An example in two dimensional fluid flow

Nets Katz
Indiana University

An important open problem in the theory of the two dimensional incompressible, unforced Euler equations is to determine the rate at which the Sobolev norms of a smooth solution may grow. The proof of the Beale-Kato-Majda theorem shows that such growth may be no faster than double exponential in time and it is unknown whether the growth can be that fast. One way of formulating the proof is that the rate at which dyadic scales can become active is at most exponential in time and the rate of growth at a given time is controlled by the number of active scales. We derive a model for growth which takes into account only the dominant low-high part of the nonlinear term in the Euler equations. We show that when $N$ scales are active, this model approximates the behavior of the equations for time at least ${\log N \over N}$.
We produce an example for the model which
exhibits growth consistent with the double
exponential bound being best.

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