# Vorticity coherence effect on energy-enstrophy bounds

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One of the main tenets in the Kolmogorov theory of 3D turbulence is the direct cascade of energy. This means that the rate of transfer of energy from one length scale to the next smallest is roughly constant over the so-called inertial range of scales. This can be indicated by a large quotient of the averages of enstrophy over energy. Similarly, the Batchelor, Kraichnan, Leith theory of 2D turbulence features an additional direct cascade, that of enstrophy, which in turn is indicated by a large quotient of averaged palinstrophy over enstrophy. In the case of the 2D NSE we have derived bounding curves for these pairwise quantities by combining estimates for their time derivatives. To do so for the 3D NSE, however, is to confront its outstanding global regularity problem.

Beirao da Veiga, following work of Constantin and Fefferman, showed that solutions to the 3D NSE are regular if one assumes that the direction of vortex filaments is Holder continuous with exponent 1/2. Under this assumption we construct in a single bounding curve whose maximal enstrophy is shown to scale as an exponential of the Grashof number. This suggests that even under this smoothness assumption solutions can display extraordinary bursts in enstrophy.

# TBA

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# Diffusions with Rough Drifts and Navier-Stokes Equation

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According to DiPerna-Lions theory, velocity fields with weak

derivatives in L^ p spaces possess weakly regular flows. When a velocity

field is perturbed by a white noise, the corresponding (stochastic) flow

is far more regular in spatial variables; a diffusion with a drift in a

suitable L p space possesses weak derivatives with exponential bounds.

As an application we show that a Hamiltonian system that is perturbed by a

white noise produces a symplectic flow for a Hamiltonian function that is

merely in W^{ 1,p} for p strictly larger than dimension. I also discuss

the potential application of such regularity bounds to study solutions of

Navier-Stokes equation with the aid of Constantin-Iyer's circulation

formula.

# Estimates for the homogeneous Landau equation with Coulomb potential

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We present conditional existence results for the Landau equation with

Coulomb potential. Despite lack of a comparison principle for the equation, the

proof of existence relies on barrier arguments and parabolic regularity theory. The

Landau equation arises in kinetic theory of plasma physics. It was derived by Landau

and serves as a formal approximation to the Boltzmann equation when grazing

collisions are predominant.

We also present long-time existence results for the isotropic version of the Landau

equation with Coulomb potential.