Existence of Axisymmetric Weak Solutions of the 3-d Euler Equations for Near-vortex-sheet Initial Data

We study the initial value problem for the 3-D Euler equation when the fluid is inviscid and incompressible, and flows with axisymmetry and without swirl. On the initial vorticity ω0, we assumed that ω0/r belongs to L(logL(R3))α with α > 1/2, where r is the distance to an axis of symmetry. To prove the existence of weak global solutions, we prove first a new a priori estimate for the solution. Introduction We consider the Euler equations for homogeneous inviscid incompressible fluid flow in R ∂v ∂t + (v · ∇)v = −∇p , div v = 0 in R+ × R 3 , (1) v(0, ·) = v0 , (2) where v(t, x) = (v1(t, x), v2(t, x), v3(t, x)) is the velocity of the fluid flow and p(t, x) is the pressure. The problem of finite-time breakdown of smooth solutions to (1)(2) for smooth initial data is a longstanding open problem in mathematical fluid mechanics. (See [6,13,14] for a detailed discussion of this problem.) The situation is similar even for the case of axisymmetry (see e.g.[11], [4]). In the case of axisymmetry without swirl velocity (θ-component of velocity), however, we have a global unique smooth solution for smooth initial data [14,17]. In this case a crucial role is played by the fact that ωθ(t, x)/r (where ω = curl v, r = √ x1 + x 2 2) is preserved along the flow, and the problem looks similar to that of the 2-D Euler equations. This apparent similarity between the axisymmetric 3-D flow without swirl and the 2-D flow for smooth initial data breaks down for nonsmooth initial data. In particular, Delort [8] found the very interesting phenomenon that for a sequence of approximate solutions to the axisymmetric 3-D Euler equations with nonnegative vortex-sheet initial data, either the sequence converges strongly in Lloc([0,∞)×R ), or the weak limit of the sequence is not a weak solution of the equations. This is in contrast with Delort’s proof of the existence of weak solutions for the 2-D Euler equations with the single-signed vortex-sheet initial data, where we have weak 1991 Mathematics Subject Classifications: 35Q35, 76C05.