Last updated: 27/Oct/2010
CKN stands for Caffarelli-Kohn-Nirenberg. The theory is about partial regularity of solutions of INS. One should probably add the name of Scheffer, who introduced the concepts that CKN improved/generalized.
The main result is that the (parabolic) 1-dimensional Hausdorff measure of the (hypothetical) singular set in space-time is zero (which means that, if singular points exist, they indeed form a “very small” set). The result is obtained in 1982, nearly 30 years ago. No major improvement since then. Basically, the result is optimal, if one uses only “generic” features of INS (quadratic nonlinear term, viscosity term, …), and not the things which are very specific to INS (but what are the things which are specific to INS and how to use them ?!)
References for this part:
– Original paper of Caffarelli-Kohn-Nirenberg (1982)
– Paper by Fanghua Lin, Commun. Pure Appl. Math. LI (1998), 241–257. [this paper has lots of typos everywhere which make it a bit annoying — I wonder why the author/editors were not more careful — but it contains a very nice proof]
– Lecture notes by Seregin (2010): Section 3 (called epsilon-regularity)
– Lecture notes by James Robinson (Campinas 2010) [these lecture notes don’t contain the proof of the main theorem, only some parts of it]
– Lecture notes by Gallavotti (2008 ?) [complicated, even in the notations; uses inequalities which don’t look very natural ?!] Gallavotti has also written a book on fluid dynamics.
– Paper by Ladyzhenskaya & Seregin (1999): contains some results similar to the ones in Lin’s paper (?). Ref: Ladyzhenskaya, O. A., Seregin, G. A., On partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations, J. math. fluid mech., 1(1999), pp. 356-387.
– Paper by AlexisVasseur ? [don’t think it contains anything conceptually new ?]
– papers by Igor Kukavica: contains some improvements of CKN results (as mentioned in the lecture notes by Robinson)
Regular point. A point is regular if is essentially bounded in a neighborhood of . Otherwise it’s called singular.
(Parabolic) Hausdorff measure. For a set in space-time and , define -dim Hausdorff measure of to be:
Suitable weak solution: is a weak solution such that the pressure and satisfies a local form of the energy inequality: If we assume that all terms are smooth and take the inner product of INS with , where is some smooth scalar cut-off function, we obtain:
If we integrate in time we obtain (the local form of the energy inequality):
Suitable solutions satisfy the above inequality for any cutoff function with compact suport in .
Another (equivalent, and probably simpler) to express the local energy inequality, also called generalized energy inequality, is as follows:
for all non-negative cutoff function , where is the direct product of an interval of time with a domain in space. (The term in the first formulation of the inequality vanishes here, because we take and ). However, the first formulation of generalized energy inequality is probably more useful.
Fact (CKN): Suitable weak solutions do exist. (Proved in the appendix of CKN paper).
Lin’s proof of CKN (1998)
We will play with the following 4 local quantities:
— max energy quantity in ball for time ,
– amount of local energy dissipation,
– loca L3 norm of velocity,
– local L3/2 norm of pressure
The powers of in the above quantities are to normalize things, to make them invariant with respect to scaling: , then the INS equation remains invariant, and becomes and so on.
Lin’s Theorem 1. (Suitable solution). Let be a sequence of weak solutions in such that, for some positive constants one has:
a) for a.e. ,
c) , and
d) satisfies the generalized energy inequality for all .
Suppose that is the weak limit of . Then is a weak suitable solution of INS.
Lin’s Theorem 2 (Holder continuity). Assume that is a suitable solution which satisfies
for some appropriate small positive constant . Then is Holder-continuous (with some Holder exponent ) in some (.
Lin’s Theorem 3 (local regularity criterion). There is a positive constant such that if a suitable weak solution satisfies
then there are $\theta_0, r_0 \in ]0,1[$ such that either
Remark. Lin’s Theorem 3 implies that, starting at a point in space-time which satisfies the condition of the theorem, we can choose a small centered at it which satisfy the conditions of Lin’s Theorem 2, which implies that the point is regular. In other words, any point which satisfies the condition of Theorem 3 is regular. (The set of points which don’t satisfy the condition of Theorem 3 is of 1-dim Huasrdorff measure 0, and so we get the CKN theorem). The proof of Lin’s Theorem 3 is based on the following lemmas and the generalized energy inequality, which together allow to “control” the quantities starting from .
Lin’s Lemma 1 (Lin’s interpolation inequality). There is a constant such that for any and any we have:
Remark. By this Lin’s Lemma 2, we can bound by , or, assuming that is very small, bound by . This
Lemma is a modification of the following Sobolev interpolation inequality:
Sobolev inequality. For (in 3-dimensional space) one has:
for all .
Lin’s Lemma 2. Let be a weak solution in (with the usual assumptions on ). Then .
Lin’s Lemma 3. Suppose that is a suitable weak solution such that for some sufficiently small . Then
for some positive constants , where is the mean value of in and is the mean value of in .
Lin’s Lemma 4. Let be a suitable weak solution in . Then, for almost all one has
for all , where is a constant which does not depend on the choice of .
Remark & proof. In Lin’s paper, the term is completely missing in his lemma (apparently a typo), which makes the lemma senseless. The proof of Lin’s Lemma 4 is relatively simple: decompose as the sum of a harmonic function and a function with the same Laplacian of but which is zero on the boundary of some appropriate , . Bound the first function to the power 3/2 by its value on using subharmonicity, and the second function by Calderon-Zygmund inequality.
Lin’s Lemma 5. For any , one has:
Remark: Lin’s Lemma 5 follows directly from Lin’s Lemma 4, and it allows one to bound by and (note the coefficient $r/\rho$, which can be chosen very small when is small).
Proof of Lin’s Lemma 3. Suppose that the conclusion of Lin’s Lemma 3 is false. Then there is a sequence of suitable weak solutions such that when , which do not satisfy the above property. Let , then is a suitable weak solution of the equation
One also notices that
It follows from the generalized energy inequality that the ‘s lie in a bounded set of , and hence they lie in a bounded set of (by an interpolation inequality).
By taking a subsequence, we may assume that converge weakly to which satisfy:
A simple estimate for the Stokes equation (i.e. the linearized Navier-Stokes equation, without the quadratic term in velocity; see Seregin’s lecture notes Section 2) –> are smooth in the spatial variable, and Holder-continuous in the time variable with, say, exponent . Thus, for a suitable one has:
(the number 4 in the above inequality is a bit arbitrary; in fact can choose any number we like). We can also assume that strongly in . Hence we have:
for all sufficiently large.
Next consider . Decompose it as: in , where
in and on (hence is harmonic in ). Using the fact that is harmonic to bound it, and the Calderon-Zygmund inequality to bound via , we get (see Lin’s paper for details):
for all sufficiently large.
Summing this inequality with the previous one for , we get a contradiction with the beginning assumption. Thus the lemma is proven.
Proof of Lin’s Theorem 2. We will use Lin’s Lemma 3. Let be a weak solution such that . Put
Then is a suitable weak solution of
Moreover, Lin’s Lemma 3 implies that
Now repeating the same arguments as in the proof of Lin’s Lemma 3 (with minor modifications), we can conclude that
By a simple iteration, we then conclude that
for all . Thus is Holder-continuous in (per Campanato Lemma about Holder continuity, see Robinson’s lecture notes, or e.g. the book: F. Schulz, Regularity for quasilinear elliptic systems and Monge-Ampere equations in two dimensions, Lecture Notes in Maths, vol. 1445, Chapter 1).
– Scheffer (Commun. Math. Phys, 1977) proved that the Hausdorff dimension of the singular set didn’t exceed 2.