ON THE MORSE–SARD THEOREM
FOR THE SHARP CASE
OF SOBOLEV MAPPINGS
AND APPLICATIONS IN FLUID MECHANICS
Korobkov M.V.
Sobolev Institute of Mathematics, Novosibirsk, Russia
1
The talk is based on joint papers Bourgain J., Korobkov M.V. and
Kristensen J.: Rev. Mat. Iberoam. V.29, no. 1 (2013), 1–23
and
Journal fur die reine und angewandte Mathematik (Crelles Journal,
Online first), DOI: 10.1515/crelle-2013-0002.
We establish Luzin N - and Morse–Sard properties for mappings
k with k = n − m + 1
f : Rn → Rm of the Sobolev–Lorentz class Wp,1
and p = n
k (this is the sharp case that guaranties the continuity
of mappings). Using these results we prove that almost all level
sets are finite disjoint unions of C1–smooth compact manifolds
of dimension n − m.
2
2(R2 ), we prove
Using the two-dimensional version, i.e., for W1
the Bernoully Law under minimal smoothness asumptions (see
[Korobkov M.V. Dokl. Math. 2011. V.83, No.1. P.107-110.]).
Further, using the last result, we prove the existence of the
solutions to the boundary problem for steady Navier-Stokes equations
(Leray’s problem) for the following cases:
3
2(R2 ), we prove
Using the two-dimensional version, i.e., for W1
the Bernoully Law under minimal smoothness asumptions (see
[Korobkov M.V. Dokl. Math. 2011. V.83, No.1. P.107-110.]).
Further, using the last result, we prove the existence of the
solutions to the boundary problem for steady Navier-Stokes equations
(Leray’s problem) for the following cases:
1) in bounded plane and axially symmetric spatial domains (see
[Korobkov M.V., Pileckas K. and Russo R. arXiv:1302.0731]).
4
2(R2 ), we prove
Using the two-dimensional version, i.e., for W1
the Bernoully Law under minimal smoothness asumptions (see
[Korobkov M.V. Dokl. Math. 2011. V.83, No.1. P.107-110.]).
Further, using the last result, we prove the existence of the
solutions to the boundary problem for steady Navier-Stokes equations
(Leray’s problem) for the following cases:
1) in bounded plane and axially symmetric spatial domains (see
[Korobkov M.V., Pileckas K. and Russo R. arXiv:1302.0731]);
2) in exterior axially symmetric spatial domains.
5
2(R2 ), we prove
Using the two-dimensional version, i.e., for W1
the Bernoully Law under minimal smoothness asumptions (see
[Korobkov M.V. Dokl. Math. 2011. V.83, No.1. P.107-110.]).
Further, using the last result, we prove the existence of the
solutions to the boundary problem for steady Navier-Stokes equations
(Leray’s problem) for the following cases:
1) in bounded plane and axially symmetric spatial domains (see
[Korobkov M.V., Pileckas K. and Russo R. arXiv:1302.0731]);
2) in exterior axially symmetric spatial domains.
(In the case of axially symmetric domains we assume that the
boundary data, etc., are axially symmetric as well). No restriction
on the size of fluxes are assumed in these results.
6
Furthermore, using some different methods (the Hopf cut-off
function, etc.) we prove the existence theorem to the boundary
problem for steady Navier-Stokes equations in exterior plane
axially symmetric domain (with symmetric data) under additional
assumption that every boundary component intersects the axis
of symmetry (see [M.V. Korobkov, K. Pileckas and R. Russo: J.
Math. Pures. Appl. Online first (2013),
DOI Number: http://dx.doi.org/10.1016/j.matpur.2013.06.002] ).
7
Furthermore, using some different methods (the Hopf cut-off
function, etc.) we prove the existence theorem to the boundary
problem for steady Navier-Stokes equations in exterior plane
axially symmetric domain (with symmetric data) under additional
assumption that every boundary component intersects the axis
of symmetry (see [M.V. Korobkov, K. Pileckas and R. Russo: J.
Math. Pures. Appl. Online first (2013),
DOI Number: http://dx.doi.org/10.1016/j.matpur.2013.06.002] ).
Here we also have no restrictions on the size on fluxes.
8
Morse-Sard Theorem for Sobolev Functions
Recall that for any F ⊂ Rk , Hs (F ) = lim Hsα(F ), where for each
α→0+
0 < α ≤ ∞,
Hsα (F ) = inf
∞
nX
s
(diam Fi )
i=1
: diam Fi ≤ α,
F ⊂
∞
[
i=1
o
Fi .
9
Theorem (Morse-Sard, 1942). Let Ω ⊂ Rn be an open set and let
f : Ω → Rm be a Ck –smooth mapping with k ≥ max(n − m + 1, 1).
Then
Hm(f (Zf )) = 0,
where Hm denotes the m-dimensional Hausdorff measure and Zf
denotes the set of critical points of f :
Zf = {x ∈ Ω : rankDf (x) < m}.
10
For example, if n = 2 and m = 1, i.e., if f : Ω ⊂ R2 → R is a
C2-smooth real function of two variables, then the Morse–Sard
theorem guarantees that
H1(f (Zf )) = 0,
where Zf = {x ∈ Ω : ∇f (x) = 0}.
11
For example, if n = 2 and m = 1, i.e., if f : Ω ⊂ R2 → R is a
C2-smooth real function of two variables, then the Morse–Sard
theorem guarantees that
H1(f (Zf )) = 0,
where Zf = {x ∈ Ω : ∇f (x) = 0}.
Whitney demonstrated (1935) that the C2–smoothness condition
in the above assertion cannot be dropped. Namely, he constructed
a C1–smooth function f : (0, 1)2 → R for which the set Zf of
critical points contains an arc on which f is not a constant
(subsequently called a Whitney arc).
12
However, some analogs of Sard’s theorem are valid for the functions
lacking the required smoothness in the classical theorem. Although
the equality Hm(f (Zf )) = 0 may be no longer valid then,
A. Ya. Dubovitskiĭ (1957) obtained some results on the structure
of level sets in the case of reduced smoothness.
13
Theorem (A. Ya. Dubovitskiĭ, 1957). Let Ω ⊂ Rn be an open set
and let f : Ω → Rm be a Ck –smooth mapping. Set s = n − m − k + 1.
Then
Hs(f −1(y) ∩ Zf ) = 0
for Hm-a.a. y ∈ Rm.
14
Theorem (A. Ya. Dubovitskiĭ, 1957). Let Ω ⊂ Rn be an open set
and let f : Ω → Rm be a Ck –smooth mapping. Set s = n − m − k + 1.
Then
Hs(f −1(y) ∩ Zf ) = 0
for Hm-a.a. y ∈ Rm.
Because of isolation of the Soviet Union, this result was not
known well by West mathematicians. It was rediscovered in the
paper [Bojarski B., Hajlasz P., and Strzelecki P., “Sard’s theorem
for mappings in Hölder and Sobolev spaces”, Manuscripta Math.,
118, 383–397 (2005)]. In this paper it was also obtained some
new results for Hölder and Sobolev spaces, for example
15
Theorem (Bojarski B., Hajlasz P., and Strzelecki P., 2005). Let
Ω ⊂ Rn be an open set and let f : Ω → Rm be a Ck,λ+–smooth
mapping for some k ≥ 1 and λ ∈ (0, 1). Put s = m − n − (k + λ) + 1.
Then
Hs(f −1(y) ∩ Zf ) = 0
for Hm-a.a. y ∈ Rm.
Here the class Ck,λ+ is the space of Ck -smooth mappings f
satisfying the following condition: for each compact set K ⊂ Ω
there exists a nondecreasing continuous function ε : R → R with
ε(0) = 0 such that for each α with |α| = k
|Dαf (x) − Dα f (y)| ≤ ε(|x − y|)|x − y|λ
for all x, y ∈ K.
16
Theorem (Federer–Dubovitskiĭ 1966). Let m ∈ {1, . . . , n}, d, k ∈ N,
and let f : Rn → Rd be a Ck –smooth mapping. Denote q◦ =
m − 1 + n−m+1
. Then
k
Hq◦ (f (Zf,m )) = 0,
(1)
where Hβ denotes the β–dimensional Hausdorff measure and
Zf,m denotes the set of m–critical points of f :
Zf,m = {x ∈ Rn :
rank∇f (x) < m}.
17
Theorem (Federer–Dubovitskiĭ 1966). Let m ∈ {1, . . . , n}, d, k ∈ N,
and let f : Rn → Rd be a Ck –smooth mapping. Denote q◦ =
m − 1 + n−m+1
. Then
k
Hq◦ (f (Zf,m )) = 0,
(2)
where Hβ denotes the β–dimensional Hausdorff measure and
Zf,m denotes the set of m–critical points of f :
Zf,m = {x ∈ Rn :
rank∇f (x) < m}.
The Morse–Sard–Federer–Dubovitskiĭ results have subsequently
