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The partial regularity theory of Caffarelli, Kohn, and Nirenberg and its sharpness. (English) Zbl 1441.35004

Advances in Mathematical Fluid Mechanics. Lecture Notes in Mathematical Fluid Mechanics. Cham: Birkhäuser (ISBN 978-3-030-26660-8/pbk; 978-3-030-26661-5/ebook). vi, 138 p. (2019).
In fluid dynamics the motion of water and in general of incompressible viscous fluids is modeled by a set of partial differential equations called the Navier-Stokes equations. Close to 200 years now, this model managed to strike a good balance in capturing the attention of engineers and that of mathematicians. The importance of the Navier-Stokes equations in fluid mechanics as well as the developments generated by their study in mathematical analysis cannot be overstated. On the “pure mathematics” side, it suffices to mention that proving or disproving the existence of global smooth solutions for the Navier-Stokes equations is one of the seven most important open problems in mathematics on the Clay Mathematics Institute, Millennium Problems list C. L. Fefferman [in: The Millennium Prize problems. Providence, RI: American Mathematical Society (AMS); Cambridge, MA: Clay Mathematics Institute. 57–67 (2006; Zbl 1194.35002)].
In the setting of three spatial dimensions, the equations arise by considerations in theoretical continuum mechanics from Newton’s second law \(F=ma\). They are written \[ \partial_t u -\nu \Delta u + (u\cdot \nabla )u +\nabla p =0,\text{ and }\mathrm{div}\, u =0,\tag{1} \] where the divergence equation expresses the incompressibility condition. This is a system of four equations on \(\mathbb{R}^3\times [0,\infty)\) for the velocity field \(u\) of the fluid, together with the scalar pressure \(p\). The constant \(\nu >0\) is the viscosity. The initial condition for this system is \[ u(x,0)=u_0(x),\tag{2} \] with \(u_0\) being a given divergence free vector field. It is known that under the requirement that the initial condition \(u_0\) is sufficiently small (in certain norms), then a smooth solution exists for all times. When this requirement is dropped, the existence of a smooth solution can be proved only for a short time.
J. Leray [Acta Math. 63, 193–248 (1934; JFM 60.0726.05)], in the entire space \(\mathbb{R}^3\), and E. Hopf [Math. Nachr. 4, 213–231 (1951; Zbl 0042.10604)], in a smooth bounded domain in \(\mathbb{R}^3\), proved the all time existence of {\em weak solutions}. For certain {\em suitable weak solutions} of the Leray-Hopf type, Scheffer proved partial regularity via innovative use of ideas from geometric measure theory and calculus of variations ([V. Scheffer, Lect. Notes Math. 565, 174–183 (1976; Zbl 0394.76029); Commun. Math. Phys. 55, 97–112 (1977; Zbl 0357.35071); Commun. Math. Phys. 73, 1–42 (1980; Zbl 0451.35048)]). Scheffer’s partial regularity theorem was improved by L. Caffarelli et al. [Commun. Pure Appl. Math. 35, 771–831 (1982; Zbl 0509.35067)] who proved that the 1-dimensional (parabolic) Hausdorff measure of the singular set of the solution is zero. The proof of L. Caffarelli et al. [Commun. Pure Appl. Math. 35, 771–831 (1982; Zbl 0509.35067)] was later simplified by F. Lin [Commun. Pure Appl. Math. 51, No. 3, 241–257 (1998; Zbl 0958.35102)]. As summarized by C. L. Fefferman [in: The Millennium Prize problems. Providence, RI: American Mathematical Society (AMS); Cambridge, MA: Clay Mathematics Institute. 57–67 (2006; Zbl 1194.35002)], this result on the partial regularity of suitable weak solutions was still the state of the art in the year 2000 when the Millennium Problems were announced, and as he writes “It appears to be very hard to go further.”
The story of the paper of L. Caffarelli et al. [Commun. Pure Appl. Math. 35, 771–831 (1982; Zbl 0509.35067)], is told beautifully by Robert Kohn in an insightful article on some of the scientific accomplishments of Louis Nirenberg, in the Springer volume [R. V. Kohn, “A few of Louis Nirenberg’s many contributions to the theory of partial differential equations”, in: The Abel Prize 2013–2017. Cham: Springer. 501–528 (2019)].
It is in this connection, with the results of Scheffer and of Caffarelli, Kohn and Nirenberg, that the subject of the book under review evolves. In four chapters, the author recasts in uniform notations and often with simpler proofs the achievements of Scheffer and of Caffarelli, Kohn and Nirenberg. Some of the closely related results that appeared in the literature since [L. Caffarelli et al., Commun. Pure Appl. Math. 35, 771–831 (1982; Zbl 0509.35067)] are also discussed from this perspective.
Chapter 1, provides a brief history of the work on the regularity of solutions of the incompressible, homogeneous, Navier-Stokes equations in three space dimensions. Here are established the concepts of {\em strong solution}, {\em Leray-Hopf weak solution}, and {\em suitable weak solution}, for the Navier-Stokes equation, and also the concept of {\em weak solution} for the Navier-Stokes {\em inequality} (NSI). In this chapter are discussed the theorems that are proved and is concluded with an overview of the structure of the book.
Chapter 2 contains “a simple proof” of the Caffarelli, Kohn and Nirenberg result, which is stated for weak solutions of the NSI. As corollaries of this theorem follow upper estimates on the Hausdorff and on the box-counting dimension of the singular set of such solutions.
Then, the author proceeds to the construction of weak solutions of NSI which are singular on sets as large as the theorem of Chapter 2 allows, confirming thus the sharpness of the partial regularity results.
In Chapter 3 is shown how to construct solutions with a single point singularity. The existence of these singular weak solutions of NSI is based on the earlier counterexample of Scheffer. From this result also follows the existence of weak solutions whose maximum at time \(t=1\) surpasses the maximum of \(u_0\) by any desired factor.
In Chapter 4 is presented, also based on Scheffer’s counterexamples, the construction of solutions of NSI that blow up at a finite time on a Cantor set with Hausdorff dimension as close to \(1\) as desired. A number of auxiliary gadgets, such as “structure” and “geometric arrangement” are introduced and analyzed in Chapters 3 and 4.
This is a well written, and this makes it easy to read, mathematical text. Part of the reason for this is the fact that it focuses on only one aspect, albeit at the heart of the matter, of one of the major open problems in mathematics. Essentially self-contained, the book can be used as a straightforward introduction to the topic of regularity of solutions of the Navier-Stokes equations.

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35B65 Smoothness and regularity of solutions to PDEs
35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
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