LMS-EPSRC Short Course: Theoretical Fluid Dynamics      
Heriot-Watt University: 29th August -- 2nd Sept 2011     
Charlie Doering    Emmanuel Leveque    James Robinson      
John Gibbon    Darryl Holm    Sergei Kuksin
   Laurent Chevillard    Masoumeh Dashti     

Summary

The notion of Theoretical Fluid Dynamics will conjure up a lot of different and diverse topics in as many different and diverse scientific minds. However, there is an undeniable centralizing issue: do solutions of the Navier-Stokes equations develop a singularity in finite time? Therefore we have to address this. However contemporary theoretical fluid dynamics is also about much more. There are other models for fluid flow, for example, perhaps arguably juxtaposed to the Navier--Stokes equations, are models for turbulence. There are also the Euler equations for an ideal incompressible fluid, and isentropic fluid flow. But perhaps, most importantly, what really strikes scientists with its complexity and sophistication, is:
  • Vortex stretching: the underlying effect producing the flow intricacies that marvel us as simple observers of nature, experimentalists and mathematicians;
  • Mathematical theory: the cornucopia of theoretical techniques, languages and analysis that have been applied to, and sprouted from, tackling and enumerating turbulent fluid flow.
Nearly eighty years ago Leray suggested that Navier--Stokes singularities may signify the onset of turbulence in a real fluid flow. With this pertinence, the flavour and range of this short course is set.

Objectives and outcomes

There are three main themes; these are to provide the knowledge base underlying the:
  1. three-dimensional, incompressible Navier--Stokes equations global regularity problem; what do we know about this problem?
  2. models for three-dimensional fluid turbulence; again what do we know, how good are they, and in what sense are they converging towards the Navier--Stokes equations?
  3. developments in theory and prescription that provide a different insight into the character of the Navier--Stokes equations and turbulence.
Ultimately the description of this knowledge requires some background in functional analysis, geometric measure theory and basic probability. We aim to include instruction of these as an embedded integral part of the course. Participants will be expected to be familiar with classical partial differential equations.
The expected outcomes for students that attend this course are that they should:
  1. Gain knowledge about the Navier--Stokes equations, its derivation, its properties, the fine balance of vortex stretching and diffusion mechanisms it encapsulates and techniques of nonlinear analysis.
  2. Understand the context and consequences of the three-dimensional Navier--Stokes global regularity: what results have been proven, what do they characterize, what are their consequences?
  3. Understand models of turbulence and their prescription of real fluid flow: what are the nuances of the leading models, how do they relate to the Navier--Stokes equations?
  4. Relate results for associated models, in particular the Euler equations and isentropic flows, and their consequences.
  5. Meet and socialize with world class researchers in this area, with the view to perhaps working with them at some juncture in the future.
  6. Be stimulated to think about conducting research in theoretical fluid dynamics and in particular, the Navier--Stokes equations and turbulence.

Academic matters and course outline

The course will consist of three lecture courses given by four lecturers together with two special guest lectures. To open the whole week, on Monday there will be a guest lecture given by John Gibbon which will include a history of important developments in the field and provide some context to the week ahead. At the end of the week, the last lecture on the Friday, will be given by Sergei Kuksin, who will look towards the future and speculation on where recent developments might lead us. In between, during the week, the lecture courses will unfold as follows--also see the timetable below. (Note that these topics are subject to minor alteration.)
  1. Introductory fluid mechanics; 3 lectures given Simon Malham: Continuum hypothesis; conservation principles; transport theorem; derivation of the Navier--Stokes equations (and some non-Newtonian fluids); exact solutions (eg. axisymmetric); convection, stretching and diffusion; vortex dynamics (filaments and sheets); nonlinear analysis and local in time existence.
  2. Navier--Stokes equations; 3+3 lectures given by Charlie Doering and James Robinson: Weak and strong solutions; small energy/large viscosity result; Serrin's regularity criterion and related results; Navier-Stokes equations in turbulence modelling; simulations and experiments; partial regularity: Caffarelli-Kohn-Nirenberg and related results.
  3. Turbulence; 5 lectures given by Emmanuel Leveque and Laurent Chevillard: turbulence in fluids from a physical viewpoint: dynamics and scalings, characteristic scales, conjectured cascades, experimental results. Numerically tractable mathematical models of turbulence.
Some additional points are as follows. The organiser (SJAM) will give the introductory lectures as shown, intended both for those attendees who do not have a background in fluid dynamics as well as those who do. SJAM has a complete set of notes in book-form to provide/email in advance. Included will be some ``primer'' material the participants are expected to know as a prerequisite for the course. There will be a total of nine tutorial/example classes. These are intended to be extremely informal and interactive---this is usual practice of the organiser who will endeavour to foster this type optimal environment for learning. Two postdoctoral assistants for the Navier-Stokes and Turbulence courses will be on hand to help:

Laurent Chevillard (ENS, Lyon) and Masoumeh Dashti (Warwick)

Printed and/or photocopied notes will be provided. It is not expected that every lecturer will provide typed notes, though in reality most of them will do so automatically. If they have handwritten notes they use for their lectures that they think are suitable for photocopying and distribution, then we will do so. Otherwise one of the post-doctoral assistants (or PhD students if they volunteer) could act as a scribe, writing neat notes onto paper for photocopying, or writing direct onto a tablet for direct electronic storage and distribution.

Administrative matters and logistics

First we have support for this short course from the following organisations, the:
  • Centre for Analysis of Nonlinear Partial Differential Equations (CANPDEs) as part of the Maxwell Institute in Edinburgh. The themes of the short course very much align alongside those of the CANPDEs. Director Professor Sergei Kuksin will give the closing lecture.
  • School of Mathematical and Computer Sciences (MACS) at Heriot--Watt, the hosting school of course;
  • International Centre for Mathematical Sciences (ICMS) in Edinburgh, who have decades of experience organising and holding conferences, workshops and summer schools, have agreed to provide advice and know-how;
  • Edinburgh Conference Centre (ECC)} at Heriot--Watt, who will provide accommodation and subsistence logistics.
These organisations offer(ed) their assistance free of charge. Second the logistical details are as follows.
  • Registration/Arrival: this is more of a formal reception and orientation on arrival for the participants. There will be a desk manned at the main entrance/reception of Heriot--Watt University on the Sunday afternoon/early evening. We will greet the arriving participants, hand them some information packs and orient them as far as the location of the accommodation, dining facilities and lecture rooms are concerned.
  • Accommodation rooms: are en-suite, on campus, and have been booked. Disabled access rooms are provided.
  • Lecture room: there will be one main lecture/tutorial room (already booked) provided for free by MACS, this will be in the Colin Maclaurin building, room G.01 (capacity 50) which has disabled access.
  • Coffee/Tea room: this will be provided nearby (in the same school). We plan for these to be in the crush area of the Earl Mountbatten building (a 30 second journey by foot).
  • Social event: these will be two events on the Wednesday afternoon. One, a trip round Edinburgh in an open top bus on which a commentary is provided on the history and important events and places in Edinburgh. Another, a tour around Edinburgh Castle.
  • Banquet: will take place at Howie's restaurant on Waterloo Place at 7:30pm on Thursday September 1st.