Dynamical Systems Weekend
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Joel Avrin (University of North Carolina-Charlotte)
Approximate Globally-Regular Solutions of the Navier-Stokes Equations via Spectral Methods
Abstract: We introduce spectral versions of the hyperviscous Navier-Stokes equations (HNSE) and the LANS equations. As in spectral eddy viscosity (SEV) and spectral vanishing viscosity (SVV) models, we allow the coefficients of the extra regularizing terms to vary as a monotonically-increasing function of the wave number k, and discuss why this is a natural generalization for these models in the context of modeling turbulence. We establish global regularity of solutions in 3-d via H1-bounds which are developed in such a way that if m is such that the significant regularizing effects occur after the first m wave numbers, then these bounds depend on m, but only as a fractional power of it. In the limit of appropriate parameters we establish weak subsequence convergence to Leray solutions of the standard Navier-Stokes equations (NSE). Thus our models can be viewed as globally-regular approximate models of the NSE.
In particular, in each model the minimal monotonic function of k is a Heaviside function 
where the constant 
represents an appropriate length scale. In this case and similarly in related generalizations the extra regularizing effects are only applied for 
, as in the way viscosity is applied in SVV, so that as in SVV the 
range essentially sees only standard NSE physics. The idea is not to interfere with the dynamics of the convection-driven energy transfer in the inertial range from larger to smaller scales and back (the latter phenomena known as "backscatter") before finally cascading to the dissipative scales. The fractional-power dependence on m of the H1-norm gives a rough idea that the complexity-growth penalty incurred by this "no-interference" policy is manageable. The mechanism of approximating the NSE is distinct from the mechanisms in the original HNSE or LANS-
models, which involve a singular limit of a fixed differential operator with coefficient going to zero. Here we can fix and let 
, reducing the domain of the extra regularizing operator inside a regime which according to the Kolmogorov theory quickly becomes of no dynamical consequence.