Does Current Density Decay to Zero at Vortex Centers in 2D Systems?

[fluidistic]

Hello, I have derived an expression for $\vec J$ in a particular, unusual physical problem in 2D. The expression is different from $\vec 0$ everwhere in the material except at at least two different points.

Can I conclude that the only way for this to occur is that $\vec J$ itself decays to $\vec 0$ the closer it gets to the center of the vortices? I assume that the current density vanishes exactly at the center of vortices of currents.

Also, what can I conclude about the magnetic field?

 

[osilmag]

Would you be able to share the expression or perhaps have an image?

I would assume that the magnetic field decays to zero since at those vortices the current goes to zero.