EuroSummer School
Instructional Conference on Mathematical
Analysis of Hydrodynamics
Edinburgh, 18-29 June 2003 Scientific Programme | Participants | Conference Arrangements | Application Forms |
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Scientific ProgrammeThe timetable and abstracts
are available further down this page.
A full list of participants can be
found by following this link.
Timetable, Abstracts and Reading ListsCourses
consist of two morning lectures of 90 minutes each by a given speaker (numbered
(1) and (2)). Tutorials will take place on two afternoons and research lectures
will be given during the remaining afternoons and the morning of the final day.
Please note that the final morning's talks do not follow the same pattern as
the other mornings.
Clicking on a name in the timetable will take you
directly to the title, abstract and suggested reading for that speaker.
Alternatively you can scroll down this page to browse through the abstracts.
The lectures
will take place in Lecture Theatre B on level 3 of the James Clerk Maxwell
Building (within the Edinburgh University Kings Buildings complex).
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Courses (Two lectures of 90 minutes) Yann
Brenier Euler equations - The Euler
equations viewed as a variational and geometric equation - 2D Euler
equations in a very thin domain and their 'hydrostatic' limit - Euler
equations and electrodynamics: links with plasma physics (Vlasov-Poisson) and
nonlinear electromagnetism (Born-Infeld) Knowledge assumed Differential
calculus and basic analysis, some familiarity with PDEs of continuum mechanics
and mathematical physics
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timetable
Peter Constantin
Navier Stokes equations We will describe a formalism for the
Euler and Navier-Stokes equations based on diffusive Lagrangian maps.
References: P Constantin, An Eulerian-Lagrangian approach for
incompressible fluids: local theory. JAMS 14 (2001), 263-78.
P Constantin, An Eulerian-Lagrangian approach to the Navier-Stokes
equations. Commun. Math. Phys. 216 (2001), 663-86
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Diego Cordoba
Quasi-Geostrophic Equations We will show that the 2D
Quasi-geostrophic equation has the same structure than 3D Euler equations. We
will cover front formation and global regularity with dissipation.
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Sergei
Kuksin/Armen Shirikyan Statistical
hydrodynamics
- FIRST LECTURE:
- 1. Dynamical system generated by 2D Navier-Stokes equations
with discrete forcing.
- 2. Homogeneous Navier-Stokes system: dissipation and
regularization.
- 3. Inhomogeneous Navier-Stokes system: a priori estimates,
stabilisation, squeezing.
- 4. Markov chain associated with a discrete time random
dynamical system.
- 5. Stationary measures and the Bogolyubov-Krylov argument.
- SECOND LECTURE:
- 6. Uniqueness on the stationary measure (without proof)
- 7. Consequences of the uniqueness: space-homogeneity, strong
law of large numbers, CLT.
- 8. The Eulerian limit
- BOOKS TO READ:
- [1] A. Babin and M. Vishik, Attractors of Evolutionary
Equations, "North-Holland", 1992. - For 2), 3)
- [2] G. Da Prato and J. Zabczyk, Ergodicity for Infinite
Dimensional Systems, "Cambridge University Press", 1996. - For 4), 5)
- PAPERS TO READ:
- For 1)-5) - [3] S. B. Kuksin and A. Shirikyan, Comm. Math.
Phys. 213, 291-330 (2000)
- For 6) - [4] S. B. Kuksin and A. Shirikyan, Comm. Math. Phys.
221, 351-366 (2001)
- For 6)-7) - [5] S. B. Kuksin, Rev. Math. Phys. 14, 585-600
(2002) The papers [3-5] can be down-loaded from
http://www.ma.hw.ac.uk/~kuksin/
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James Robinson Attractors and
finite dimensional behaviour in the Navier-Stokes equations I will
discuss various approaches to finite-dimensional behaviour in both the 2d and,
under the assumption of regularity, the 3d Navier-Stokes equations. I intend to
cover dimension estimates for the attractor (and their physical
interpretation), determining modes and nodes, and perhaps a breif discussion of
approximate inertial manifolds. The lectures will rely heavily on the material
presented in the two lectures given by Titi.
