Axisymmetric solutions to the 3D3D Euler equations

Here we consider the 3D3D incompressible Euler equations with axisymmetric velocity without swirl. First we will show that if u0∈Cs∩L2,s>1, and ω0(x)≤Cx12+x22, then there exists a unique u∈C([0,∞);Cs)u∈C([0,∞);Cs) that solves the equation. This conclusion improves on the related results given by Majda [A.L. Bertozzi, A. Majda, Vorticity and Incompressible Flow, in: Cambridge Texts in Applied Mathematics, vol. 27, 2002; A. Majda, Vorticity and the mathematical theory of incompressible fluid flow, Comm. Pure Appl. Math. 39 (1986) S187–S220] and by Raymond [X. Saint Raymond, Remarks on axisymmetric solutions of the incompressible Euler system, Comm. Partial Differential Equations 19 (1994) 321–334].On the other hand, if u0∈L2,ω0∈L∞u0∈L2,ω0∈L∞ and ω0x12+x22∈L∞, then there exists a unique quasilipschitzian solution u∈C([0,∞);C∗1). This improves on the corresponding results due to Raymond [X. Saint Raymond, Remarks on axisymmetric solutions of the incompressible Euler system, Comm. Partial Differential Equations, 19 (1994) 321–334] and to Chae and Kim [D. Chae, N. Kim, Axisymmetric weak solutions of the 3D3D Euler equations for incompressible fluid flows, Nonlinear Anal. 29(12) (1997) 1393–1404].