The Partial Regularity Theory of Caffarelli, Kohn, and Nirenberg and its Sharpness

By Wojciech S. Ożański

This monograph focuses on the partial regularity theorem, as developed by Caffarelli, Kohn, and Nirenberg (CKN), and offers a proof of the upper bound on the Hausdorff dimension of the singular set of weak solutions of the Navier-Stokes inequality, while also providing a clear and insightful presentation of Scheffer’s constructions showing their bound cannot be improved. A short, complete, and self-contained proof of CKN is presented in the second chapter, allowing the remainder of the book to be fully dedicated to a topic of central importance: the sharpness result of Scheffer. Chapters three and four contain a highly readable proof of this result, featuring new improvements as well. Researchers in mathematical fluid mechanics, as well as those working in partial differential equations more generally, will find this monograph invaluable.

• Language: English
• Publisher: Birkhäuser
• Release date: Sep 16, 2019
• ISBN: 9783030266615

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© Springer Nature Switzerland AG 2019

W. S. OżańskiThe Partial Regularity Theory of Caffarelli, Kohn, and Nirenberg and its SharpnessAdvances in Mathematical Fluid Mechanicshttps://doi.org/10.1007/978-3-030-26661-5_1

1. Introduction

Wojciech S. Ożański¹  

(1)

Department of Mathematics, University of Southern California, Los Angeles, CA, USA

Wojciech S. Ożański

Email: [e-mail]

Abstract

We present the context of the partial regularity theory of the three-dimensional incompressible Navier-Stokes equations due to Caffarelli, Kohn and Nirenberg.

We present the context of the partial regularity theory of the three-dimensional incompressible Navier-Stokes equations due to Caffarelli, Kohn and Nirenberg. We discuss the concept of weak solutions of the Navier-Stokes inequality and introduce the constructions of such solutions (due to Scheffer) which show the sharpness of the Caffarelli–Kohn–Nirenberg theorem.

The Navier–Stokes equations,

$$\begin{aligned} \begin{aligned} \partial _t u -\nu \Delta u + (u\cdot \nabla )u +\nabla p&=0,\\ \mathrm {div}\, u&=0, \end{aligned} \end{aligned}$$

(1.1)

where u denotes the velocity of a fluid, p the scalar pressure and $$\nu >0$$ the viscosity, comprise a fundamental model for viscous, incompressible flows. The equations are usually supplemented with an initial condition $$u(0)=u_0$$ , where a divergence-free vector field $$u_0$$ is given.

The fundamental mathematical theory of the Navier–Stokes equations goes back to the pioneering work of Leray (1934) (see Ożański & Pooley, 2018 for a comprehensive modern review of this paper), who used an elegant Picard iteration scheme to prove existence and uniqueness of local-in-time strong solutions on the whole space  $$\mathbb {R}^3$$ .

Definition 1.1

(Strong solution of the NSE) We say that u is a strong solution of the Navier–Stokes equations (1.1) on

$$\Omega \times (0,T)$$

with the initial condition $$u_0\in V$$ if

(i)

(regularity)

$$u\in L^\infty ((0,T);V) \cap L^2 ((0,T);H^2 (\Omega ) )$$

,

(ii)

(the equation) u satisfies

$$ \int _0^T \int _\Omega \left( - u\cdot \partial _t \phi + \nu \nabla u :\nabla \phi + (u \otimes u ) :\nabla \phi \right) = \int _\Omega u_0 \cdot \phi (0) $$

for all divergence-free

$$\phi \in C_0^\infty (\Omega \times [0,T);\mathbb {R}^3 )$$

Here

$$\begin{aligned} V := \{ v\in H^1 (\mathbb {R}^3 ; \mathbb {R}^3 ) :\mathrm {div}\, v=0\} \end{aligned}$$

if $$\Omega = \mathbb {R}^3$$ , where $$H^1$$ denotes the Sobolev space of functions $$v\in L^2$$ such that $$\nabla v \in L^2$$ . In the case $$\Omega \ne \mathbb {R}^3$$ we set V to be the closed subspace of

$$\{ v\in H^1 (\Omega ; \mathbb {R}^3 ) : \mathrm {div}\, v=0\}$$

which takes into account appropriate boundary conditions. For example, if $$\Omega \subset \mathbb {R}^3$$ is a bounded, smooth domain then

$$V:= \{ v\in H^1 (\Omega ; \mathbb {R}^3 ) :\mathrm {div}\, v=0,\, \left. v \right| _{\partial \Omega } =0\}$$

