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Topic: 21st Century 'Hilbert' problems
Replies: 25   Last Post: Feb 10, 1999 1:11 AM

 Messages: [ Previous | Next ]
 Stephen Montgomery-Smith Posts: 97 Registered: 12/6/04
Re: 21st Century 'Hilbert' problems
Posted: Feb 7, 1999 11:55 AM

Hi,

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?

Thanks, Stephen

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 stephen@math.missouri.edu
307 Math Science Building stephen@showme.missouri.edu
Department of Mathematics stephen@missouri.edu
University of Missouri-Columbia
Columbia, MO 65211
USA

Phone (573) 882 4540
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http://math.missouri.edu/~stephen