You seem to know more about this problem then I do. I was thinking about Dirichlet boundary conditions on a smooth bounded domain. I have been thinking quite a bit about periodic boundary conditions on the cube. I got interested in this about 3 years ago, and have since looked at the books by Temam, Constantin and Foias, and Doerhing and Gibbons.
In the cases I am looking at, the problem really is to establish reqularity of the solutions, that is, if initial data and the forcing term is smooth, then so is the solution for all time.
None of these books mentioned the non-uniqueness of the flow between rotating cylinders. Could you tell me more about that?
RobXXVIII wrote: > > Under what constraints? > > It's been proved that if the velocity is given everywhere > on the boundary of some region and the Reynold number > is smaller than pi sqrt(3) there is a unique solution. > > The flow between rotating cylinders has been observed to be > non-unique. > > Some 2-d problems have multiple solutions. By choosing > appropriate boundary conditions 3-d problems can be > created with multiple solutions. > > The question when does the Navier-Stokes equation have > unique solutions may be interesting.
Stephen Montgomery-Smith firstname.lastname@example.org 307 Math Science Building email@example.com Department of Mathematics firstname.lastname@example.org University of Missouri-Columbia Columbia, MO 65211 USA