Notes on INS (14): CKN theory

 

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)

- Scheffer

- 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 (t,x) is regular if u  is essentially bounded in a neighborhood of (t,x). Otherwise it’s called singular.

(Parabolic) Hausdorff measure. For a set X in space-time and k \geq 0, define k-dim Hausdorff measure of X to be:

P^k(X) = \lim_{\delta \to 0+} P^k_\delta(X), where

P^k_\delta (X) = \inf \{ \sum_i r_i^k: X \subset \cup_i Q^*_{r_i} (t_i, x_i), \ r_i < \delta\}

and Q^*_{r} (t, x) = ]t - r^2, t+r^2[ \times B_r(x), Q_{r} (t, x) = ]t - r^2, t[ \times B_r(x)

Suitable weak solution: is a weak solution u such that the pressure p \in L^{5/3}(]0,T[ \times \Omega) 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 u\phi, where \phi is some smooth scalar cut-off function, we obtain:

(1/2) (d/dt) \int |u|^2 \phi - (1/2) \int |u|^2 \phi_t + \int |\nabla u|^2 \phi - (1/2) \int |u|^2 \Delta \phi - (1/2) \int |u|^2 (u.\nabla) \phi - \int p (u.\nabla) \phi = 0

If we integrate in time we obtain (the local form of the energy inequality):

\int |u|^2 \phi|_t + 2\int \int  |\nabla u|^2 \phi \leq \int \int [|u|^2 [\phi_t + \Delta \phi] + (|u|^2 + 2p) (u.\nabla) \phi]

Suitable solutions satisfy the above inequality for any cutoff function \phi with compact suport in ]0,T[ \times \Omega.

Another (equivalent, and probably simpler) to express the local energy inequality, also called generalized energy inequality, is as follows:

2\int_{Q}|\nabla v|^2\phi dx dt \leq \int_{Q}|v|^2(\phi_t  +\Delta \phi) dx dt + \int_{Q}(|v|^2 + 2P) v. \nabla \phi dx dt

for all non-negative cutoff function \phi \in C^\infty(Q), where Q is the direct product of an interval of time with a domain in space. (The term \int |u|^2 \phi|_t dx in the first formulation of the inequality vanishes here, because we take t=T and \phi(T,x) = 0). 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:

A(r) = \sup_{-r^2 \leq t \leq 0} r^{-1} \int_{\{t\} \times B_r}|v|^2 dx — max energy quantity in ball B_r for time t \in [-r^2,0],

B(r) = r^{-1} \int_{Q_r}|\nabla v|^2 dxdt – amount of local energy dissipation,

C(r) = 1/r^{-2} \int_{Q_r}|v|^3 dxdt – loca L3 norm of velocity,

D(r) = 1/r^{-2} \int_{Q_r}|P|^{3/2} dxdt – local L3/2 norm of pressure

The powers of r in the above quantities are to normalize things, to make them invariant with respect to scaling: x = rx_1, v = v_1/r, t = r^2t_1, P = P_1/r^2, then the INS equation remains invariant, and A(r) becomes A(1) and so on.

Lin’s Theorem 1. (Suitable solution). Let (v_n, P_n) be a sequence of weak solutions in Q_1 such that, for some positive constants E, E_0, E_1 < \infty one has:

a) \int_{\{t\} \times B_1}|v_n|^2 dx \leq E_0 for a.e. -1 < t <0,

b) \int_{Q_1} |\nabla v_n|^2 dx dt \leq E_1,

c) \int_{Q_1} |P_n|^{3/2} dx dt \leq E, and

d) (v_n,p_n) satisfies the  generalized energy inequality for all n.

Suppose that (v,p) is the weak limit of (v_n,p_n). Then (v,P) is a weak suitable solution of INS.

Lin’s Theorem 2 (Holder continuity). Assume that (v,P) is a suitable solution which satisfies

\int_{Q_1} (|v|^3 + |P|^{3/2}) dx dt < \epsilon_0 for some appropriate small positive constant \epsilon_0. Then v is Holder-continuous (with some Holder exponent \alpha > 0) in some Q_r (0 < r < 1).

