Axisymmetric solutions to the 3D Euler equations

作者：Shen Gang; Zhu Xiangrong

来源：Nonlinear Analysis-Theory Methods & Applications, 2007, 66(9): 1938-1948.

DOI：10.1016/j.na.2006.02.034

Here we consider the 3D incompressible Euler equations with axisymmetric velocity without swirl. First we will show that if u(0) is an element of C-s boolean AND L-2, s > 1, and omega(0)(x) <= C root x(1)(2) + x(2)(2), then there exists a unique u is an element of C([0, infinity); C-s) 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 u(0) is an element of L-2, omega(0) is an element of L-infinity and omega(0)/root x(1)(2)+x(2)(2) is an element of L-infinity, then there exists a unique 2 2 quasilipschitzian solution u is an element of C([0, infinity); 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 3D Euler equations for incompressible fluid flows, Nonlinear Anal. 29(12) (1997) 1393-1404].

• 出版日期2007-5-1
• 单位浙江大学