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The Navier-Stokes Problem in the 21st Century

Pierre Gilles Lemarié-Rieusset
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Michael Berg
, on

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: — 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. 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.

Presentation of the Clay Millennium Prizes
Regularity of the three-dimensional fluid flows: a mathematical challenge for the 21st century
The Clay Millennium Prizes
The Clay Millennium Prize for the Navier–Stokes equations
Boundaries and the Navier–Stokes Clay Millennium Problem


The physical meaning of the Navier–Stokes equations
Frames of references
The convection theorem
Conservation of mass
Newton's second law
The equations of hydrodynamics
The Navier–Stokes equations
Boundary terms
Blow up


History of the equation
Mechanics in the Scientific Revolution era
Bernoulli's Hydrodymica
Laplacian physics
Navier, Cauchy, Poisson, Saint-Venant, and Stokes
Oseen, Leray, Hopf, and Ladyzhenskaya
Turbulence models


Classical solutions
The heat kernel
The Poisson equation
The Helmholtz decomposition
The Stokes equation
The Oseen tensor
Classical solutions for the Navier–Stokes problem
Small data and global solutions
Time asymptotics for global solutions
Steady solutions
Spatial asymptotics
Spatial asymptotics for the vorticity
Intermediate conclusion


A capacitary approach of the Navier–Stokes integral equations
The integral Navier–Stokes problem
Quadratic equations in Banach spaces
A capacitary approach of quadratic integral equations
Generalized Riesz potentials on spaces of homogeneous type
Dominating functions for the Navier–Stokes integral equations
A proof of Oseen's theorem through dominating functions
Functional spaces and multipliers


The differential and the integral Navier–Stokes equations
Uniform local estimates
Heat equation
Stokes equations
Oseen equations
Very weak solutions for the Navier–Stokes equations
Mild solutions for the Navier–Stokes equations
Suitable solutions for the Navier–Stokes equations


Mild solutions in Lebesgue or Sobolev spaces
Kato's mild solutions
Local solutions in the Hilbertian setting
Global solutions in the Hilbertian setting
Sobolev spaces
A commutator estimate
Lebesgue spaces
Maximal functions
Basic lemmas on real interpolation spaces
Uniqueness of L3 solutions


Mild solutions in Besov or Morrey spaces
Morrey spaces
Morrey spaces and maximal functions
Uniqueness of Morrey solutions
Besov spaces
Regular Besov spaces
Triebel–Lizorkin spaces
Fourier transform and Navier–Stokes equations


The space BMO-1 and the Koch and Tataru theorem
Koch and Tataru's theorem
A special subclass of BMO-1
Further results on ill-posedness
Large data for mild solutions
Stability of global solutions
Small data


Special examples of solutions
Symmetries for the Navier–Stokes equations
Two-and-a-half dimensional flows
Axisymmetrical solutions
Helical solutions
Brandolese's symmetrical solutions
Self-similar solutions
Stationary solutions
Landau's solutions of the Navier–Stokes equations
Time-periodic solutions
Beltrami flows


Blow up?
First criteria
Blow up for the cheap Navier–Stokes equation
Serrin's criterion
Some further generalizations of Serrin's criterion


Leray's weak solutions
The Rellich lemma
Leray's weak solutions
Weak-strong uniqueness: the Prodi–Serrin criterion
Weak-strong uniqueness and Morrey spaces on the product space R × R3
Almost strong solutions
Weak perturbations of mild solutions


Partial regularity results for weak solutions
Interior regularity
Serrin's theorem on interior regularity
O'Leary's theorem on interior regularity
Further results on parabolic Morrey spaces
Hausdorff measures
Singular times
The local energy inequality
The Caffarelli–Kohn–Nirenberg theorem on partial regularity
Proof of the Caffarelli–Kohn–Nirenberg criterion
Parabolic Hausdorff dimension of the set of singular points
On the role of the pressure in the Caffarelli, Kohn, and Nirenberg regularity theorem


A theory of uniformly locally L2 solutions
Uniformly locally square integrable solutions
Local inequalities for local Leray solutions
The Caffarelli, Kohn, and Nirenberg ε-regularity criterion
A weak-strong uniqueness result


The L3 theory of suitable solutions
Local Leray solutions with an initial value in L3
Critical elements for the blow up of the Cauchy problem in L3
Backward uniqueness for local Leray solutions
Seregin's theorem
Known results on the Cauchy problem for the Navier–Stokes equations in presence of a force
Local estimates for suitable solutions
Uniqueness for suitable solutions
A quantitative one-scale estimate for the Caffarelli–Kohn–Nirenberg regularity criterion
The topological structure of the set of suitable solutions
Escauriaza, Seregin, and Šverák's theorem


Self-similarity and the Leray–Schauder principle
The Leray–Schauder principle
Steady-state solutions
Statement of Jia and Šverák's theorem
The case of locally bounded initial data
The case of rough data
Non-existence of backward self-similar solutions


Global existence, uniqueness and convergence issues for approximated equations
Leray's mollification and the Leray-α model
The Navier–Stokes α -model
The Clark- α model
The simplified Bardina model
Reynolds tensor


Other approximations of the Navier–Stokes equations
Faedo–Galerkin approximations
Frequency cut-off
Ladyzhenskaya's model
Damped Navier–Stokes equations


Artificial compressibility
Temam's model
Vishik and Fursikov's model
Hyperbolic approximation


Energy inequalities
Critical spaces for mild solutions
Models for the (potential) blow up
The method of critical elements


Notations and glossary