flows in the full plane (with special emphasis on rearrangement invariant regularity spaces), and obtain new existence results of solutions without energy concentrations in any number of spatial dimensions. Our results identify the 'critical' regularity which prevents concentra- tions, regularity which is quantified in terms of Lebesgue, Lorentz, Orlicz and Morrey spaces. Thus, for example, the strong convergence criterion cast in terms of circulation logarithmic decay rates due to DiPerna and Majda is simplified (removing the weak control of the vorticity at infin- ity) and extended (to any number of space dimensions). Our approach relies on using a generalized div-curl lemma (interesting for its own sake) to replace the role that elliptic regularity theory has

http://www.numdam.org/item/AIHPC_2000__17_3_371_0.pdf

Approximate solutions of the incompressible Euler equations with no concentrations

We prove the existence of a classical weak solution for the 2-D incompressible Euler equations with initial vorticity ω0=ω0′ + ω0″, where ω0′ is inL1(R2)⌢H−1(R2), compactly supported, and ω0″ is a compactly supported positive Radon measure inH−1(R2).

Italo Vecchi

Papers

On L1-vorticity for 2-D incompressible flow