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The Navier-Stokes equation is a nonlinear equation with a well understood
linear part, natural energy estimates, and a non-trivial scaling symmetry. In
two space dimensions, the energy is invariant under the scaling. In the theory
of non-linear PDE such a situation is often called "the critical case". Due to
fundamental contributions of many researchers, for parabolic and elliptic
equations this situation is quite well-understood. In three space dimensions,
the natural Navier-Stokes energy decreases if we scale towards smaller lengths.
This is often called "the super-critical case". Questions regarding existence
of regular solutions seem to become much harder, and the specifics of the
equation come much more into play. Perhaps studying some simpler super-critical
equations, and trying to view Navier-Stokes as a member of a suitable family of
super-critical equations might be useful. In the lecture I will talk about some
results in this direction.
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