coupled PDEs

Hello,

I want to find the profiles of two fields (u1(x,y) and u2(x,y)) on a 2d geometry with known boundary conditions that satisfy the following equation:

[(u2y*u1x-u1y*u2x)^2]x + [(u2y*u1x-u1y*u2x)^2]y = 0

The geometry is a square box with the following boundary conditions:
left edge: u1=1, u2=no flux
right edge: u1=0.01, u2=no flux
top: u1=no flux, u2=1
bottom: u1=no flux, u2=0.01

I know (at least one) solution which is u1(x,y)=1-0.99*x and u2(x,y)=0.01+0.99*y. If I give COMSOL this solution as initial condition (or actually even exponential profiles satisfying the boundary conditions) it returns it as expected. However when I start e.g. with a random distribution it doesn't find it. What can be the problem? And what solver configurations make sense? I am quite new to COMSOL, so I have no idea what to change...

And the future idea is to extend the problem to different shapes, so trying it on the square is trivial but I need to make it work there before going to a geometry of which I don't know the answer.

Thank you - I am really happy for any hints/ideas/... :)
Kathrin