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Mathematical Fluid Dynamics
Mathematical Fluid Dynamics
Organizer:Igor Kukavica, University of Southern California
James Robinson, University of Warwick
Boundary layer for the Navier-Stokes-alpha
model of uid turbulence
Alexey Cheskidov
Indiana University, USA
email:acheskid@indiana.edu
We study a boundary layer problem for the Navier-
Stokes-alpha model obtaining a generalization of the
Prandtl equations which we conjecture to represent
the averaged ow in a turbulent boundary layer. We
study the equations for the semi-innite plate, both
theoretically and numerically. Solutions agree with
some experimental data in a part of the turbulent
boundary layer.

Inertial manifolds and Gevrey regularity for
the MooreGreitzer model of turbomachine
engine
Yeojin Chung
University of California, Irvine, USA
email:ychung@math.uci.edu
Edriss S. Titi
We study the regularity and long-time behavior of
the solutions to the MooreGreitzer model of turbo-
machine engine. In particular, we prove that this
dissipative system of evolution equations possesses
a global invariant inertial manifold, and therefore
its underlying long-time dynamics reduces to that
of an ordinary dierential system. Furthermore, we
show that the solutions of this model belong to a
Gevrey class of regularity (real analytic in the spatial
variables). As a result, one can show the exponen-
tially fast convergence of the Galerkin approximation
method to the exact solution, an evidence of the re-
liability of the Galerkin method as a computational
scheme in this case. The rigorous results presented
here justify the readily available low dimensional nu-
merical experiments and control designs for stabiliz-
ing certain states and travelling wave solutions for
this model.

Remarks on rotating uids
Peter Constantin
University of Chicago, USA
email:const@math.uchicago.edu
We will discuss bounds for transport and spectra in
rotating Navier-Stokes equations forced at the bound-
ary.

On interpolation between algebraic and geo-
metric conditions for smoothness of the vor-
ticity in the 3D NSE
Zoran Grujic
University of Virginia, USA
email:zg7c@virginia.edu
We formulate some sucient conditions for
smoothness of the vorticity consisting of space-time
integrability mixed with the regularity of the vortic-
ity directions.

Recurrent estimates for the Navier-Stokes
equations
Michael Jolly
Indiana University, USA
email:msjolly@indiana.edu
C. Foias and O. P. Manley
In the Kraichnan theory of two-dimensional turbu-
lence it is assumed that the enstrophy, rather than the
energy that is the most relevant quantity. We briey
recall recent rigorous results which show that indeed
the enstrophy cascade is more pronounced than that
of energy, in the 2-D case. These results pertain to
averages of the enstrophy transfer, or ux, through
a specied wavenumber, and thus do not provide
any information on how these quantities can uctu-
ate in time. Toward answering this question, we also
present two estimates relating the enstrophy beyond
a given wavenumber (the so-called high modes) and
the enstrophy ux through that wavenumber. They
are recurrent in that they hold at least once within
certain bounded time intervals. The rst eectively
provides a bound on how long the enstrophy ux can
remain negative, the second a bound on the enstro-
phy of the high modes, valid for wavenumbers on the
order of the square root of the Grashof number. Some
1 numerical results are presented to illustrate these es-
timates. This work is joint with C. Foias and O.P.
Manley, in that much of it was completed before
O.P.M. passed away.

The number of determining modes in 2D tur-
bulence: A computational study
Eric Olson
University of Nevada, Reno, USA
University of California, Irvine, USA
email:ejolson@unr.edu
Edriss Titi
The method of continuous data assimilation from
weather forecasting is used to study the number of de-
termining modes for the two-dimensional incompress-
ible NavierStokes equations. Our focus is on how
the body forcing aects the rate of continuous data
assimilation and the number of determining modes.
These quantities are shown to depend strongly on the
length scales present in the forcing.

Finite-dimensional dynamics on global attrac-
tors
James Robinson
University of Warwick, United Kingdom
email:jcr@maths.warwick.ac.uk
C. Foias
The talk will discuss approaches to reproducing
the dynamics on a nite-dimensional attractor using
a nite-dimensional dynamical system. In particular,
the aim is to obtain something akin to an inertial
form for the 2d Navier-Stokes equations even though
the existence of an inertial manifold is still an open
question.

Nonlinear wave equations and the Melnikov
problem
Alain Schenkel
University of Helsinki, Finland
email:Alain.Schenkel@Helsinki.FI
Bricmont and A. Kupiainen
I will describe a new proof of the Melnikov problem
in innite dimensional systems, namely, persistence
of quasi-periodic, low dimensional elliptic tori. Our
result covers situations in which the so-called normal
frequencies are multiple. In particular, it provides a
new proof of the existence of small amplitude, quasi-
periodic solutions of nonlinear wave equations with
periodic boundary conditions.

Sharp interface limits and global existence for
the phase eld Navier-Stokes equations
Steve Shkoller
University of California, Davis, USA
email: shkoller@math.ucdavis.edu
We will introduce the phase-eld Navier-Stokes
equations, and prove that they possess Leray global
weak solutions. We will then show that for smooth
initial data, solutions of the phase-eld model con-
verge weakly to solutions of the sharp-interface
Navier-Stokes equations. In the convergence proof,
an auxiliary PDE is introduced which couples the
Navier-Stokes equations with the classical geometric
problem of motion by mean curvature.

Attracting
xed
points
for
Kuramoto-
Sivashinsky equation - a computer assisted
proof
Piotr Zgliczynski
Jagiellonian University, Poland
email: zgliczyn@im.uj.edu.pl
We present a computer assisted proof of an ex-
istence of several attracting xed points for the
Kuramoto-Sivashinsky equation
u
t
= (u
2
)
x
u
xx
u
xxxx
,
u(x, t) = u(x + 2, t),
u(x, t) = u(x, t),
where > 0.
The approach based on the con-
cept of self-consistent a priori bounds introduced in
[?]. The method is general and can be applied to
other dissipative PDEs, for example Navier-Stokes or
Ginzburg-Landau equations, not only to obtain xed
points, but also more complex dynamical objects like
periodic orbits and hopefully topological horseshoes.
The partial results concerning a rigorous steady-state
bifurcation diagram for Kuramoto-Sivashinsky equa-
tion will be also mentioned. References
[Z]
P. Zgliczynski, Attracting xed points for Kuramoto-
Sivashinsky equation - a computer assisted proof, sub-
mitted, http://www.im.uj.edu.pl/zgliczyn
[ZM] P. Zgliczynski and K.
Mischaikow, Rigorous Numer-
ics for Partial Dierential Equations: the Kuramoto-
Sivashinsky equation, Foundations of Computational
Mathematics, (2001) 1:255-288