been generalized to mappings in more refined scales of spaces,
including Hölder and Sobolev spaces. For Hölder spaces we mention
in particular [Bates93, Bojarski-Hajlasz-Strzelecki2002, Moreira2001,
Norton1986, Yomdin1983] where essentially sharp results were
obtained.
18
They constructed examples showing that the smoothness assumption
on f in Federer–Dubovitskiĭ theorem cannot be weakened within
the scale of Cj spaces. However, it follows from [Bates93] that
the conclusion Hq◦ (f (Zf,m )) = 0 remains valid for Ck−1,1 mappings
f , and according to [Moreira2001] it fails in general for Ck−1,α
mappings whenever α < 1. (For k ∈ N0 and α ∈ (0, 1] we say
that the mapping f is of class Ck,α when f is Ck and the k–
th order derivative of f is locally α–Hölder continuous.) One
interpretation of these results is that for the validity of
Hq◦ (f (Zf,m )) = 0
one must assume existence of k derivatives of f in a suitably
strong sense.
19
The main problem: to replace assumption f ∈ C k (Rn , Rd) in the
Morse–Sard–Federer–Dubovitskiĭ Theorem by assumption f ∈
Wpk (Rn , Rd).
20
The main problem: to replace assumption f ∈ C k (Rn , Rd) in the
Morse–Sard–Federer–Dubovitskiĭ Theorem by assumption f ∈
Wpk (Rn , Rd).
What we can say about p?
21
The main problem: to replace assumption f ∈ C k (Rn , Rd) in the
Morse–Sard–Federer–Dubovitskiĭ Theorem by assumption f ∈
Wpk (Rn , Rd).
What we can say about p? The first Morse–Sard result in the
Sobolev context that we are aware of is [DePascale2001].
22
Theorem (De Pascale, 2001). Let Ω ⊂ Rn be an open set and
k
let f : Ω → Rm belongs to the Sobolev class Wp,loc
(Ω, Rm) with
k = n − m + 1 ≥ 2 and p > n. Then
Hm(f (Zf )) = 0.
Note that because of Sobolev Imbedding Theorem, f ∈ Ck−1
under the above assumptions on k, p. Thus Zf can be defined in
a usual way.
23
Further we consider in detail the case n = 2 and m = 1, i.e., if
f : Ω ⊂ R2 → R is a real function of two variables.
In this case we have richer theory. The following Sard–type
theorem was obtained by A.V. Pogorelov.
24
Theorem (A.V. Pogorelov, 1956). For a real function f ∈ C1(Ω)
on a plane domain Ω, the equality
H1(f (Zf )) = 0
(1)
holds if for any linear map L : R2 → R the sum f (x)+L(x) satisfies
the maximum principle, i.e., f (x) + L(x) has no points of strict
extremum.
25
Theorem (A.V. Pogorelov, 1956). For a real function f ∈ C1(Ω)
on a plane domain Ω, the equality
H1(f (Zf )) = 0
(1)
holds if for any linear map L : R2 → R the sum f (x)+L(x) satisfies
the maximum principle, i.e. f (x) + L(x) has no points of strict
extremum.
This result was formulated in geometrical form. Thus not so
many specialists in functions theory knew this theorem. Pogorelov’s
Theorem was rediscovered in [K06], where the proof was analytical
and it was based on the Coarea formula.
In particular, the equality (1) holds if the gradient range ∇f (Ω)
has no interior points (see also [K06,K09,K07]).
26
For the n = 2 and m = 1, i.e., if f : Ω ⊂ R2 → R is a real function
of two variables, De Pascale’s theorem guarantees that under
2,p
the assumption f ∈ Wloc (Ω) with p > 2 the equality
H1(f (Zf )) = 0
holds.
27
For the n = 2 and m = 1, i.e., if f : Ω ⊂ R2 → R is a real function
of two variables, De Pascale’s theorem guarantees that under
2,p
the assumption f ∈ Wloc (Ω) with p > 2 the equality
H1(f (Zf )) = 0
holds.
In the paper [Bucur D., Giacomini A., and Trebeschi P., “Whitney
property in two dimensional Sobolev spaces”, Proc. Amer. Math. Soc.
136, No. 7, (2008), 2535–2545] it was proved that for functions
2,p
f ∈ Wloc(R2) with p > 1 there are no Whitney arcs.
28
Landis (1951) proved that the equality H1(f (Zf )) = 0 holds if
f : Ω → R is a difference of two convex functions (sometimes
called a d.c.-function), a result which answered a question raised
previously by A.V. Pogorelov.
29
Landis (1951) proved that the equality H1(f (Zf )) = 0 holds if
f : Ω → R is a difference of two convex functions (sometimes
called a d.c.-function), a result which answered a question raised
previously by A.V. Pogorelov.
D. Pavlica and L. Zajíček (2006) presented the detailed and
modern proof of the Landis result. Moreover, they proved that
the equality H1(f (Zf )) = 0 holds for Lipschitz functions f ∈
BV2,loc(Ω), where BV2,loc(Ω) is the space of functions f ∈ L1
loc (Ω)
such that all its partial (distributional) derivatives of the second
order are R-valued Radon measures on Ω.
30
Landis (1951) proved that the equality H1(f (Zf )) = 0 holds if
f : Ω → R is a difference of two convex functions (sometimes
called a d.c.-function), a result which answered a question raised
previously by A.V. Pogorelov.
D. Pavlica and L. Zajíček (2006) presented the detailed and
modern proof of the Landis result. Moreover, they proved that
the equality H1(f (Zf )) = 0 holds for Lipschitz functions f ∈
BV2,loc(Ω), where BV2,loc(Ω) is the space of functions f ∈ L1
loc (Ω)
such that all its partial (distributional) derivatives of the second
order are R-valued Radon measures on Ω.
In the paper [Bourgain, K, and Kristensen 2010] we extend the
last result to the case of any BV2-function defined on a planar
domain (without the additional Lipschitz assumption).
31
Also we studied the N -property and the structure of the level
sets for these functions BV2(R2).
32
Also we studied the N -property and the structure of the level
sets for these functions BV2(R2).
We also obtain the similar results for Wn,1(Rn) and BVn(Rn)
[Bourgain, K, and Kristensen 2011].
33
The scalar case f ∈ W1n(Rn).
Proposition 1 (see [Dorronsoro1989]). Let f ∈ W1n(Rn ). Then
there exists a set Af such that H1(Af ) = 0 and for all x ∈
Rn \ Af the function f is differentiable (in the classical sense) at
the point x, furthermore, the classical derivative coincides with
R
R
∇f (x) = lim −Br (x) ∇f (z)dz, where lim −Br (x) |∇f (z) − ∇f (x)|n dz = 0.
r→0
r→0
34
The scalar case f ∈ W1n(Rn).
Proposition 1 (see [Dorronsoro1989]). Let f ∈ W1n(Rn ). Then
there exists a set Af such that H1(Af ) = 0 and for all x ∈
Rn \ Af the function f is differentiable (in the classical sense) at
the point x, furthermore, the classical derivative coincides with
R
R
∇f (x) = lim −Br (x) ∇f (z)dz, where lim −Br (x) |∇f (z) − ∇f (x)|n dz = 0.
r→0
r→0
Put Zf = {x ∈ Rn \ Af : ∇f (x) = 0}.
35
The scalar case f ∈ W1n(Rn).
Proposition 1 (see [Dorronsoro1989]). Let f ∈ W1n(Rn ). Then
there exists a set Af such that H1(Af ) = 0 and for all x ∈
Rn \ Af the function f is differentiable (in the classical sense) at
the point x, furthermore, the classical derivative coincides with
R
R
∇f (x) = lim −Br (x) ∇f (z)dz, where lim −Br (x) |∇f (z) − ∇f (x)|n dz = 0.
r→0
r→0
Put Zf = {x ∈ Rn \ Af : ∇f (x) = 0}.
Theorem 1 [BKKr2011]. Let f ∈ W1n(Rn). Then H1(f (Zf )) = 0.
36
The scalar case f ∈ W1n(Rn).
Proposition 1 (see [Dorronsoro1989]). Let f ∈ W1n(Rn ). Then
there exists a set Af such that H1(Af ) = 0 and for all x ∈
Rn \ Af the function f is differentiable (in the classical sense) at
the point x, furthermore, the classical derivative coincides with
R
R
∇f (x) = lim −Br (x) ∇f (z)dz, where lim −Br (x) |∇f (z) − ∇f (x)|n dz = 0.
r→0
r→0
Put Zf = {x ∈ Rn \ Af : ∇f (x) = 0}.
Theorem 1 [BKKr2011]. Let f ∈ W1n(Rn). Then H1(f (Zf )) = 0.
What we can say about H1(f (Af ))?
37
Theorem 2 [BKKr2011]. Let f ∈ W1n(Rn ). Then for every ε > 0
