#nonlinear functional analysis

[Navier-Stokes Equations and Nonlinear Functional Analysis. Roger Temam. 2nd Edition. 1 January 1987. CBMS-NSF Regional Conference Series in Applied Mathematics, Series Number 66.  Publisher; Society for Industrial and Applied Mathematics. Paperback: 155 pages. ISBN-10: 0898713404. ISBN-13 9780898713404. Dimensions: 17.15 x 1.27 x 24.77 cm. Thanks to Amazon for technical publication data. First Edition 1983]

A couple of years ago my wife and I watched an utterly charming film called Gifted, starring Chris Evans and an adorable girl named McKenna Grace.  The story revolved around the little girl, who was a mathematics prodigy being raised by her uncle (Evans) after her mother committed suicide some time prior.  The uncle wants his niece to have the benefit of a normal childhood despite her extraordinary gift, whereas his mother (her grandmother), herself a mathematician of some renown, dearly wants to exploit the little girl’s talent, perhaps as a way of making up for the lack of a connection with the daughter she lost.

At the center of this drama is a putative proof of one of the famous Millennium Problems put forth and sponsored by The Clay Mathematics Institute. These seven problems are regarded as the most difficult ‘important’ math problems in the world.  There is a cash prize of $1,000,000 to anyone who can either prove one of the assertions or provide a convincing counterexample disproving the statement.  To date, only one of them has been solved, the Poincare Conjecture (now the Poincare Theorem).

The fourth stated assertion involves the Navier-Stokes (N-S) equations, which are a set of partial differential equations that describe the evolution of fluid flow in 2 and 3 dimensions.  Fluid here means either a gas or a liquid, the latter generally being considered incompressible.  There is a curious issue with the N-S equations in that it should be possible to state very precise yet general circumstances under which the equations admit a solution that is unique.  Such a statement addresses existence and uniqueness of solutions based on initial fluid flow conditions and the shape of the domain in which the fluid flows.  However, to date, such existence and uniqueness of solutions has only been definitively shown for 2 dimensions.  There is something about going up to 3 dimensions that gives a bit too much wiggle room to easily show that, given arbitrary initial conditions and domain description, one can either guarantee (perhaps by direct solving) a unique solution or show that one cannot exist.

In the aforementioned film, the little girl’s dead mother apparently had written a proof of the N-S equations for 3 dimensions, and thus could have laid claim to the Clay Institute prize for solving one of the Millennium problems.  The trouble is that no one seems to know where she left the proof.  The film moves forward with this premise and the dramatic tensions between the desires of the girl’s uncle and the girl’s grandmother, each with their own view on how best to nurture (or exploit) the little girl’s phenomenal potential.

Back in the real world, this N-S problem continues to be an active area of research well more than a century after it was first posed.  Prof Temam is a leading researcher in this area of applied and computational fluid dynamics.  The monograph reviewed here is an excellent compendium of the many fundamental issues facing the scientist who wishes to work with the N-S equations in a practical setting.

Temam first essays the questions related to existence, uniqueness and regularity (solutions can’t have physically impossible whirlpools, for example) of solutions to the N-S equations.  He points out the paradox between the two- and three-dimensional versions of these equations.  He describes the setting and general solution framework for the equations, drawing on both classical and newer a priori results (these latter are similar to the ansatz approach to problems in physics).  He then surveys some issues in functional analysis related to the analyticity of the time-evolved equations as well as more over-arching representational elements to the problem.

The second half of the monograph deals with somewhat more general functional analysis questions, most of which are directly applicable to the Navier-Stokes equations but are by no means limited to them.  A brief third section deals with computational aspects of N-S equations, including such minutiae as grid size, convergence of short- and long-term solutions (and the marrying up of same), and a brief appendix that describes inertial manifolds (a functional space description in which the solutions to the N-S equations should naturally reside).

This monograph is very well-written in my opinion, and I say this as someone who has not done very much fluid dynamics work as a mathematician – I learned a great deal from it.  As one might expect, the level of mathematics is graduate/post-doctoral level, and requires a strong background in measure theory, functional analysis, and topology, as well as a healthy acquaintanceship with numerical analysis, especially techniques designed for classes of partial differential equations.

[Cover © 1987 Society for Industrial and Applied Mathematics; portrait of  Roger Temam © Indiana University Bloomington, USA]

Kevin Gillette

Words Across Time

2 February 2022