Lin’s Theorem 3 (local regularity criterion). There is a positive constant \epsilon_0 such that if a suitable weak solution satisfies

\limsup_{r \to 0} B(r) \leq \epsilon_0

then there are $\theta_0, r_0 \in ]0,1[$ such that either

A^{3/2}(\theta_0 r) + D^2 (\theta_0 r) \leq (1/2) (A^{3/2} (r) + D^2(r))

or A^{3/2} (r) + D^2(r) \leq \epsilon_1 << 1

where 0 < r < r_0

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 Q_r 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 A,B,C,D starting from  \limsup_{r \to 0} B(r) \leq \epsilon_0.

Lin’s Lemma 1 (Lin’s interpolation inequality). There is a constant C such that for any v and any 0 < r  \leq \rho we have:

C(r) \leq C[(r/\rho)^3 A^{3/2}(\rho) + (\rho/r)^3  A^{3/4}(\rho) B^{3/4} (\rho)]

Remark. By this Lin’s Lemma 2, we can bound C(r) by A(\rho), B(\rho), or, assuming that B(\rho) is very small, bound C(r) by (r/\rho)^3 A(\rho). This

Lemma is a modification of the following Sobolev interpolation inequality:

Sobolev inequality. For v \in H^1(B_r) (in 3-dimensional space) one has:

\int_{B_r}|v|^q dx \leq c(\int_{B_r}|\nabla v|^2 dx)^a (\int_{B_r}|v|^2 dx)^{q/2 -a} + \frac{c}{r^{2a}}(\int_{B_r}|v|^2 dx)^{q/2}

for all 2 \leq q \leq 6, a = 3(q-2)/4.

Lin’s Lemma 2. Let (v,P) be a weak solution in Q_1 (with the usual assumptions on v). Then P \in L^{5/3}(Q_1).

Lin’s Lemma 3. Suppose that (v,P) is a suitable weak solution such that \int_{Q_1} (|v|^3 + |P|^{3/2}) dx dt \leq \epsilon_0 for some sufficiently small \epsilon_0. Then

\theta^{-5} \int_{Q_\theta} \theta^{-\alpha_0}[|v-v_\theta|^3 + |P - P_\theta(t)|^{3/2}] dx dt \leq \frac{1}{2} \int_{Q_1} [|v|^3 + |P |^{3/2}] dx dt

for some positive constants \theta, \alpha_0 \in ]0, 1/2[, where v_\theta is the mean value of v in Q_\theta and P_\theta(t) is the mean value of P in \{t\} \times B_\theta.

Lin’s Lemma 4. Let (v,P) be a suitable weak solution in Q_1. Then, for almost all t \in ]-1/2,0[  one has

\theta^{-3} \int_{\{t\} \times B_\theta}|P|^{3/2}dx \leq C_{\theta_0} \int_{\{t\} \times B_1}|v - \bar{v}|^{3/2}dx + C_0\int_{\{t\} \times B_1}|P|^{3/2}dx

for all \theta \in ]\theta_0, 1/2[, where C_0 is a constant which does not depend on the choice of \theta_0.

Remark & proof. In Lin’s paper, the term \theta^{-3} 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 P as the sum of a harmonic function and a function with the same Laplacian of P but which is zero on the boundary of some appropriate  B_\rho, 1/2 < \rho < 1. Bound the first function to the power 3/2 by its value on B_\rho using subharmonicity, and the second function by Calderon-Zygmund inequality.

Lin’s Lemma 5. For any r \in (\theta_0\rho, \rho/2), \rho \leq 1, one has:

D(r) \leq C_{\theta_0} \frac{1}{\rho^2} \int_{Q_ \rho} |v - v_\rho|^3 dx dt + C_0 \frac{r}{\rho} D(\rho)

Remark: Lin’s Lemma 5 follows directly from Lin’s Lemma 4, and it allows one to bound D(r) by C(\rho) and \frac{r}{\rho} D(\rho) (note the coefficient $r/\rho$, which can be chosen very small when r is small).

Proof of Lin’s Lemma 3. Suppose that the conclusion of Lin’s Lemma 3 is false. Then there is a sequence (v_k, P_k) of suitable weak solutions such that \|v_k\|_{L^3} + \|P_k\|_{L^{3/2}} = \epsilon_k \to 0 when k \to \infty, which do not satisfy the above property. Let (u_k,\tilde{P}_k) = (v_k/\epsilon_k, P_k/\epsilon_k), then (u_k,\tilde{P}_k) is a suitable weak solution of the equation

\partial u_k / \partial t + \epsilon_k u_k \nabla u_k + \nabla \tilde{P}_k = \Delta u_k.