there exists δ > 0 such that for any set U with H1
∞ (U ) < δ the
inequality H1(f (U )) < ε holds.
38
Theorem 2 [BKKr2011]. Let f ∈ W1n(Rn ). Then for every ε > 0
there exists δ > 0 such that for any set U with H1
∞ (U ) < δ the
inequality H1(f (U )) < ε holds.
Corollary 1 [BKKr2011]. H1(f (E)) = 0 whenever H1(E) = 0. In
particular, H1(f (Af )) = 0.
39
An Approximation Theorem for W1k (Rn ) .
Theorem [BKKr2013]. Let k, l ∈ {1, . . . , n}, l ≤ k. Then for any
f ∈ Wk,1(Rn ) and for each ε > 0 there exist an open set U ⊂ Rn
and a function g ∈ C l (Rn ) such that Hn−k+l
(U ) < ε and f ≡ g,
∞
∇mf ≡ ∇mg on Rn \ U for m = 1, . . . , l.
40
An Approximation Theorem for W1k (Rn ) .
Theorem [BKKr2013]. Let k, l ∈ {1, . . . , n}, l ≤ k. Then for any
f ∈ Wk,1(Rn ) and for each ε > 0 there exist an open set U ⊂ Rn
and a function g ∈ C l (Rn ) such that Hn−k+l
(U ) < ε and f ≡ g,
∞
∇mf ≡ ∇mg on Rn \ U for m = 1, . . . , l.
For f ∈ Wpk , p > 1, the set U is small with respect to the Bessel
capacity (see, e.g., Chapter 3 in [Ziemer1989]).
41
Level sets of W1n(Rn) .
Theorem 3 [BKKr2011]. Let f ∈ W1n(Rn). Then for H1–almost
all y ∈ f (Rn ) ⊂ R the preimage f −1(y) is a finite disjoint family
of C 1–smooth compact manifolds (without boundary) Sj , j =
1, 2, . . . , N (y).
42
Level sets of W1n(Rn) .
Theorem 3 [BKKr2011]. Let f ∈ W1n(Rn). Then for H1–almost
all y ∈ f (Rn ) ⊂ R the preimage f −1(y) is a finite disjoint family
of C 1–smooth compact manifolds (without boundary) Sj , j =
1, 2, . . . , N (y).
All above results are valid for wider class of functions of bounded
variation BVn (Rn ) whose n-derivative is a Radon measure.
43
The case of vector-valued mappings f ∈ Wpk (Rn , Rd).
Of course, it is very natural to assume, that the inclusion f ∈
Wpk (Rn, Rd) should guarantee at least the continuity of f . For
values k ∈ {1, . . . , n − 1} it is well–known that f ∈ Wpk (Rn , Rd) is
n . So
and
could
be
discontinuous
for
p
≤
continuous for p > n
k
k
the borderline case is p = p◦ = n
k . It is well–known that really f ∈
Wpk◦ (Rn, Rd) is continuous if the derivatives of k-th order belong
to the Lorentz space Lp◦,1, we will denote the space of such
mappings by Wpk◦,1(Rn, Rd).
44
The case of vector-valued mappings f ∈ Wpk (Rn , Rd).
Of course, it is very natural to assume, that the inclusion f ∈
Wpk (Rn, Rd) should guarantee at least the continuity of f . For
values k ∈ {1, . . . , n − 1} it is well–known that f ∈ Wpk (Rn , Rd) is
n . So
and
could
be
discontinuous
for
p
≤
continuous for p > n
k
k
the borderline case is p = p◦ = n
k . It is well–known that really f ∈
Wpk◦ (Rn, Rd) is continuous if the derivatives of k-th order belong
to the Lorentz space Lp◦,1, we will denote the space of such
mappings by Wpk◦,1(Rn, Rd).
We prove the precise analog of the above Federer–Dubovitskiĭ
theorem for mappings f : Rn → Rd locally of Sobolev class Wpk◦ ,
k ∈ {2, . . . , n}, m ∈ {2, . . . , n}.
45
The case of vector-valued mappings f ∈ Wpk (Rn , Rd).
Of course, it is very natural to assume, that the inclusion f ∈
Wpk (Rn, Rd) should guarantee at least the continuity of f . For
values k ∈ {1, . . . , n − 1} it is well–known that f ∈ Wpk (Rn , Rd) is
n . So
and
could
be
discontinuous
for
p
≤
continuous for p > n
k
k
the borderline case is p = p◦ = n
k . It is well–known that really f ∈
Wpk◦ (Rn, Rd) is continuous if the derivatives of k-th order belong
to the Lorentz space Lp◦,1, we will denote the space of such
mappings by Wpk◦,1(Rn, Rd).
We prove the precise analog of the above Federer–Dubovitskiĭ
theorem for mappings f : Rn → Rd locally of Sobolev class Wpk◦ ,
k ∈ {2, . . . , n}, m ∈ {2, . . . , n}.
46
We prove N -property with respect to Hq , q ∈ (p◦, n], Hq -almost
everywhere differentiability and classical properties of level sets
for mappings of Sobolev–Lorentz class f ∈ Wpk◦,1(Rn, Rd).
The Federer–Dubovitskiĭ Theorem for f ∈ Wpk◦ (Rn , Rd).
Lemma (see, e.g., Chapter 3 in [Ziemer1989]). Let k ∈ {2, . . . , n}
and f ∈ Wpk◦ (Rn). Then there exists a Borel set Af ⊂ Rn such that
Hq (Af ) = 0 for every q ∈ (p◦, n] and all points of Rn \ Af are LnLebesgue points for ∇f , i.e.,
Z
−
B(x,r)
|∇f (z) − ∇f (x)|n dz → 0
as
r ց 0.
47
The Federer–Dubovitskiĭ Theorem for f ∈ Wpk◦ (Rn , Rd).
Lemma (see, e.g., Chapter 3 in [Ziemer1989]). Let k ∈ {2, . . . , n}
and f ∈ Wpk◦ (Rn). Then there exists a Borel set Af ⊂ Rn such that
Hq (Af ) = 0 for every q ∈ (p◦, n] and all points of Rn \ Af are LnLebesgue points for ∇f , i.e.,
Z
−
B(x,r)
|∇f (z) − ∇f (x)|n dz → 0
as
r ց 0.
Put Zf,m = {x ∈ Rn \ Af : rank∇f (x) < m}.
48
The Federer–Dubovitskiĭ Theorem for f ∈ Wpk◦ (Rn , Rd).
Lemma (see, e.g., Chapter 3 in [Ziemer1989]). Let k ∈ {2, . . . , n}
and f ∈ Wpk◦ (Rn). Then there exists a Borel set Af ⊂ Rn such that
Hq (Af ) = 0 for every q ∈ (p◦, n] and all points of Rn \ Af are LnLebesgue points for ∇f , i.e.,
Z
−
B(x,r)
|∇f (z) − ∇f (x)|n dz → 0
as
r ց 0.
Put Zf,m = {x ∈ Rn \ Af : rank∇f (x) < m}.
The Morse–Sard–Federer–Dubovitskiĭ Theorem for Sobolev case
[KKr2013].
If
k, m
∈
{2, . . . , n}
and
f
∈
Wpk◦ (Rn , Rd),
then
n−m+1
Hq◦ (f (Zf,m )) = 0. (Recall that p◦ = n
,
q
=
m
−
1
+
.)
◦
k
k
49
The Morse–Sard Theorem for f ∈ Wpk◦ (Rn , Rd).
Theorem 1 implies the following analog of the classical Morse–
Sard Theorem:
50
The Morse–Sard Theorem for f ∈ Wpk◦ (Rn , Rd).
Theorem 1 implies the following analog of the classical Morse–
Sard Theorem:
Corollary 1 [KKr2013]. If m ∈ {1, . . . , n} and f ∈ Wpk◦ (Rn, Rm) with
m
k = n − m + 1 and p◦ = n
k , then L (f (Zf,m )) = 0.
51
The Morse–Sard Theorem for f ∈ Wpk◦ (Rn , Rd).
Theorem 1 implies the following analog of the classical Morse–
Sard Theorem:
Corollary 1 [KKr2013]. If m ∈ {1, . . . , n} and f ∈ Wpk◦ (Rn, Rm) with
m
k = n − m + 1 and p◦ = n
k , then L (f (Zf,m )) = 0.
Recall, that Zf,m = {x ∈ Rn \ Af : rank∇f (x) < m}. Can we estimate
the measure of f (Af )?
52
The Morse–Sard Theorem for f ∈ Wpk◦ (Rn , Rd).
Theorem 1 implies the following analog of the classical Morse–
Sard Theorem:
Corollary 1 [KKr2013]. If m ∈ {1, . . . , n} and f ∈ Wpk◦ (Rn, Rm) with
m
k = n − m + 1 and p◦ = n
k , then L (f (Zf,m )) = 0.
Recall, that Zf,m = {x ∈ Rn \ Af : rank∇f (x) < m}. Can we estimate
the measure of f (Af )? No for Sobolev case (moreover, f could
be discontinuous here for m > 1)!
53
The Lorentz norm
If k < n, then it is well-known that functions from Sobolev spaces
Wpk (Rn) are continuous for p > n
k and could be discontinuous for
n
k
k
n
p ≤ p◦ = n
k . The Sobolev–Lorentz space Wp◦,1(R ) ⊂ Wp◦ (R )
is a refinement of the corresponding Sobolev space that for
our purposes turns out to be convenient. Among other things
functions that are locally in Wpk◦,1 on Rn are in particular continuous.
54
The Lorentz norm
Given a measurable function f : Rn → R, denote by f∗ : (0, ∞) → R
its distribution function
f∗(s) := Ln{x ∈ Rn : |f (x)| > s},
and by f ∗ the nonincreasing rearrangement of f , defined for t > 0
by
f ∗(t) = inf{s ≥ 0 : f∗(s) ≤ t}.
Since f and f ∗ are equimeasurable, we have for every 1 ≤ p < ∞,
Z
Rn
|f (x)|p dx
!1/p
=
+∞
Z
!1/p
f ∗(t)p dt
0
.
55
The Lorentz norm
The Lorentz space Lp,q (Rn) for 1 ≤ p < ∞, 1 ≤ q < ∞ can be
defined as the set of all measurable functions f : Rn → R for
which the expresssion
kf kLp,q =
is finite.









