The Navier-Stokes Problem in the 21st Century

Navier-Stokes, or, more properly, the Navier-Stokes equation, is one of those things in our racket that evokes immediate reactions along the lines of, “Yeah, that’s going hunting for really big game,” or something isomorphic thereto. The Clay Institute placed one of its $\(10^6\)-sized bounties on it, and you could probably argue that now that the Poincaré Conjecture has bitten the dust, the only problem that eclipses Navier-Stokes in the mathematical imagination is the Riemann Hypothesis — and, yes, of course, the Clay folks also offer a million-dollar reward for its capture, dead or alive, so to speak, i.e. a proof or a Gegenbeispiel: smart money seems to bet it’s true. In all fairness,

it’s probably the case that some of our merry band would hold that the P/NP business is even more relevant than Navier-Stokes to the modern world and should therefore edge it out, if only by a nose, and it should also eclipse RH for similar reasons. I wouldn’t give them the time of day (on my stubbornly analog watch): it pains me, really, that I am working on a computer — OK, not really, but you get my drift: the inner life of the primes is just too evocative and mysterious a business to pass up — it is truly the sexiest problem in all of mathematics. Still, Navier-Stokes is undeniably sexy: solving this Clay problem — see below for what is called for — is obviously not just very important for reasons involving DEs, hard (and maybe soft) analysis, hydrodynamics (which is to say, the real world in no uncertain terms), global analysis, and so on, and so on, but it’s clearly titanically difficult.

So, first of all, what do the Navier-Stokes equations look like? Well, see p. xvi of the book under review for the real skinny, but let’s just say for now that, quoting Lemarié-Rieusset (on p. xii) quoting Constantin quoting (or paraphrasing) Winston Churchill: “The Navier-Stokes equations are a viscous regularization of the Euler equations, which are still an enigma. Turbulence [the phenomenon with which Navier-Stokes is concerned] is a riddle wrapped in a mystery in an enigma.” No, really, what does Navier-Stokes say? Well, on p. 27 (and also on p. xvi), we encounter the following, slightly paraphrased here: if \(u=u(t,x)\) is a velocity field for particles in a “fluid parcel” in the sense of Euler, with the independent variables obviously standing for time and space coordinates, then with \(p\) the reduced pressure, \(v\) the kinematic viscosity and \(f\) the kinematic pressure, \[\partial_t u + (u\cdot\nabla)u = - \nabla p + v\Delta u + f\] and \[\mathrm{div}(u)=0.\] Here we have it, then: a profound amount of physical information in one set of equations, the heavy lifting being done by a PDE the full (global) analysis of which has resisted attacks for a long time. I am reminded of an aphorism which manifestly applies in spades to all the Clay problems: “a problem worthy of attack proves its worth by fighting back.”

And so the book under review, at over 700 pages, is an account of the problem, its history, its various avatars and incarnations, the plethora of attacks , rebuffs, and, e.g., mild solutions (see below), and much else besides: it is an encyclopædic account of almost all things Navier-Stokes, and then some. These 700 pages are split up into nineteen chapters necessarily covering a lot of ground in all the categories mentioned above. After introductory material on the Clay Institute Millennium Prize angle, Lemarié-Rieusset hits physics (hydrodynamics, vorticity, turbulence), a chunk of history (Euler is there, as are Navier, Cauchy, Poisson, and Stokes (and others), and even Leray, Hopf [Eberhard, not Heinz], and Olga Ladyshenskaya. (In connection with the latter, read this: http://www.ams.org/notices/200411/fea-olga.pdf — there is some good stuff specifically about Navier-Stokes in this article, to boot).

Subsequently we get to a goodly amount of hard analysis: the rest of this big book deals with different approaches, ranging from a treatment of the classical solutions to the associated Cauchy IVP for a restricted form of Navier-Stokes to … well, here is what the author himself says at the beginning of Chapter 7:

The search for solutions to the Navier-Stokes equations has known three eras. The first one was based on explicit formulas for hydrodynamic potentials, given by Lorentz and Oseen, and … used by Leray [yes, the same guy who discovered sheaves!] in his seminal work introducing weak solutions. Then, in the fifties, a second approach was developed by Hopf and Ladashenskaya … who turned the PDEs into the study of an ODE in a finite-dimensional space … The third period began in the mid-sixties, when the theory of accretive operators was developed, leading to the theory of semi-groups of operators … solutions obtained [in this context] were called mild solutions by Browder and Kato.

Subsequently Lemarié-Rieusset goes at such things as BMO spaces (actually we’re playing with BMO^-1 — see Chapter 10) and then goes on to the 21st century with his 11th chapter, titled “Blow Up?” (with apologies to Antonioni, I guess). And here we get the specifics regarding the Clay Institute’s carrot: does the IVP for Navier-Stokes, as above, taken to live in \(\mathbb{R}^3\), admit a mild solution which has global existence (i.e. with unlimited “maximal time”) when \(f=0\)? (And note that it’s now p. 300 and we have eight dense chapters to go).

Well, it’s clear as vodka (cf. https://www.youtube.com/watch?v=tHxZBwZsbss and go to \(\pi+e\) (or 3:18, actually)) that this big book is enthusiastically serious about its theme, Navier-Stokes, and is correspondingly impressive: even though the author tells us on pp. xiii about five things the book is not about (hydraulics, turbulence, general fluids, fluids in bounded domains, computational fluid dynamics), he goes on to delineate a great number of deep things that it is about: I’ve already given an indication of some of these themes above. It’s all very, very impressive, and modulo the quintet of omitted themes just mentioned, it’s fair to say that the book is encyclopædic, or near enough to fool anyone who’s not a deep insider. 

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.