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.

 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.

Abstract: We consider the nonlinear instability of a steady state
$v_*$ of the Euler equation in an $n$-dim fixed smooth bounded domain. When
considered in $H^s$, $s>1$, at the linear level, the stretching of the
steady fluid trajectories induces unstable essential spectrum which
corresponds to linear instability at small spatial scales and the
corresponding growth rate depends on the choice of the space $H^s$.
More physically interesting linear instability relies on the unstable
eigenvalues which correspond to large spatial scales. In the case when
the linearized Euler equation at $v_*$ has an exponential dichotomy of
center-stable and unstable (from eigenvalues) directions, most of the
previous results obtaining the expected nonlinear instability in $L^2$
(the energy space, large spatial scale) were based on the vorticity
formulation and therefore only work in 2-dim. In this talk, we prove,
in any dimensions, the existence of the unique local unstable manifold
of $v_*$, under certain conditions, and thus its nonlinear
instability. Our approach is based on the observation that the Euler
equation on a fixed domain is an ODE on an infinite dimensional
manifold of volume preserving maps in function spaces. This is a
joint work with Zhiwu Lin.

Understanding the incompressible/compressible fluid is a fundamental, but
challenging, project not only in numerical analysis, but also in
theoretical analysis, especially when extra effects, such as the elastic
deformation or the magnetic field, interact with the flow. In this talk,
the incompressible fluid and its associated flow map will be reviewed first.
The main object of this talk devotes to a recent work in understanding
incompressible/compressible magnetohydrodynamic fluids with zero magnetic
diffusivity (which is equivalent to infinite conductivity). This is a
joint work with Fanghua Lin.