q
p
+∞
Z
1/p ∗
(t
0
q dt
f (t))





1/p f ∗ (t)


 sup t
t>0
t
!1/q
if 1 ≤ q < ∞
if q = ∞
56
The Lorentz norm
kf kLp,q =









q
p
+∞
Z
(t1/p f ∗(t))q
0





1/p f ∗ (t)


 sup t
t>0
dt
t
!1/q
if 1 ≤ q < ∞
if q = ∞
In view of the definition of k · kLp,q and the equimeasurability of
f and f ∗ we have an identity kf kLp = kf kLp,p so that in particular
Lp,p(Rn ) = Lp(Rn). Further, for a fixed exponent p and q1 < q2
we have an estimate kf kLp,q ≤ kf kLp,q , and, consequently, an
2
1
embedding Lp,q1 (Rn ) ⊂ Lp,q2 (Rn) . Finally we recall that k · kLp,q is
a norm on Lp,q (Rn) for all q ∈ [1, p].
57
The Lorentz norm
Here we shall mainly be concerned with the Lorentz space Lp,1,
and in this case one may rewrite the norm as
kf kp,1 =
+∞
Z h
0
n
n
i1
L ({x ∈ R : |f (x)| > t})
p
dt.
58
N -property
k (Rn ) the space of all functions f ∈ Wk (Rn ) such
Denote by Wp,1
p
that in addition the Lorentz norm k∇f k kLp,1 is finite.
59
N -property
k (Rn ) the space of all functions f ∈ Wk (Rn ) such
Denote by Wp,1
p
that in addition the Lorentz norm k∇f k kLp,1 is finite.
Theorem (N -property, [KKr2013]). Let k ∈ {2, . . . , n}, q ∈ (p◦, n],
and f ∈ Wpk◦,1(Rn, Rd). Then for each ε > 0 there exists δ > 0 such
that for any set E ⊂ Rn
if
q
q
H∞(E) < δ, then H∞(f (E)) < ε. In
particular, Hq (f (E)) = 0 whenever Hq (E) = 0.
60
N -property
k (Rn ) the space of all functions f ∈ Wk (Rn ) such
Denote by Wp,1
p
that in addition the Lorentz norm k∇f k kLp,1 is finite.
Theorem (N -property, [KKr2013]). Let k ∈ {2, . . . , n}, q ∈ (p◦, n],
and f ∈ Wpk◦,1(Rn, Rd). Then for each ε > 0 there exists δ > 0 such
that for any set E ⊂ Rn
if
q
q
H∞(E) < δ, then H∞(f (E)) < ε. In
particular, Hq (f (E)) = 0 whenever Hq (E) = 0.
1 (Rn , Rd ), the similar result (N -property
For the case k = 1, f ∈ Wn,1
with respect to the measure Hn ), was received in the paper
[Kauhanen J., Koskela P., and Maly J., 1999], see also [Romanov
A.S. 2008].
61
Differentiability properties
Theorem (Stein, Maly, Ziemer, etc.). Let k ∈ {2, . . . , n} and f ∈
Wpk◦,1(Rn , Rd). Then there exists a Borel set Af ⊂ Rn such that
Hq (Af ) = 0 for every q ∈ (p◦, n] and for any x ∈ Rn \Af the function f
is differentiable (in the classical Fréchet sense) at x, furthermore,
the classical derivative coincides with ∇f (x), where
Z
lim −
rց0 B(x,r)
|∇f (z) − ∇f (x)|n dz = 0.
62
Differentiability properties
Theorem (Stein, Maly, Ziemer, etc.). Let k ∈ {2, . . . , n} and f ∈
Wpk◦,1(Rn , Rd). Then there exists a Borel set Af ⊂ Rn such that
Hq (Af ) = 0 for every q ∈ (p◦, n] and for any x ∈ Rn \Af the function f
is differentiable (in the classical Fréchet sense) at x, furthermore,
the classical derivative coincides with ∇f (x), where
Z
lim −
rց0 B(x,r)
|∇f (z) − ∇f (x)|n dz = 0.
Further, for every ε > 0 and q ∈ (p◦, n] there exist an open set
q
U ⊃ Af and a function g ∈ C1(Rn) such that H∞(U ) < ε and f ≡ g,
∇f ≡ ∇g on Rn \ U .
63
Level sets
Theorem ([KKr2013]. Let k, m ∈ {2, . . . , n} and f ∈ Wpk◦,1(Rn, Rm).
Then for Lm–almost all y ∈ f (Rn ) the preimage v −1(y) is a finite
disjoint family of (n−m)–dimensional C1-smooth compact manifolds
(without boundary) Sj , j = 1, . . . , N (y).
64
Euler equation
In this section we prove some properties of a solution to the
Euler system
( w · ∇ w + ∇p = 0,
div w = 0.
Let Ω ⊂ R2 be a bounded domain with Lipschitz boundary.
Assume that w ∈ W 1,2(Ω) and p ∈ W 1,s(Ω), s ∈ [1, 2), satisfy the
R
Euler equations for almost all x ∈ Ω and let Γi w · n dS = 0, i =
1, 2, . . . , N , where Γi are connected components of the boundary
∂Ω. Then there exists a stream function ψ ∈ W 2,2(Ω) such that
∇ψ = (−w2, w1).
65
|w |2
Denote by Φ = p +
the total head pressure corresponding to
2
the solution (w, p). Obviously, Φ ∈ W 1,s(Ω) for all s ∈ [1, 2). By
direct calculations one easily gets the identity
∂w2 ∂w1 ∇Φ ≡
−
w2, −w1 = (∆ψ)∇ψ.
∂x1
∂x2
(2)
If all functions are smooth, then from (2) the classical Bernoulli
law follows immediately:
66
|w |2
Denote by Φ = p +
the total head pressure corresponding to
2
the solution (w, p). Obviously, Φ ∈ W 1,s(Ω) for all s ∈ [1, 2). By
direct calculations one easily gets the identity
∂w2 ∂w1 ∇Φ ≡
−
w2, −w1 = (∆ψ)∇ψ.
∂x1
∂x2
(2)
If all functions are smooth, then from (2) the classical Bernoulli
law follows immediately:
The total head pressure Φ(x) is constant along any streamline
of the flow.
67
Regularity properties of Φ. There exists a set Aw ⊂ Ω such that:
(i)
H1(Aw ) = 0;
(ii) for all x ∈ Ω \ Aw we have
R
−
lim Br (x) |Φ(z) − Φ(x)|2dz = 0.
r→0
68
Regularity properties of Φ. There exists a set Aw ⊂ Ω such that:
(i)
H1(Aw ) = 0;
(ii) for all x ∈ Ω \ Aw we have
R
−
lim Br (x) |Φ(z) − Φ(x)|2dz = 0.
r→0
Theorem 7 [K2010]. Under above conditions on Ω, w, p, for any
compact connected set K ⊂ Ω̄ such that
the assertion
ψ
Φ(x) = Φ(y)
K
= const,
for all x, y ∈ K \ Aw
holds.
69
Using Bernoulli Law for Sobolev solutions we prove the existence
of the solutions to steady Navier–Stokes equations for plane
case and axial-symmetric spatial case (see [K, Pileckas K. and
Russo R. Solution of Leray’s problem for stationary Navier-Stokes
equations in plane and axially symmetric spatial domains //
arXiv:1302.0731, [math-ph], 4 Feb 2013]).
70
Steady Navier–Stokes equations
Let Ω be a bounded multiply connected domain in Rn, n = 2, 3,
with Lipschitz boundary ∂Ω consisting of N disjoint components
Γj : ∂Ω = Γ1 ∪ . . . ∪ ΓN and Γi ∩ Γj = ∅, i 6= j. Consider in Ω the
stationary Navier–Stokes system with nonhomogeneous boundary
conditions



 −ν∆u + u · ∇ u + ∇p = 0



div u = 0
u = a
in Ω,
in Ω,
on ∂Ω.
(N S)
71
Starting from the famous paper of J. Leray published in 1933,
problem (NS) was a subject of investigation in many papers. The
continuity equation div u = 0 implies the necessary compatibility
condition for the solvability of problem (NS):
Z
∂Ω
a · n dS =
N Z
X
a · n dS = 0,
(f lux)
j=1Γ
j
where n is a unit vector of the outward (with respect to Ω)
normal to ∂Ω.
72
Starting from the famous paper of J. Leray published in 1933,
problem (NS) was a subject of investigation in many papers. The
continuity equation div u = 0 implies the necessary compatibility
condition for the solvability of problem (NS):
Z
∂Ω
a · n dS =
N Z
X
a · n dS = 0,
(f lux)
j=1Γ
j
where n is a unit vector of the outward (with respect to Ω)
normal to ∂Ω.
However, for a long time the existence of a weak solution u ∈
W 1,2(Ω) to problem (NS) was proved only under the condition
Fj =
Z
a · n dS = 0,
j = 1, 2, . . . , N.
(f lux0)
Γj
73
There are many partial results on Leray’s problem (see, e.g.,
[Leray33], [Hopf41], [Finn61], [Ladyzhenskaya69], [Lions69], [Amick84
[Galdi91], etc). But in all previous papers they assumed smallness
of the net flux
Fj =
Z
a · n dS,
j = 1, 2, . . . , N,
Γj
or symmetry conditions on the domain and boundary values. In
the joint paper [K., Pileckas K. and Russo R. // arXiv:1302.0731,
2013] we obtained the first result without any additional assumptions
(necessary and sufficient condition of solvability of stationary NS
system).
74
Steady Navier–Stokes equations: plane case
Theorem 8 [KPR2013]. Let Ω be a bounded multiply connected
domain in R2 with C 2-smooth boundary ∂Ω. If f ∈ W 1,2(Ω) and
a ∈ W 3/2,2(∂Ω) satisfies the necessary condition
Z
a · n dS = 0,
(flux)
∂Ω
then the steady Navier–Stokes system



