Homogenization of Hamilton-Jacobi equations in dynamic random environments

Speaker: 

Wenjia Jing

Institution: 

University of Chicago

Time: 

Thursday, April 9, 2015 - 3:00pm to 4:00pm

Host: 

Location: 

RH306

Abstract: We discuss some homogenization problems of Hamilton-Jacobi equations in time-dependent (dynamic) random environments, where the  coefficients of PDEs are highly oscillatory in the space and time variables. We consider both first order and second order equations, and
 emphasize how to overcome the difficulty imposed by the lack of coercivity in the time derivative. In the first order case with linear growing Hamiltonian, periodicity in either the space or the time variable is  assumed; in the second order case with at most quadratic growing  Hamiltonian, uniform ellipticity of the second order term is assumed.
 

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.

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