, and if $$\Omega = \mathbb {T}^3$$ (the three-dimensional flat torus) then

$$V:= \{ v\in H^1 (\mathbb {T}^3 ; \mathbb {R}^3 ) :\mathrm {div}\, v=0,\, \int _{\mathbb {T}^3} v =0\}$$

.

We note that the above definition does not include the pressure function p , which can be recovered from the equation

$$-\Delta p = \partial _{ij} (u_i u_j)$$

(which in turn can be derived by calculating the divergence of (1.1); see Chap. 5 of Robinson, Rodrigo & Sadowski, 2016 for details). It can be shown in the case $$\Omega = \mathbb {R}^3$$ that p is given uniquely by

$$\begin{aligned} p = \sum _{i, j=1}^3 \partial _{ij} \Psi *(u_iu_j) \end{aligned}$$

(1.2)

at every time (see (6.47) in Ożański & Pooley, 2018 for details), where

$$\Psi (x) := (4\pi |x|)^{-1}$$

is the fundamental solution of the Laplace equation.

In addition to the notion of strong solutions, one often considers the so-called Leray–Hopf weak solutions.

Definition 1.2

(Leray–Hopf weak solutions of the NSE) A vector field

$$u\in L^2 ((0,T);V )$$

is a Leray–Hopf weak solution  of the Navier–Stokes equations (1.1) on

$$\Omega \times (0,T)$$

with the divergence-free initial condition

$$u_0\in L^2(\Omega )$$

if it satisfies condition (ii) of Definition 1.1 and the energy inequality

$$\begin{aligned} \Vert u(t) \Vert ^2_{L^2} + 2\nu \int _s^t \Vert \nabla u (\tau )\Vert ^2_{L^2} \mathrm {d}\tau \le \Vert u(s) \Vert ^2_{L^2} \end{aligned}$$

(1.3)

for almost every $$s\ge 0$$ and every $$t>s$$ .

Solutions defined above are named after Leray (1934) and Hopf (1951), who were the first to construct such solutions that are global in time in the case of the whole space $$\mathbb {R}^3$$ (Leray)  and a bounded, smooth domain $$\Omega \subset \mathbb {R}^3$$ (Hopf). They can be thought of as weak continuations of a strong solution, due to a certain weak-strong uniqueness result, see, for example, Theorem 6.10 in Robinson et al. (2016) or Lemma 6.39 in Ożański and Pooley (2018). We refer the reader to Robinson et al. (2016) for a more comprehensive introduction to the theory of existence and uniqueness of solutions as well as other topic related to the Navier–Stokes equations (1.1).

Although the fundamental question of global-in-time existence and uniqueness of strong solutions remains unresolved (as does the question of uniqueness of Leray-Hopf weak solutions; however, see Buckmaster and Vicol (2019) for nonuniqueness of (non-Leray–Hopf) weak solutions) , many significant results contributed to the theory of the Navier–Stokes equations during the second half of the twentieth century.

The Caffarelli–Kohn–Nirenberg Theorem

One such contribution is the partial regularity theory introduced by Scheffer(1976a, b, 1977, 1978, 1980) and subsequently developed by Caffarelli, Kohn and Nirenberg (1982); see also Lin (1998), Ladyzhenskaya and Seregin (1999), Vasseur (2007) and Kukavica (2009b) for alternative approaches.

This theory is concerned with local behaviour of weak solutions, which gives rise to the notion of a suitable weak solution.

