On the global well-posedness for the axisymmetric Euler equations

Abstract

This paper deals with the global well-posedness of the 3D axisymmetric Euler equations for initial data lying in critical Besov spaces B

Inviscid limit for axisymmetric flows without swirl in a critical Besov space

“…we can obtain similar results by similar arguments as in [1,7] where the method used is different from this paper.…”

“…Uniform persistence of the initial regularityBy similar arguments as in Proposition A.2 in[1], we can obtaine −CV (t) ω(t) τ ) L ∞ e −CV (τ ) ω(τ )…”

“…The optimal result in Sobolev spaces is done by Shirota and Yanagisawa [12] who proved global existence in H s with s > 5 2 . Recently, in [1] the global well-posedness for the 3-D axisymmetric Euler equations with the initial data lying in the critical Besov spaces B …”

“…Making use of completely similar arguments as in[1], we can prove the global well-posedness for the 3-D axisymmetric Navier-Stokes equations in the critical Besov spaces B…”

A remark on the Lipschitz estimates of solutions to Navier-Stokes equations

“…We denote by f p the L p (R n ) norm of a function f . We shall use L p,q to denote the Lorentz spaces (see [6] for the definition and some basic properties of this space).…”