Axisymmetric solutions to the image Euler equations Original Research Article

Here we consider the 3D3D incompressible Euler equations with axisymmetric velocity without swirl. First we will show that if View the MathML sourceu0∈Cs∩L2,s>1, and View the MathML sourceω0(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 View the MathML sourceω0x12+x22∈L∞, then there exists a unique quasilipschitzian solution View the MathML sourceu∈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].