Definition 1.3

(Suitable weak solution of the NSE) A pair (u, p) is a suitable weak solution  of the Navier–Stokes equations with viscosity  $$\nu >0$$ on an open set

$$U \times (a, b)$$

if

(i)

(regularity of u and p)

$$u\in L^\infty ((a, b);L^2(U ))$$

,

$$\nabla u \in L^2 (U \times (a, b ) )$$

, u(t) is divergence-free for almost every $$t\in (a, b )$$ , and

$$p\in L^{3/2}_{loc} (U \times (a, b ) )$$

,

(ii)

(relation between u and p) the equation

$$-\Delta p = \sum _{i, j=1}^3 \partial _i\partial _j (u_i u_j) $$

holds in the sense of distributions in U for almost every $$t\in (a, b)$$ ,

(iii)

(the local energy inequality) the inequality

$$\begin{aligned} \begin{aligned} \int _{U} |u(t) |^2 \phi (t) + 2 \nu&\int _a^t \int _{U} | \nabla u |^2\phi \\&\le \int _a^t \int _{U } \left( |u|^2 (\partial _t \phi + \nu \Delta \phi ) +(|u|^2+2p)(u\cdot \nabla )\phi \right) \end{aligned}\end{aligned}$$

(1.4)

is valid for every

$$\phi \in C_0^\infty (U \times (a, b ); [0,\infty ))$$

and $$t\in (a, b )$$ .

(iv)

(the equation) the Navier–Stokes equation (1.1) holds in the sense of distributions on

$$U \times (a, b )$$

, that is

$$\begin{aligned} \int _a^b \int _U \left( u \cdot ( \partial _t \phi + \nu \Delta \phi )- \phi \cdot ((u\cdot \nabla )u) + p\,\mathrm {div}\,\phi \right) =0 \end{aligned}$$

(1.5)

for all

$$\phi \in C_0^\infty (U \times (a, b );\mathbb {R}^3)$$

.

Note that, in contrast to (ii) in Definition 1.1, (1.5) must hold for all $$\phi $$ (i.e. not only for divergence-free $$\phi $$ ), and in particular the pressure term in (1.5) does not vanish. Note also that the regularity assumptions on u (i.e. boundedness in time of the $$L^2$$ norm and the space-time $$L^2$$ integrability of the gradient) is the same as the regularity of Leray-Hopf weak solutions that can be deduced from the energy inequality (1.3). Using the regularity of u from (i) and the Lebesgue interpolation one obtains that

$$\begin{aligned} u\in L^{10/3} (U \times (a, b )), \end{aligned}$$

(1.6)

see Lemma 3.5 in Robinson et al. (2016), for example. Thus (i) implies that all terms on the right-hand side of the local energy inequality (1.4) are well defined. Moreover, in the case when $$U=\mathbb {R}^3$$ and p is given by (1.2) (ii) is satisfied and

$$\begin{aligned} p\in L^{5/3} (\mathbb {R}^3 \times (a, b)), \end{aligned}$$

(1.7)

which can be deduced from (1.6) using the Calderón-Zygmund inequality (see (2.​39) below). In other words in this case the regularity of p required in (i) (i.e.

$$p\in L^{3/2}_{loc} (U \times (a, b ) )$$

) follows from the regularity of u.

An important difference between suitable weak solutions and Leray-Hopf weak solutions is that the former is a distributional solution of the NSE, while the latter is a solution of the initial value problem (i.e. (1.1) with $$u(0)=u_0$$ ). In addition to this, suitable weak solutions satisfy the local energy inequality (1.4), which is an interior regularity assumption not included in the definition of Leray-Hopf weak solutions (Definition 1.2). However, given divergence-free initial data

$$u_0 \in L^2 (\mathbb {R}^3)$$

, there exist Leray-Hopf weak solutions that are suitable, as was proved by Scheffer (1977) (and by Caffarelli et al. (1982) in the case of a bounded domain). In fact, the Leray–Hopf weak solutions on $$\mathbb {R}^3$$ constructed by Leray (1934) are suitable, which can be deduced from Theorem 2.1 in Biryuk, Craig and Ibrahim (2007).

The central result of the partial regularity theory is the following theorem, which was proved by Caffarelli et al. (1982).

