# Multivalued Navier Stokes Solution

1. Aug 9, 2012

### facettedlemon

I was messing around with the Navier-Stokes equations a while ago and I found a time dependent 2D solution. The force I used was periodic, bounded, and smooth. The question I have is with regards to the time functions in the solution. The solution is spatially periodic and has the form:

u = Arctan[(√2/2)*Tan[exp(-t)]]*f(x,y)
v = Arctan[(√2/2)*Tan[exp(-t)]]*g(x,y)
p= p(x,y,t)

t=time, x,y = coordinate directions.

The solution is unique in the "classical sense" because they are the only functions to satisfy the PDE with the given force and IC. However, they are multivalued with respect to time because arctan is a multivalued function. This where I'm confused. Yes, if you follow one arctan curve the solution is single valued, but what's to prevent the solution from jumping from one arctan value to another? The IC doesn't prevent it because arctan can jump to a different curve at any time greater than zero. Any help would be much appreciated. Thanks!