EuroSummer School

Instructional Conference on
Mathematical Analysis of Hydrodynamics

Edinburgh, 18-29 June 2003

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Scientific Programme

The 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 Lists

Courses 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.

Timetable

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).

	0915- 1045	1045- 1115	1115- 1245	1245- 1415	1415- 1515	1515- 1545	1545- 1645
Thu 19	Titi (1)	Coffee	Titi (2)	Lunch	Plotnikov	Tea
Fri 20	Toland (1)	Coffee	Robinson (1)	Lunch	Kupiainen	Tea	Gallavotti
Sat 21	Toland (2)	Coffee	Robinson (2)	Lunch	Free Afternoon
Mon 23	Brenier (1)	Coffee	Constantin (1)	Lunch	Doering	Tea	Shnirelman
Tue 24	Wayne (1)	Coffee	Brenier (2)	Lunch	Tutorials	Tea	Grenier
Wed 25	Constantin (2)	Coffee	Wayne (2)	Lunch	Vladamirov	Tea	J. Wu
Thu 26	Cordoba (1)	Coffee	Kuksin (1)	Lunch	Tutorial Afternoon
Fri 27	Cordoba (2)	Coffee	Kuksin (2)	Lunch	Babine	Tea	S. Wu
	0915- 1015	1015- 1115	1115- 1145	1145- 1245	1245- 1415
Sat 28		Fursikov	Coffee	Craig	Lunch	Free Afternoon

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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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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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