| Abstract: |
| In this talk we will review classical results on determining modes for fluid
equations and present a slightly different approach where we start with a
time-dependent determining wavenumber defined for each individual
trajectory and then study its dependence on the force. While in some cases
this wavenumber has a uniform upper bound, it may blow up when the
equation is supercritical. A bound on the determining wavenumber provides
determining modes, which in some sense measure the number of
degrees of freedom of the flow, or resolution needed to describe a
solution.
For the 3D
Navier-Stokes equations, we obtain
a uniform bound on the time average of this wavenumber, which we estimate
in terms of the Kolmogorov
dissipation number and Grashof constant. |
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