 −ν∆u + u · ∇ u + ∇p =



f
div u = 0
u = a
in Ω,
in Ω,
on ∂Ω
(N S)
admits at least one weak solution.
75
Remark. It is well know (see [Lad]) that under hypothesis of
above Theorem every weak solution u of problem (NS) is more
3,2
regular, i.e, u ∈ W 2,2(Ω) ∩ Wloc (Ω).
76
Steady Navier–Stokes equations: axially symmetric case
Let Ox1 , Ox2 , Ox3 be coordinate axis in R3. Denote P = {(0, x2, x3) :
x2 ∈ R, x3 ∈ R}, P+ = {(0, x2, x3) : x3 ∈ R, x2 > 0}.
Let us introduce cylindrical coordinates θ = arctg(x2/x1), r =
2 1/2 , z = x and denote by v , v , v the projections of the
(x2
3
θ r z
1 + x2 )
vector v on the axes θ, r, z.
A function f is said to be axially symmetric if it does not depend
on θ. A vector-valued function h = (hθ , hr , hz ) is called axially
symmetric if hθ , hr and hz do not depend on θ. A vector-valued
function h = (hθ , hr , hz ) is called axially symmetric without swirl if
hθ = 0 while hr and hz do not depend on θ.
77
Steady Navier–Stokes equations: axially symmetric case
Theorem 9 [KPR2013]. Assume that Ω ⊂ R3 is a bounded axially
symmetric domain with C 2-smooth boundary ∂Ω (consisting of
several connected components). If f ∈ W 1,2(Ω), a ∈ W 3/2,2(∂Ω)
are axially symmetric and a satisfies the necessary condition
Z
a · n dS = 0,
(flux)
∂Ω
then the steady Navier–Stokes system



 −ν∆u + u · ∇ u + ∇p =



f
div u = 0
u = a
in Ω,
in Ω,
on ∂Ω
(NS)
admits at least one weak axially symmetric solution. Moreover,
if f and a are axially symmetric without swirl, then (NS) admits
at least one weak axially symmetric solution without swirl.
78
Steady Navier–Stokes equations: axially symmetric case
Remark. It is well know (see [Lad]) that under hypothesis of
above Theorem every weak solution u of problem (NS) is more
3,2
regular, i.e, u ∈ W 2,2(Ω) ∩ Wloc (Ω).
79
Steady Navier–Stokes equations:
axially symmetric spatial case in exterior domain
Consider the Navier–Stokes problem


−ν∆u + u · ∇ u + ∇p = f






div u = 0







lim
|x|→+∞
u = a
u(x) = u0
in Ω,
in Ω,
on ∂Ω,
(N S)
in the exterior domain of R3
Ω = R3 \
N
S
j=1
Ω̄j ,
(∗)
where Ωi are bounded domains with connected C 2-smooth boundaries
Γi and Ω̄j ∩ Ω̄i = ∅ for i 6= j.
80
Consider the Navier–Stokes problem


−ν∆u + u · ∇ u + ∇p = f






div u = 0







lim
|x|→+∞
in the exterior domain
in Ω,
in Ω,
on ∂Ω,
u = a
u(x) = u0
Ω = R3 \
N
S
Ω̄j ,
(N S)
(*) where Ωi are
j=1
bounded domains with connected C 2-smooth boundaries Γi and
Ω̄j ∩ Ω̄i = ∅ for i 6= j.
Theorem 10 [KPR2013].
Assume that Ω ⊂ R3 is an exterior
axially symmetric domain of type (∗) with C 2-smooth boundary
∂Ω, and u0 ∈ R3, f ∈ W 1,2(Ω), a ∈ W 3/2,2(∂Ω) are axially symmetric.
Then (N S) admits at least one weak axially symmetric solution u
R
satisfying Ω |∇u|2 dx < +∞.
81
Consider the Navier–Stokes problem


−ν∆u + u · ∇ u + ∇p = f






div u = 0







lim
|x|→+∞
in the exterior domain
in Ω,
in Ω,
on ∂Ω,
u = a
u(x) = u0
Ω = R3 \
N
S
Ω̄j ,
(N S)
(*) where Ωi are
j=1
bounded domains with connected C 2-smooth boundaries Γi and
Ω̄j ∩ Ω̄i = ∅ for i 6= j.
Theorem 10 [KPR2013].
Assume that Ω ⊂ R3 is an exterior
axially symmetric domain of type (∗) with C 2-smooth boundary
∂Ω, and u0 ∈ R3, f ∈ W 1,2(Ω), a ∈ W 3/2,2(∂Ω) are axially symmetric.
Then (N S) admits at least one weak axially symmetric solution u
R
satisfying Ω |∇u|2 dx < +∞. Moreover, if
a and f are axially symmetric
with no swirl, then the above solution u has no swirl too.
82
Steady Navier–Stokes equations:
axially symmetric plane case in exterior domain
We will say that a domain Ω ⊂ R2 and a function g = (g1, g2)
defined on Ω are symmetric, if
(x1, x2) ∈ Ω ⇔ (x1, −x2) ∈ Ω
(sym1)
and g1 is an even function of x2 while g2 is an odd function of
x2 :
g1(x1, x2) = g1(x1, −x2),
g2(x1, x2) = −g2(x1, −x2)
(sym2)
(such solutions were considered by Amick).
83
Axially symmetric plane case in exterior domain
Consider the Navier–Stokes problem



 −ν∆u + u · ∇ u + ∇p =
f
div u = 0
u = a



in Ω,
in Ω,
on ∂Ω,
(N S)
in the symmetric exterior domain of R2
Ω = R2 \
N
S
j=1
Ω̄j ,
(dom)
where Ωi are bounded domains with connected Lipschitz boundaries
Γi and Ω̄j ∩ Ω̄i = ∅ for i 6= j.
84
Axially symmetric plane case in exterior domain



 −ν∆u + u · ∇ u + ∇p =
f
div u = 0
u = a



in Ω,
in Ω,
on ∂Ω.
(N S)
Theorem 11 [KPR2013]. Let Ω ⊂ R2 be a symmetric exterior
domain with multiply connected Lipschitz boundary ∂Ω consisting
of N disjoint components Γj with Γj ∩Ox1 6= ∅, j = 1, . . . , N . Assume
that f is a symmetric distribution such that the corresponding
linear functional H(Ω) ∋ η 7→
R
f · η is continuous (with respect
Ω
to the norm k · kH(Ω)), and a is a symmetric field in W 1/2,2(∂Ω).
Then problem (NS) admits at least one symmetric weak solution
u with
2
k∇uk2
≤
c
k
a
k
2
L (Ω)
W 1/2,2 (∂Ω)
+ kak4
W 1/2,2(∂Ω)
+ kf k2
∗ .
85
Sketch of the proofs (bounded plane domain)
The proofs are based on the classical “reductio ad absurdum”
Leray argument. Namely, if we suppose, that the initial plane
(NS) problem has no weak solution, then by the Leray–Schauder
theorem we obtain that there exist sequences of functions uk ∈
W 1,2(Ω), pk ∈ W 1,q (Ω) and numbers νk → 0+, λk → λ0 > 0 such
that the norms kuk kW 1,2(Ω), kpk kW 1,q (Ω) are uniformly bounded for
every q ∈ [1, 2), the pairs (uk , pk ) satisfy