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Edriss Titi Basics of the mathematical theory of the
Navier-Stokes equations. In these lectures I will introduce the basics
of the mathematical of the Navier-Stokes equations. Topics to be covered:
1. Steady state solutions to the Navier-Stokes equations and their
regularity. 2. Time dependent weak solutions. 3. Global regularity of
strong solutions for the two dimensional case. 4. Short time existence of
strong solutions in the three dimensional case. 5. Other regularity issues
(if time allows). Textbooks: P. Constantin and C. Foias,
Navier-Stokes Equations, University of Chicago Press, 1988. C.
Doering and J. Gibbon, Applied Analysis of the Navier-Stokes Equations,
Cambridge University Press. R. Temam, Navier-Stokes Equations: Theory
and Numerical Analysis, North-Holand. New print published by the AMS 2001.
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John
Toland Modern complex methods in free boundary
problems These lectures will consider the Eulerian formulation of the
free-boundary problem for time dependent inviscid, irrotational 2-dimensional
waves. The emphasis will be on the use of complex variable methods in a
formulation as a Hamiltonian system due to Zakharov.
Questions of the
relevance of these equations to the hydrodynamics will be examined in the
context of complex Hardy spaces and some answers will be offered in the simple
case of complex travelling waves.
Some relevant preprints are available
at http://www.maths.bath.ac.uk/~jft
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Gene
Wayne Stability of vortices in fluid flow
I will describe recent work with Th. Gallay of the University of Grenoble
on how one can use ideas from dynamical systems theory and kinetic theory to
understand the long-time asymptotic behavior of solutions of the Navier-Stokes
equation. This leads to a quite complete understanding of the dynamics of
two-dimensional fluids, showing that any solution whose initial vorticity
distribution is somewhat localized will converge to an Oseen vortex solution.
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Lectures on current research topics Anatoli
Babine Renormalisation method in the
mathematical theory of rotating fluids Walter Craig On surface water waves Charlie
Doering Bounds on energy dissipation in
body-forced turbulence - link to
abstract A Fursikov Stabilization
of solutions to the Navier-Stokes equations - link
to abstract (dvi file) Giovanni Gallavotti Equivalence conjecture between fluid
equations and numerical experiments to test them John
Gibbon Intermittency in the 3D Navier-Stokes
equations Emmanuel Grenier Some
boundary layers in rotating fluids Antii, Kupiainen Lagrangean flow in self-similar
velocity ensembles - link to
abstract Pavel Plotnikov Free
boundary problems for the Euler equations A. Shnirelman Inverse cascade solutions of 2D
Euler equations Vladimir Vladimorov
Virial equality in fluid dynamics J Wu Dissipation and global existence -
link to abstract S Wu
Recent progress in the mathematical analysis of vortex sheets
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Abstracts for lectures on current research topics
DOERING The bulk rate of
energy dissipation is the power required to maintain a flow state, steady or
unsteady, laminar or turbulent. We consider the idealized situations of (1)
body-forced flows without boundaries, and (2) flow in a channel with
slippery walls driven by an imposed shearing force.
In the first case
we present rigorous estimates on solutions of the Navier-Stokes equations that
are in qualitative agreement with the fundamental scaling theories in the
laminar and turbulent limits; that is joint work with Ciprian Foias. In the
second case, the Navier-Stokes equations are used to derive a mini-max problem
for an upper limit to the long-time averaged bulk power consumption. This
mini-max problem can be solved exactly in the high Reynolds number limit.
Quantitative results are compared to the results of direct numerical solutions
of the Navier-Stokes equations for a particular "shape" of the driving force.
Curiously, a component of the high Reynolds number solution of the variational
problem is reminiscent of statistical aspects of the turbulent flow; this is
joint work with Bruno Eckhardt and Joerg Schumacher.
C. R. Doering
& C. Foias, Energy dissipation in body-forced turbulence, Journal of
Fluid Mechanics 467, 289-306 (2002).
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KUPIAINEN Velocity fields occurring in
fully developed turbulence are spatially rough and give rise to the phenomenon
of nonuniqueness of trajectories in the motion of fluid particles even in the
absence of noise. We review progress in understanding of this phenomenon in the
context of synthetic Gaussian velocity ensembles
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J. WU The talk is divided into two parts. The
first part reviews existence and regularity results for several partial
differential equations arising in fluid mechanics and attention is focused on
how regularity is modified by dissipation. The second part is devoted to the 2D
dissipative quasi-geostrophic equation. Old results are summarized and new
developments are presented.
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