Theorem 1.4

(Partial regularity of the Navier–Stokes equations) There exist $$\varepsilon _0, \varepsilon _1>0$$ with the following properties. Let (u, p) be a suitable weak solution of the Navier–Stokes equations on

$$\mathbb {R}^3 \times (0,\infty )$$

, and let

$$ Q_r=Q_r(z):= \{ (x,t) :|x-y|< r, \, t\in (s-r^2 , s] \} \subset \mathbb {R}^3 \times (0,\infty ), $$

where $$z=(y, s)$$ , denote a cylinder in space-time. Then

(i)

if

$$\begin{aligned} \frac{1}{r^2} \int _{Q_r} \left( |u|^3 + |p|^{3/2} \right) \le \varepsilon _0 \end{aligned}$$

(1.8)

then

$$u\in L^{\infty } (Q_{r/2} )$$

;

(ii)

if

$$\begin{aligned} \limsup _{r\rightarrow 0} \frac{1}{r} \int _{Q_r} |\nabla u|^2 \le \varepsilon _1 \end{aligned}$$

(1.9)

then

$$u\in L^\infty (Q_\rho )$$

for some $$\rho >0$$ .

We note that alternative approaches to partial regularity have been developed by Lin (1998), Ladyzhenskaya and Seregin (1999), Vasseur (2007) and Kukavica (2009b). In fact, the results of Lin (1998), Ladyzhenskaya and Seregin (1999) are a little different, as instead of local boundedness (as in the theorem above) they show a stronger property, namely local Hölder continuity (in space-time).

In short, the above theorem provides sufficient conditions on the local (in space-time) behaviour of suitable weak solutions that guarantee boundedness. The partial regularity theorem is also a key ingredient in the $$L_{3,\infty }$$ regularity criterion for the three-dimensional Navier–Stokes equations (see Escauriaza, Seregin & Šverák, 2003) and in the uniqueness of Lagrangian trajectories for suitable weak solutions (Robinson & Sadowski, 2009); similar ideas have also been used for other models, such as the surface growth model

$$\begin{aligned} \partial _t u+ u_{xxxx}+\partial _{xx} u_x^2=0 \end{aligned}$$

(Ożański & Robinson, 2019), which is a one-dimensional model of the Navier–Stokes equations (Blömker & Romito 2009, 2012).

A remarkable feature of this partial regularity result is that the quantities involved (namely $$|u|^3$$ , $$|p|^{3/2}$$ , $$|\nabla u|^2$$ ) are globally (in space-time) integrable for any Leray-Hopf weak solution. This leads to a corollary of Theorem 1.4 which provides upper bounds on the dimension of a putative singular set.

The Singular Set of a Suitable Weak Solution of the Navier–Stokes Equations

Theorem 1.4 implies that, given a suitable weak solution (u, p), there cannot be too many singular points. This point can be made precise by considering the singular set , i.e.

$$\begin{aligned} S:= \{ (x,t) \in U \times (a,b ) :u \text { is unbounded in any neighbourhood of }(x, t)\}. \end{aligned}$$

(1.10)

In other words, $$(x, t) \not \in S$$ if and only if u is bounded in a neighbourhood of (x, t). In fact, it then follows from a local Serrin condition that u is smooth in the spatial variables in such a neighbourhood (see Theorem 13.4 in Robinson et al. 2016 for a proof). Theorem 1.4 lets us estimate the dimension of S.

Corollary 1.5

Let (u, p) be a suitable weak solution of the Navier–Stokes equations on

$$U\times (a, b)$$

. Then

$$\begin{aligned} d_H(S)\le 1. \end{aligned}$$

(1.11)

Moreover, if

$$p\in L^{5/3}_{loc} (U\times (a, b))$$

then for every compact

$$K\subset U\times (a, b)$$$$ d_B(S\cap K) \le 5/3. $$

Here $$d_B$$ denotes the (upper) box-counting dimension and $$d_H$$ denotes the Hausdorff dimension. Namely, given a compact set K

$$\begin{aligned} d_B(K) := \limsup _{r \rightarrow 0} \frac{M(K, r )}{-\log r }, \end{aligned}$$

(1.12)

where M(K, r) stands for the maximal number of r-balls with centres in K (or, equivalently, for the minimal number of r-balls required to cover K, see Exercise 3.1 in Robinson (2011)