In this talk I discuss the different types of models which originate from a lubrication approximation of viscous coating flow dynamics on the outer surface of a rotating cylinder, that is in the presence of a gravitational field. Analytical and numerical results related to existence, uniqueness and stability of solutions will be presented.

The existence and uniqueness of Gevrey regularity solution for a class of nonlinear
bistable gradient flows, with the energy decomposed into purely convex and concave parts,
such as epitaxial thin film growth and square phase field crystal models, are discussed in this talk.
The polynomial pattern of the nonlinear terms in the chemical potential enables one to derive a
local in time solution with Gevrey regularity, with the existence time interval length dependent
on certain functional norms of the initial data. Moreover, a detailed Sobolev estimate for the gradient
equations results in a uniform in time bound, which in turn establishes a global in
time solution with Gevrey regularity. An extension to a system of gradient flows,
such as the three-component Cahn-Hilliard equations, is also addressed in this talk.

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.