The Cauchy problem to the Euler equations is considered: \[ \begin{aligned} &\frac{\partial v}{\partial t}+(v\cdot\nabla)v+\nabla p=0, \quad \text{div}\,v=0 \quad x\in \mathbb R^3,\quad t\in(0,\infty),\\ &v(x,0)=v_0(x),\quad x\in \mathbb R^3. \end{aligned}\tag{1} \] Here \(v=(v_1,v_2,v_3)\) is the velocity of a fluid, \(p\) is the pressure. If \(v\) does not depend on the angular coordinate \(\theta\) of cylindrical coordinates \((r,\theta,x_3)\) we have axisymmetric velocity. Here \(r=\sqrt{x_1^2+x_2^2}\). The angular component \(v_\theta\) is called the swirl velocity.
Two main results are proved. First: if \(v_0\) is a divergence free, axisymmetric vector without swirl such that \(v_0\in C^s\) \((s>1)\), \(| \text{curl}\,v_0| \leq Cr\) then the problem (1) has a unique global solution \(v\in C([0,\infty);C^s)\). Second: if \(v_0\) is a divergence free, axisymmetric vector without swirl such that \(v_0\in L^2\), \(\text{curl}\,v_0\in L^\infty\), \(\frac{\text{curl}\,v_0}{r(x)}\in L^\infty\) then the problem (1) has a unique global axisymmetric without swirl solution \(v\in C([0,\infty);C_*^1)\).