We consider a sequence of Leray-Hopf weak solutions of the 2D Navier-Stokes equations on a bounded domain, in the vanishing viscosity limit. We provide sufficient conditions on the associated vorticity measures, away from the boundary, which ensure that as the viscosity vanishes the sequence converges to a weak solution of the Euler equations. These assumptions are consistent with vortex sheet solutions of the Euler equations.Date: September 12, 2018.

In this article we consider the α-Euler equations in the exterior of a small fixed disk of radius ε. We assume that the initial potential vorticity is compactly supported and independent of ε, and that the circulation of the unfiltered velocity on the boundary of the disk does not depend on ε. We prove that the solution of this problem converges, as ε → 0, to the solution of a modified α-Euler equation in the full plane where an additional Dirac located at the center of the disk is imposed in the potential vorticity.

Cheskidov et al. (2016 Commun. Math. Phys. 348 , 129–143. ( doi:10.1007/s00220-016-2730-8 )) proved that physically realizable weak solutions of the incompressible two-dimensional Euler equations on a torus conserve kinetic energy. Physically realizable weak solutions are those that can be obtained as limits of vanishing viscosity. The key hypothesis was boundedness of the initial vorticity in L p , p > 1 . In this work, we extend their result, by adding forcing to the flow. This article is part of the theme issue ‘Scaling the turbulence edifice (part 2)’.

Lei and Lin [Comm. Pure Appl. Math. 64 (2011), pp. 1297–1304] have recently given a proof of a global mild solution of the three-dimensional Navier-Stokes equations in function spaces based on the Wiener algebra. An alternative proof of existence of these solutions was then developed by Bae [Proc. Amer. Math. Soc. 143 (2015), pp. 2887–2892], and this new proof allowed for an estimate of the radius of analyticity of the solutions at positive times. We adapt the Bae proof to prove existence of the Lei-Lin solution in the spatially periodic setting, finding an improved bound for the radius of analyticity in this case.