One also notices that

\Delta \tilde{P}_k = -\epsilon_k \sum_{i,j} (\partial u_k^i/\partial x_j)(\partial u_k^j/\partial x_i)

It follows from the generalized energy inequality that the u_k‘s lie in a bounded set of L^\infty([-1,0], L^2_{loc}(B_1)) \cap L^2([-1,0], H^1_{loc}(B_1)), and hence they lie in a bounded set of L^{10/3}([-1,0]; L^{10/3}_{loc}(B_1)) (by an interpolation inequality).

By taking a subsequence, we may assume that (u_k,\tilde{P_k}) converge weakly to (u,P) which satisfy:

\partial u/\partial t + \nabla P = \Delta u; div u = 0, \int_{Q_1}|u|^3 dx dt \leq 1, \int_{Q_1}|P|^{3/2} dx dt \leq 1

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) –> u,P are smooth in the spatial variable, and Holder-continuous in the time variable with, say, exponent 2 \alpha_0. Thus, for a suitable \theta \in ]0, 1/2[ one has:

\theta^{-5} \int_{Q_\theta} |u-u_\theta|^3 dx dt \leq \theta^{\alpha_0}/4

(the number 4 in the above inequality is a bit arbitrary; in fact can choose any number we like). We can also assume that u_k \to u strongly in L^3([-1,0],L^3_{loc}(B_1)). Hence we have:

\theta^{-5} \int_{Q_\theta} |u_k-u_{k,\theta}|^3 dx dt \leq  \theta^{\alpha_0}/3 for all k sufficiently large.

Next consider \tilde{P_k}. Decompose it as: \tilde P_k = h_k + g_k in ]0,1[ \times B_{2/3}, where

\Delta g_k = \Delta \tilde{P_k} in B_{2/3} and g_k = 0 on \partial B_{2/3} (hence h_k is harmonic in B_{2/3}).  Using the fact that h_i is harmonic to bound it, and the Calderon-Zygmund inequality to bound g_k via u_k, we get (see Lin’s paper for details):

\theta^{-5} \int_{Q_\theta} |\tilde{P}_k- \tilde{P}_{k,\theta}|^{3/2} dx dt \leq   \theta^{\alpha_0}/3 for all k sufficiently large.

Summing this inequality with the previous one for |u_k-u_{k,\theta}|, 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 (v,P) be a weak solution such that \int_{Q_1} (|v|^3 + |P|^{3/2}) dx dt \leq \epsilon_0. Put

v_1(t,x) = \frac{v - v_\theta}{\theta^{\alpha_0/3}}(\theta^2 t, \theta x),

P_1(t,x) = \theta^{1-\alpha_0/3} (P(\theta^2  t, \theta x) - P_\theta(t)).

Then (v_1,P_1) is a suitable weak solution of

\partial v_1 / \partial t + \theta (v_\theta + \theta^{\alpha_0/3} v_1). \nabla v_1 + \nabla P_1 = \Delta v_1 in Q_1

Moreover, Lin’s Lemma 3 implies that

\int_{Q_1} (|v_1|^3 + |P_1|^{3/2}) dx dt \leq \epsilon_0/2

Now repeating the same arguments as in the proof of Lin’s Lemma 3 (with minor modifications), we can conclude that

\theta^{-5} \int_{Q_\theta} \theta^{-\alpha_0}[|v_1-v_{1,\theta}|^3 +  |P_1 - P_{1,\theta(t)}|^{3/2}] dx dt \leq \frac{1}{2} \int_{Q_1} [|v_1|^3 + |P_1|^{3/2}] dx dt \leq \epsilon_0/4

By a simple iteration, we then conclude that

r^{-5} \int_{Q_r} |v - v_r|^3 dx dt \leq C \epsilon_0 r^{\alpha_0}

for all r \in ]0,1/2[. Thus v is Holder-continuous in (t,x) (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).

Various Remarks

- Scheffer (Commun. Math. Phys, 1977) proved that the Hausdorff dimension of the singular set didn’t exceed 2.

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