Under what assumptions are planar vortices understood/solved? Under what conditions are the fluid dynamics equations known and in some sense solved for planar ($2$-dimensional), stationary vortices?
Are the equations even solved for a single vortex,
with positive viscous friction $\mu$?
Or maybe only for ideal / inviscid fluids?

          


          

NASA image of Jupiter.


 A: There is a vast body of knowledge pertaining to vortex motion in fluid dynamics. I can only begin to scratch the surface with a few comments that hopefully may be helpful to you. 
Vorticity $\mathbf{\omega} = \nabla \times \mathbf{u}$ is the curl of the velocity field. At a point $\mathbf{x}$, the vorticity $\mathbf{\omega}(\mathbf{x})$ measures the rate of local fluid rotation. A vortex is commonly defined by a region of concentrated vorticity in the fluid.
For an incompressible fluid, the vorticity satifies the PDE 
$$\frac{\partial \mathbf{\omega}}{\partial t} + \mathbf{u}\cdot \nabla \mathbf{\omega} = \mathbf{\omega} \cdot \nabla \mathbf{u} + \nu \nabla^2 \mathbf{\omega},$$
which governs the spatial and temporal evolution of vorticity as it is convected with the flow, diffused by viscosity and intensified by the stretching and tilting of vortex lines. Here the parameter $\nu$ is the kinematic viscosity.
A two-dimensional flow is only an approximate model of fluid motion, typically away from solid boundaries where vorticity may be generated by shear stresses.  In this case, there can be only one component of vorticity $\omega_z(x,y)$, normal to the plane of motion, that does not vanish. This is easily seen by taking the curl of a velocity field of the form $(u_x(x,y), u_y(x,y),0)$. The vorticity equation then reduces to 
$$\frac{\partial \omega_z}{\partial t} + u_x\frac{\partial \omega_z}{\partial x}+ u_y\frac{\partial \omega_z}{\partial y}   = \nu\nabla^2 \omega_z $$
There can be no amplification of vorticity since the term $\mathbf{\omega} \cdot \nabla \mathbf{u}$ accounting for vortex-line stretching and tilting vanishes.  The concentration of vorticity in a two-dimensional vortex will be convected and subject to viscous dissipation.
The simplest exact solution for inviscid flow corresponds to a single straight vortex filament normal to the plane of flow -- a point vortex where the velocity field decays with radial distance from the filament. For a point vortex located at the origin, the velocity field in terms of polar coordinates has only a non-zero azimuthal component $u_\theta = \frac{\Gamma}{2\pi r}$ where $\Gamma$ measures the circulation around a closed contour encircling the origin.
Multiple point vortices in an inviscid fluid will not remain stationary, but will interact and be convected with the flow. 
When viscosity is present the point vortex must decay as vorticity diffuses outward. This is described by another exact solution -- the Lamb-Oseen vortex where
$$u_\theta = \frac{\Gamma}{2\pi r}\left(1 - e^{-r^2/4 \nu t} \right), \quad \omega_z(r,t)  = \frac{\Gamma}{4\pi \nu t}e^{-r^2/4\nu t}$$
For a filament-like vortical structure to persist through time in steady-state there must be  a mechanism for amplification like vortex-line stretching. In the simplest configuration with axial symmetry, we have the exact solution for Burgers vortex:
$$u_r = -\alpha r,\,\, u_z= 2\alpha z, \, \, u_\theta = \frac{\Gamma}{2\pi r}\left( 1- e^{-\alpha r^2/2\nu}\right), \, \, \omega_z =  \frac{\Gamma}{2\pi \nu}e^{-\alpha r^2/2\nu}$$
This is no longer a two-dimensional flow. Here we have  inflow in the radial direction balanced by outflow in the axial direction to conserve mass.  The axial velocity $u_z$ streches fluid elements to sustain the  vorticity, and  diffusion away from the axis is offset by inward convection due to the radial velocity $u_r$. 