 −νk ∆uk + uk · ∇ uk + ∇pk =



div uk
uk
λk νk2
with fk = ν 2 f ,
k∇uk kL2(Ω) → 1,
ak =
λk νk
ν
fk
= 0
= ak
in Ω,
in Ω,
on ∂Ω
(NSk )
a, and
uk ⇀ v in W 1,2(Ω),
pk ⇀ p in W 1,q (Ω)
∀ q ∈ [1, 2),
86
Sketch of the proofs (bounded plane domain)
k∇uk kL2(Ω) → 1,
uk ⇀ v in W 1,2(Ω),
pk ⇀ p in W 1,q (Ω)
∀ q ∈ [1, 2),
where the limiting functions (v, p) satisfy the Euler system
 

 v · ∇ v + ∇p = 0



div v = 0
v = 0
in Ω,
in Ω,
on ∂Ω.
(E)
87
Sketch of the proofs (bounded plane domain)
k∇uk kL2(Ω) → 1,
uk ⇀ v in W 1,2(Ω),
pk ⇀ p in W 1,q (Ω)
∀ q ∈ [1, 2),
where the limiting functions (v, p) satisfy the Euler system
 

 v · ∇ v + ∇p = 0



div v = 0
v = 0
in Ω,
in Ω,
on ∂Ω.
(E)
For this limit solution to the Euler equations we can use the
above version of the Bernoulli Law, etc.
88
Sketch of the proofs (bounded plane domain)
k∇uk kL2(Ω) → 1,
uk ⇀ v in W 1,2(Ω),
pk ⇀ p in W 1,q (Ω)
∀ q ∈ [1, 2),
where the limiting functions (v, p) satisfy the Euler system
 

 v · ∇ v + ∇p = 0



div v = 0
v = 0
in Ω,
in Ω,
on ∂Ω.
(E)
For this limit solution to the Euler equations we can use the
above version of the Bernoulli Law, etc. Our purpose — to get
a contradiction.
89
We consider the case of the plane annulus-type domain Ω =
Ω0 \ Ω̄1 with the boundary ∂Ω = Γ0 ∪ Γ1.
Since for the limit solution to the Euler system v|∂Ω ≡ ∇⊥ψ|∂Ω ≡ 0,
by Morse-Sard Theorem we have ∃ξj ∈ R such that ψ|Γj = ξj ,
j = 0, 1. Then by Bernoulli Law ∃p̂j ∈ R such that Φ|Γj ≡ p|Γj ≡ p̂j ,
j = 0, 1.
90
We consider the case of the plane annulus-type domain Ω =
Ω0 \ Ω̄1 with the boundary ∂Ω = Γ0 ∪ Γ1.
Since for the limit solution to the Euler system v|∂Ω ≡ ∇⊥ψ|∂Ω ≡ 0,
by Morse-Sard Theorem we have ∃ξj ∈ R such that ψ|Γj = ξj ,
j = 0, 1. Then by Bernoulli Law ∃p̂j ∈ R such that Φ|Γj ≡ p|Γj ≡ p̂j ,
j = 0, 1.
Taking a limit in the Navier-Stokes systems (NSk ), one have a
well-known identity.
Z
1
1
X
X
ν
=
−
pbj a · n ds =
pbj Fj .
λ0
j=0 Γ
j=0
j
(lim1)
91
We consider the case of the plane annulus-type domain Ω =
Ω0 \ Ω̄1 with the boundary ∂Ω = Γ0 ∪ Γ1.
Since for the limit solution to the Euler system v|∂Ω ≡ ∇⊥ψ|∂Ω ≡ 0,
by Morse-Sard Theorem we have ∃ξj ∈ R such that ψ|Γj = ξj ,
j = 0, 1. Then by Bernoulli Law ∃p̂j ∈ R such that Φ|Γj ≡ p|Γj ≡ p̂j ,
j = 0, 1.
Taking a limit in the Navier-Stokes systems (NSk ), one have a
well-known identity.
Z
1
1
X
X
ν
−
=
pbj a · n ds =
pbj Fj .
λ0
j=0 Γ
j=0
j
(lim1)
So if F0 = F1 = 0 (the Leray case) or pb0 = pb1, we immediately
obtain the required contradiction. Thus, assume that pb0 < pb1 = 0.
92
Take a number t0 ∈ (0, −pb0) such that the sequence ti = 2−it0
has the following properties. For each ti there exists a C 1-cycle
(i.e., a C 1-curve homeomorphic to the circle S1) Ai such that Ai
separates Γ0 from Γ1, ψ|Ai ≡ const, Φ|Ai ≡ −ti,
93
Take a number t0 ∈ (0, −pb0) such that the sequence ti = 2−it0
has the following properties. For each ti there exists a C 1-cycle
(i.e., a C 1-curve homeomorphic to the circle S1) Ai such that Ai
separates Γ0 from Γ1, ψ|Ai ≡ const, Φ|Ai ≡ −ti, Ai+1 lies between
Ai and Γ1,
94
Take a number t0 ∈ (0, −pb0) such that the sequence ti = 2−it0
has the following properties. For each ti there exists a C 1-cycle
(i.e., a C 1-curve homeomorphic to the circle S1) Ai such that Ai
separates Γ0 from Γ1, ψ|Ai ≡ const, Φ|Ai ≡ −ti, Ai+1 lies between
Ai and Γ1, and
Φk |Ai ⇉ Φ|Ai = −ti.
(1)
95
Take a number t0 ∈ (0, −pb0) such that the sequence ti = 2−it0
has the following properties. For each ti there exists a C 1-cycle
(i.e., a C 1-curve homeomorphic to the circle S1) Ai such that Ai
separates Γ0 from Γ1, ψ|Ai ≡ const, Φ|Ai ≡ −ti, Ai+1 lies between
Ai and Γ1, and
Φk |Ai ⇉ Φ|Ai = −ti.
(1)
The existence of such ti , Ai follows from the regularity properties
of the level sets of the stream function ψ obtained by the MorseSard theorem.
96
Take a number t0 ∈ (0, −pb0) such that the sequence ti = 2−it0
has the following properties. For each ti there exists a C 1-cycle
(i.e., a C 1-curve homeomorphic to the circle S1) Ai such that Ai
separates Γ0 from Γ1, ψ|Ai ≡ const, Φ|Ai ≡ −ti, Ai+1 lies between
Ai and Γ1, and
Φk |Ai ⇉ Φ|Ai = −ti.
(1)
From (1) we have that ∀i ∃ki such that ∀k ≥ ki
7
Φk |Ai < − ti,
8
Φk |Ai+1
5
> − ti .
8
97
Take a number t0 ∈ (0, −pb0) such that the sequence ti = 2−it0
has the following properties. For each ti there exists a C 1-cycle
(i.e., a C 1-curve homeomorphic to the circle S1) Ai such that Ai
separates Γ0 from Γ1, ψ|Ai ≡ const, Φ|Ai ≡ −ti, Ai+1 lies between
Ai and Γ1, and
Φk |Ai ⇉ Φ|Ai = −ti.
(1)
From (1) we have that ∀i ∃ki such that ∀k ≥ ki
7
Φk |Ai < − ti,
8
Φk |Ai+1
5
> − ti .
8
Then
h5
7 i
∀t ∈ ti , ti ∀k ≥ ki
8 8
Φk |Ai < −t,
Φk |Ai+1 > −t.
98
Take a number t0 ∈ (0, −pb0) such that the sequence ti = 2−it0
has the following properties. For each ti there exists a C 1-cycle
(i.e., a C 1-curve homeomorphic to the circle S1) Ai such that Ai
separates Γ0 from Γ1, ψ|Ai ≡ const, Φ|Ai ≡ −ti, Ai+1 lies between
Ai and Γ1, and
Φk |Ai ⇉ Φ|Ai = −ti.
(1)
From (1) we have that ∀i ∃ki such that ∀k ≥ ki
7
Φk |Ai < − ti,
8
Φk |Ai+1
5
> − ti .
8
Then
h5
7 i
∀t ∈ ti , ti ∀k ≥ ki
8 8
Φk |Ai < −t,
Φk |Ai+1 > −t.
h
i
5
7
So for k ≥ ki and for almost all t ∈ 8 ti , 8 ti there exists a C 1-cycle
Sik (t) which separates Ai and Ai+1 with Φk |Sik (t) ≡ −t.
99
Take a number t0 ∈ (0, −pb0) such that the sequence ti = 2−it0
has the following properties. For each ti there exists a C 1-cycle
(i.e., a C 1-curve homeomorphic to the circle S1) Ai such that Ai
separates Γ0 from Γ1, ψ|Ai ≡ const, Φ|Ai ≡ −ti, Ai+1 lies between
Ai and Γ1, and
Φk |Ai ⇉ Φ|Ai = −ti.
(1)
From (1) we have that ∀i ∃ki such that ∀k ≥ ki
7
Φk |Ai < − ti,
8
5
Φk |Ai+1 > − ti .
8
Then
h5
7 i
∀t ∈ ti , ti ∀k ≥ ki
8 8
Φk |Ai < −t,
Φk |Ai+1 > −t.
h
i
5
7
So for k ≥ ki and for almost all t ∈ 8 ti , 8 ti there exists a C 1-cycle
Sik (t) which separates Ai and Ai+1 with Φk |Sik (t) ≡ −t. Moreover,
the gradient of Φk is directed toward Γ1.
100
h
7t
So for k ≥ ki and for almost all t ∈ 5
t
,
i
8
8 i
i
there exists a C 1-cycle
Sik (t) which separates Ai and Ai+1 with Φk |Sik (t) ≡ −t. Moreover,
the gradient of Φk is directed toward Γ1.
Fk < -t
G0
F k = -t
n
ÑF k
Fk > -t
Ai0
Gh
G1
W ik (t )
k
Ai0+1
Sik (t )
Figure 1. The case of an annulus–like domain (N = 1).
101
h
7t
So for k ≥ ki and for almost all t ∈ 5
t
,
i
8
8 i
i
there exists a C 1-cycle
Sik (t) which separates Ai and Ai+1 with Φk |Sik (t) ≡ −t. Moreover,
the gradient of Φk is directed toward Γ1.
The key step (in deriving a contradiction) is the following estimate.
Lemma 1. For any i ∈ N there exists k(i) ∈ N such that the
inequality
Z
|∇Φk | ds < F t
(∗)
Sik (t)
7 t ], where the
holds for every k ≥ k(i) and for almost all t ∈ [ 5
t
,
i
8
8 i
constant F is independent of t, k and i.
102
h
7t
So for k ≥ ki and for almost all t ∈ 5
t
,
i
8
8 i
i
there exists a C 1-cycle
Sik (t) which separates Ai and Ai+1 with Φk |Sik (t) ≡ −t. Moreover,
the gradient of Φk is directed toward Γ1.
The key step (in deriving a contradiction) is the following estimate.
Lemma 1. For any i ∈ N there exists k(i) ∈ N such that the
inequality
Z
|∇Φk | ds < F t
(∗)
Sik (t)
7 t ], where the
holds for every k ≥ k(i) and for almost all t ∈ [ 5
t
,
i
8
8 i
constant F is independent of t, k and i.
The estimate (*) leads to a required contradiction easily.
103
Z
Sik (t)
|∇Φk | ds < F t
5 7
∀k ≥ k(i) for a.a. t ∈ [ ti , ti]
8 8
(∗)
104
Z
Sik (t)
|∇Φk | ds < F t
5 7
∀k ≥ k(i) for a.a. t ∈ [ ti , ti]
8 8
For i ∈ N and k ≥ k(i) put Ei =
S
7
t∈[ 5
8 ti , 8 ti ]
(∗)
Sik (t).
105
Z
|∇Φk | ds < F t
Sik (t)
5 7
∀k ≥ k(i) for a.a. t ∈ [ ti , ti]
8 8
For i ∈ N and k ≥ k(i) put Ei =
formula,
Z
Ei
|∇Φk |2 dx =
5t
Z8 iZ
7t
8 i
Sik (t)
S
7
t∈[ 5
8 ti , 8 ti ]
(∗)
Sik (t). By the Coarea
|∇Φk |(x) dH1(x) dt ≤
5t
Z8 i
F t dt = F ′t2
i
7t
8 i
3 F is independent of i.
where F ′ = 16
106
Z
5 7
∀k ≥ k(i) for a.a. t ∈ [ ti , ti]
8 8
|∇Φk | ds < F t
Sik (t)
For i ∈ N and k ≥ k(i) put Ei =
formula,
Z
|∇Φk |2 dx =
5t
Z8 iZ
7t
8 i
Ei
Sik (t)
S
7
t∈[ 5
8 ti , 8 ti ]
(∗)
Sik (t). By the Coarea
|∇Φk |(x) dH1(x) dt ≤
5t
Z8 i
F t dt = F ′t2
i
7t
8 i
3 F is independent of i. Now, applying again the
where F ′ = 16
Coarea formula and using the Hölder inequality, we have
5t
Z8 i
7t
8 i
1
H
Sik (t) dt =
Z
Ei
|∇Φk | dx ≤
Z
Ei
2
|∇Φk | dx
!1
2
1
meas(Ei)
2
≤
q
′
F ti meas(E
(1)
107
5t
Z8 i
7t
8 i
1
H
Sik (t) dt ≤
q
′
1
F ti meas(Ei)
2
.
(1)
108
5t
Z8 i
7t
8 i
1
H
Sik (t) dt ≤
q
′
1
F ti meas(Ei)
2
.
(1)
7 t ] the set S (t) is a
By construction, for almost all t ∈ [ 5
t
,
i
ik
8
8 i
cycle separating Ai from Ai+1. Thus, each set Sik (t) separates
1
Γ0 from Γ1. In particular, H (Sik (t)) ≥ min diam(Γ0), diam(Γ1) .
Hence, the left integral in (1) is greater than Cti, where C > 0
does not depend on i. On the other hand, evidently, meas(Ei) → 0
as i → ∞ (since Ei lies between cycles Ai, Ai+1, and the sequence
of these cycles is monotone).
109
So we demonstrate how to obtain a contradiction from the above
mentioned estimate
Z
Sik (t)
|∇Φk | ds < F t
5 7
∀k ≥ k(i) for a.a. t ∈ [ ti , ti]
8 8
(∗)
Thus, to finish the proof, we have to check (*).
110
So we demonstrate how to obtain a contradiction from the above
mentioned estimate
Z
Sik (t)
|∇Φk | ds < F t
5 7
∀k ≥ k(i) for a.a. t ∈ [ ti , ti]
8 8
(∗)
Thus, to finish the proof, we have to check (*).
To establish (*), we need to integrate the classical identity
1
1
div(Φk uk ) − fk · uk
∆Φk = ωk2 +
ν
ν
k
k
over a subdomain bounded by curves Sik (t) and Γhk = {x ∈ Ω :
dist(x, Γ1) < hk for a suitable small hk ,
111
So we demonstrate how to obtain a contradiction from the above
mentioned estimate
Z
|∇Φk | ds < F t
Sik (t)
5 7
∀k ≥ k(i) for a.a. t ∈ [ ti , ti]
8 8
(∗)
Thus, to finish the proof, we have to check (*).
To establish (*), we need to integrate the classical identity
1
1
div(Φk uk ) − fk · uk
∆Φk = ωk2 +
ν
ν
k
k
over a subdomain bounded by curves Sik (t) and Γhk = {x ∈ Ω :
dist(x, Γ1) < hk for a suitable small hk , and to use the smallness
of the boundary data
kuk kL2(∂Ω) ∼ νk → 0
as k → ∞.
112
Sketch of the proofs (axially symmetric spatial exterior domain)
The proofs are based on the classical “reductio ad absurdum”
Leray argument. Namely, consider the sequence of problems



