Vorticity coherence effect on energy-enstrophy bounds

Speaker: 

Michael Jolly

Institution: 

Indiana University

Time: 

Thursday, December 4, 2014 - 3:00pm

Location: 

RH 440R

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.

Diffusions with Rough Drifts and Navier-Stokes Equation

Speaker: 

Fraydoun Rezakhanlou

Institution: 

UC Berkeley

Time: 

Wednesday, November 12, 2014 - 3:00pm to 4:00pm

Host: 

Location: 

RH440R

 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

Speaker: 

Maria Gualdani

Institution: 

George Washington University

Time: 

Tuesday, January 20, 2015 - 3:00am to 4:00am

Host: 

Location: 

RH306

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.

 

 

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