Determining Modes for Some Fluid Equations

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.

Event Topic

Stochastic & Multiscale Modeling and Computation