 −ν∆u + u · ∇ u + ∇p =



f
div u = 0
u = a
in Ωk ,
in Ωk ,
on ∂Ωk ,
(N Sk )
where Ω is the axially symmetric exterior domain in R3 and
Ωk = Ω ∩ B(0, k) = {x ∈ Ω : |x| < k}.
We know from the results for bounded domains that (N Sk ) has a
b k . We have to prove that Jk = k∇u
b kk 2
solution u
L (Ω) are uniformly
b k , by J 2
bounded. Suppose it is false. Then divide (N Sk ), u = u
k
and obtain
113



 −νk ∆uk + uk · ∇ uk + ∇pk =



div uk
uk
with νk = J1 k → 0,
k∇uk kL2(Ω) ≡ 1,
fk
= 0
= ak
νk2
1
b k , fk = 2 f ,
uk = J u
ν
k
uk ⇀ v in W 1,2(Ω),
ak =
in Ωk ,
in Ωk ,
on ∂Ωk
νk
ν
(NSk )
a, and
pk ⇀ p in W 1,q (Ω)
∀ q ∈ [1, 2)
114



 −νk ∆uk + uk · ∇ uk + ∇pk =
div uk
uk



with νk = J1 k → 0,
k∇uk kL2(Ω) ≡ 1,
in Ωk ,
in Ωk ,
on ∂Ωk
fk
= 0
= ak
νk2
1
b k , fk = 2 f ,
uk = J u
ν
k
uk ⇀ v in W 1,2(Ω),
ak =
νk
ν
(NSk )
a, and
pk ⇀ p in W 1,q (Ω)
∀ q ∈ [1, 2)
where the limiting functions (v, p) satisfy the Euler system
 

 v · ∇ v + ∇p = 0



div v = 0
v = 0
in Ω,
in Ω,
on ∂Ω.
(E)
115



 −νk ∆uk + uk · ∇ uk + ∇pk =
div uk
uk



with νk = J1 k → 0,
k∇uk kL2(Ω) ≡ 1,
in Ωk ,
in Ωk ,
on ∂Ωk
fk
= 0
= ak
νk2
1
b k , fk = 2 f ,
uk = J u
ν
k
uk ⇀ v in W 1,2(Ω),
ak =
νk
ν
(NSk )
a, and
pk ⇀ p in W 1,q (Ω)
∀ q ∈ [1, 2)
where the limiting functions (v, p) satisfy the Euler system
 

 v · ∇ v + ∇p = 0



div v = 0
v = 0
in Ω,
in Ω,
on ∂Ω.
(E)
For this limit solution to the Euler equations we can use the
above version of the Bernoulli Law, etc.
116



 −νk ∆uk + uk · ∇ uk + ∇pk =
div uk
uk



with νk = J1 k → 0,
k∇uk kL2(Ω) ≡ 1,
in Ωk ,
in Ωk ,
on ∂Ωk
fk
= 0
= ak
νk2
1
b k , fk = 2 f ,
uk = J u
ν
k
uk ⇀ v in W 1,2(Ω),
ak =
νk
ν
(NSk )
a, and
pk ⇀ p in W 1,q (Ω)
∀ q ∈ [1, 2)
where the limiting functions (v, p) satisfy the Euler system
 

 v · ∇ v + ∇p = 0



div v = 0
v = 0
in Ω,
in Ω,
on ∂Ω.
(E)
For this limit solution to the Euler equations we can use the
above version of the Bernoulli Law, etc. We need to prove
that the above properties of the systems (E)-(NSk ) lead to a
contradiction.
117
We consider the case of the exterior domain Ω = R3 \ (Ω̄0 ∪ Ω̄1)
with the boundary ∂Ω = Γ0 ∪ Γ1, Γ0 ∩ Ox3 6= ∅, Γ1 ∩ Ox3 = ∅, where,
recall, Ox3 is an axis of symmetry.
Since for the limit solution to the Euler system v|∂Ω ≡ ∇⊥ψ|∂Ω ≡ 0,
it is well-known ([Amick],[Kapitanskii-Pileckas],etc.) that ∃p̂j ∈ R
such that
Φ|Γj ≡ p|Γj ≡ p̂j ,
j = 0, 1.
118
Sketch of the proofs
Φ|Γj ≡ p|Γj ≡ p̂j ,
j = 0, 1.
Taking a limit in the Navier-Stokes systems (NSk ), one have a
well-known identity.
−ν =
1
X
j=0
pbj
Z
Γj
a · n dS =
1
X
j=0
pbj Fj .
(lim1)
119
Sketch of the proofs
Φ|Γj ≡ p|Γj ≡ p̂j ,
j = 0, 1.
Taking a limit in the Navier-Stokes systems (NSk ), one have a
well-known identity.
−ν =
1
X
j=0
pbj
Z
Γj
a · n dS =
1
X
j=0
pbj Fj .
(lim1)
So if F0 = F1 = 0 (the Leray case) or pb0 = pb1, we immediately
obtain the required contradiction. Thus, assume that pb0 > pb1.
120
Sketch of the proofs
Φ|Γj ≡ p|Γj ≡ p̂j ,
j = 0, 1.
Taking a limit in the Navier-Stokes systems (NSk ), one have a
well-known identity.
−ν =
1
X
j=0
pbj
Z
Γj
a · n dS =
1
X
j=0
pbj Fj .
(lim1)
So if F0 = F1 = 0 (the Leray case) or pb0 = pb1, we immediately
obtain the required contradiction. Thus, assume that pb0 > pb1.
First of all we proved that pb0 = 0, i.e., pb0 coincides with the
value of p at infinity (we use here the fact that the symmetry
axis Ox3 can be approximated by stream lines where Φ = const).
Thus, below 0 = pb0 > pb1.
121
Take numbers t1, t2 ∈ (pb1, 0), t1 < t2 such that there exist C 1-tori
Ai, i = 1, 2 with the following properties: Ai separates Γ1 from Γ0
and from infinity,
122
Take numbers t1, t2 ∈ (pb1, 0), t1 < t2 such that there exist C 1-tori
Ai, i = 1, 2 with the following properties: Ai separates Γ1 from Γ0
and from infinity, the identities ψ|Ai ≡ const, Φ|Ai ≡ ti hold,
123
Take numbers t1, t2 ∈ (pb1, 0), t1 < t2 such that there exist C 1-tori
Ai, i = 1, 2 with the following properties: Ai separates Γ1 from Γ0
and from infinity, the identities ψ|Ai ≡ const, Φ|Ai ≡ ti hold, and
Φk |Ai ⇉ Φ|Ai = ti .
(1)
124
Take numbers t1, t2 ∈ (pb1, 0), t1 < t2 such that there exist C 1-tori
Ai, i = 1, 2 with the following properties: Ai separates Γ1 from Γ0
and from infinity, the identities ψ|Ai ≡ const, Φ|Ai ≡ ti hold, and
Φk |Ai ⇉ Φ|Ai = ti .
(1)
The existence of such Ai follows from the regularity properties
of the level sets of the stream function ψ obtained by the MorseSard theorem.
125
Take numbers t1, t2 ∈ (pb1, 0), t1 < t2 such that there exist C 1-tori
Ai, i = 1, 2 with the following properties: Ai separates Γ1 from Γ0
and from infinity, the identities ψ|Ai ≡ const, Φ|Ai ≡ ti hold, and
Φk |Ai ⇉ Φ|Ai = ti .
(1)
Let t1 < t′ < t′′ < t2. From (1) we have that ∃k0 = k0(t1, t2, t′, t′′)
such that ∀k ≥ k0 Φk |A1 < t′,
Φk |A2 > t′′.
126
Take numbers t1, t2 ∈ (pb1, 0), t1 < t2 such that there exist C 1-tori
Ai, i = 1, 2 with the following properties: Ai separates Γ1 from Γ0
and from infinity, the identities ψ|Ai ≡ const, Φ|Ai ≡ ti hold, and
Φk |Ai ⇉ Φ|Ai = ti .
(1)
Let t1 < t′ < t′′ < t2. From (1) we have that ∃k0 = k0(t1, t2, t′, t′′)
such that ∀k ≥ k0 Φk |A1 < t′,
Φk |A2 > t′′. Then
h
′
′′
∀t ∈ t , t
i
∀k ≥ k0
Φk |A1 < t,
Φk |A2 > t.
127
Take numbers t1, t2 ∈ (pb1, 0), t1 < t2 such that there exist C 1-tori
Ai, i = 1, 2 with the following properties: Ai separates Γ1 from Γ0
and from infinity, the identities ψ|Ai ≡ const, Φ|Ai ≡ ti hold, and
Φk |Ai ⇉ Φ|Ai = ti .
(1)
Let t1 < t′ < t′′ < t2. From (1) we have that ∃k0 = k0(t1, t2, t′, t′′)
such that ∀k ≥ k0 Φk |A1 < t′,
Φk |A2 > t′′. Then
h
′
′′
∀t ∈ t , t
i
∀k ≥ k0
Φk |A1 < t,
h
Φk |A2 > t.
i
So for k ≥ k0 and for almost all t ∈ t′ , t′′ there exists a C 1-torus
Sk (t) which separates A1 and from A2 with Φk |Sk (t) ≡ t.
128
Take numbers t1, t2 ∈ (pb1, 0), t1 < t2 such that there exist C 1-tori
Ai, i = 1, 2 with the following properties: Ai separates Γ1 from Γ0
and from infinity, the identities ψ|Ai ≡ const, Φ|Ai ≡ ti hold, and
Φk |Ai ⇉ Φ|Ai = ti .
(1)
Let t1 < t′ < t′′ < t2. From (1) we have that ∃k0 = k0(t1, t2, t′, t′′)
such that ∀k ≥ k0 Φk |A1 < t′,
Φk |A2 > t′′. Then
h
′
′′
∀t ∈ t , t
i
∀k ≥ k0
Φk |A1 < t,
h
Φk |A2 > t.
i
So for k ≥ k0 and for almost all t ∈ t′ , t′′ there exists a C 1-torus
Sk (t) which separates A1 and from A2 with Φk |Sk (t) ≡ t.
Moreover, the gradient of Φk is directed toward Γ0.
129
h
So for k ≥ k0 and for almost all t ∈ t′ , t′′
i
there exists a C 1-torus
Sk (t) which separates A1 and from A2 with Φk |Sk (t) ≡ t. Moreover,
the gradient of Φk is directed toward Γ0.
The key step (in deriving a contradiction) is the following estimate.
130
h
So for k ≥ k0 and for almost all t ∈ t′ , t′′
i
there exists a C 1-torus
Sk (t) which separates A1 and from A2 with Φk |Sk (t) ≡ t. Moreover,
the gradient of Φk is directed toward Γ0.
The key step (in deriving a contradiction) is the following estimate.
Lemma 1. There exists k1 ∈ N such that the inequality
Z
|∇Φk | dS < F |t|
(∗)
Sk (t)
holds for every k ≥ k1 and for almost all t ∈ [t′ , t′′], where the
constant F is independent of t, k.
131
h
So for k ≥ k0 and for almost all t ∈ t′ , t′′
i
there exists a C 1-torus
Sk (t) which separates A1 and from A2 with Φk |Sk (t) ≡ t. Moreover,
the gradient of Φk is directed toward Γ0.
The key step (in deriving a contradiction) is the following estimate.
Lemma 1. There exists k1 ∈ N such that the inequality
Z
|∇Φk | dS < F |t|
(∗)
Sk (t)
holds for every k ≥ k1 and for almost all t ∈ [t′ , t′′], where the
constant F is independent of t, k.
The estimate (*) leads to a required contradiction easily (using
the Coarea formula, isoperimetric inequality, etc.).
132
Sketch of the proofs
Z
|∇Φk | dS < F |t|
∀k > k1, t ∈ [t′ , t′′]
(∗)
Sk (t)
Thus, to finish the proof, we have to check (*).
133
Sketch of the proofs
Z
|∇Φk | dS < F |t|
∀k > k1, t ∈ [t′ , t′′]
(∗)
Sk (t)
Thus, to finish the proof, we have to check (*).
To establish (*), we need to integrate the classical identity
1
1
2
∆Φk = ωk +
div(Φk uk ) − fk · uk
νk
νk
over a subdomain Ωk (t) with Ωk (t) with ∂Ωk (t) ⊃ Sk (t) such that
the corresponding boundary integrals
Z
∇Φk · n dS ,
∂Ωk (t)\Sk (t)
are negligible.
Z
1
Φk uk dS νk ∂Ωk (t)\Sk (t)
134
Sketch of the proofs
Z
|∇Φk | dS < F |t|
∀k > k1, t ∈ [t′ , t′′]
(∗)
Sk (t)
Thus, to finish the proof, we have to check (*).
To establish (*), we need to integrate the classical identity
1
1
2
div(Φk uk ) − fk · uk
∆Φk = ωk +
νk
νk
over a subdomain Ωk (t) with Ωk (t) with ∂Ωk (t) ⊃ Sk (t) such that
the corresponding boundary integrals
Z
∇Φk · n dS ,
∂Ωk (t)\Sk (t)
Z
1
Φk uk dS νk ∂Ωk (t)\Sk (t)
are negligible. It is possible to construct such Ωk (t) since kuk kL2(∂Ωk ) ∼
νk → 0 as k → ∞.
135
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