18 March 2016
The dimension of singular sets in the 3D incompressible
Navier-Stokes equations and in a related 1D surface
growth model
Wojciech Ożański
5th Annual CCA - MASDOC - OxPDE Conference,
Cambridge, 18 March 2016
Wojciech Ożański
Singular sets in NSE
Cambridge, 18 March 2016
1 / 14
Introduction
Let K ⊂ Rn . What is the dimension of K?
1) Hausdorff dimension
For δ > 0 let
(
Hδs (K)
:= inf
∞
∞
X
[
(diam Ui )s : K ⊂
Ui , diam Ui ≤ δ
i=1
)
.
i=1
and
H s (K) := lim Hδs (K)
δ→0
← this is s-dimensional Hausdorff measure
The Hausdorff dimension is
dH (K) := inf {s ≥ 0 : H s (K) = 0} .
Wojciech Ożański
Singular sets in NSE
Cambridge, 18 March 2016
2 / 14
Introduction
2) Box-counting dimension
dB (K) := lim sup
ε→0
log M (K, ε)
,
− log ε
where M (K, ε) is the maximal number of disjoint ε-ball with centers in K.
In particular if d < dB (K) then there exists a sequence εj → 0 such that
M (K, εj ) > ε−d
j .
3) Assouad dimension
K is called to be (C, s)-homogeneous iff for any R-ball BR one can cover
K ∩ BR by at most C(R/ρ)s many ρ-balls.
dA (K) := inf {s : K is (C, s)-homogeneous for some C ≥ 1}
Wojciech Ożański
Singular sets in NSE
Cambridge, 18 March 2016
3 / 14
Introduction
Examples:
1) If K = [0, 1]m then dH (K) = dB (K) = dA (K) = m,
2) For any compact K ⊂ Rn we have
dH (K) ≤ dB (K) ≤ dA (K),
∞
3) If K = {0} ∪ {1/m}m=1 then dH (K) = 0, dB (K) = 1/2, dA (K) = 1,
4) For any ξ ∈ (0, 1) there exists a ”Cantor-like” set C ⊂ R such that
dH (C) = dB (C) = dA (C) = ξ.
Wojciech Ożański
Singular sets in NSE
Cambridge, 18 March 2016
4 / 14
Introduction
The 3D incompressible Navier-Stokes equations are
ut − ∆u + (u · ∇)u + ∇p = 0,
div u = 0,
u(0) = u0 ,
where (typically) u0 ∈ L2 or H 1 (on some domain, we will take T3 - the torus in
R3 ) and div u0 = 0.
The essentials:
1) If u0 ∈ H 1 then there exists a unique local smooth solution until at least
T := M k∇u0 k−4
L2 ,
2) If u0 ∈ L2 then there exists a (nonunique) global weak solution (understood
as a function u(x, t) such that u ∈ L∞ ((0, T ); L2 ) ∩ L2 ((0, T ); H 1 ) and u
satisfies the weak form of the Navier-Stokes equations (tested against
divergence free test functions))
If u is a weak solution then u ∈ L10/3 ((0, T ) × T3 ).
Wojciech Ożański
Singular sets in NSE
Cambridge, 18 March 2016
5 / 14
Introduction
Moreover, for u0 ∈ L2 there exists a global weak solution u such that the pair
(u, p) satisfies
a) −∆p = ∂xi ∂xj (ui uj ) for every t ∈ (0, T ),
b) p ∈ L5/3 ((0, T ) × T3 ),
c) the strong energy inequality
1
ku(t)kL2 +
2
ˆ
t
k∇uk2 ≤
s
1
ku(s)kL2
2
for a.e. s ≥ 0 and all t > s,
d) the local energy inequality
ˆ
ˆ tˆ
|u(t)|2 φ dx + 2
|∇u|2 φ dx ds
3
3
T
0
T
ˆ tˆ
ˆ tˆ
2
≤
|u| (φt + ∆φ)dx ds +
(|u|2 + 2p)u · ∇φ dx ds
0
T3
0
T3
for all t ∈ (0, T ) and for all φ ∈ Cc∞ ((0, T ) × T3 ) with φ ≥ 0.
This is called a suitable Leray-Hopf solution.
Wojciech Ożański
Singular sets in NSE
Cambridge, 18 March 2016
6 / 14
Partial regularity results
The parabolic cylinder Qr :
Qr (a, s) = (x, t) : |x − a| < r, t ∈ (s − r2 , s) .
Qr (a, s)
Qr/2 (a, s)
(a, s)
Wojciech Ożański
Singular sets in NSE
Cambridge, 18 March 2016
7 / 14
Partial regularity results
Partial regularity results, Navier-Stokes equations
Theorem (First Local Regularity Theorema )
Let (u, p) be a suitable Leray-Hopf solution of the Navier-Stokes equations. There
exists an ε0 > 0 such that whenever
ˆ 1
3
3/2
|u|
+
|p|
< ε0
r2 Qr
then u ∈ L∞ (Qr/2 ).
Let S denote the singular set of the (u, p) (i.e. (x, t) ∈ S ⇔ u(x, t) is unbounded
in any neighbourhood of (x, t)).
Corollary (First Partial Regularity Theorem)
For any compact K ⊂ (0, T ) × T3 we have dB (S ∩ K) ≤ 5/3.
(recall dB (S ∩ K) := lim supε→0 − log M (S ∩ K, ε)/ log ε)
a L. Caffarelli, R. Kohn, L. Nirenberg, Partial Regularity of Suitable Weak Solutions of the
Navier-Stokes Equations, Comm. Pure Appl. Math., Vol XXXV, 771-831 (1982)
Wojciech Ożański
Singular sets in NSE
Cambridge, 18 March 2016
8 / 14
Partial regularity results
Partial regularity results, Navier-Stokes equations
Theorem (Second Local Regularity Theorema )
Let (u, p) be a suitable Leray-Hopf solution of the Navier-Stokes equations on
some cylinder QR . There exists an ε1 > 0 such that whenever
ˆ
1
lim sup
|∇u|2 < ε1
r→0 r Qr
then u ∈ L∞ (Qρ ) for some ρ ∈ (0, R).
Corollary (Second Partial Regularity Theorem)
We have
dH (S) ≤ 1.
a L. Caffarelli, R. Kohn, L. Nirenberg, Partial Regularity of Suitable Weak Solutions of the
Navier-Stokes Equations, Comm. Pure Appl. Math., Vol XXXV, 771-831 (1982)
Wojciech Ożański
Singular sets in NSE
Cambridge, 18 March 2016
9 / 14
Surface growth model
The 1D surface growth equation is
ut + uxxxx + |ux |2
xx
= 0,
u(0) = u0 .
The essentials:
1) If u0 ∈ H 1 (0, 2π) then there exists a unique local smooth solution until some
time T > 0,
2) If u0 ∈ L2 (0, 2π) then there exists a (nonunique) global weak solution
(understood as a function u(x, t) such that
u ∈ L∞ ((0, T ); L2 ) ∩ L2 ((0, T ); H 2 ) and u satisfies the weak form of the
equation).
Moreover, there exists a global weak solution satisfying the local energy inequality
ˆ
ˆ t ˆ 2π
1 2π
2
|u(t)| φ dx +
u2xx φ dx ds
2 0
0
0
ˆ t ˆ 2π 5 3
1
2
2
2
≤
(φt + φxxxx )|u| + 2ux φxx − ux φx − ux u φxx dx ds.
2
3
0
0
Wojciech Ożański
Singular sets in NSE
Cambridge, 18 March 2016
10 / 14
Partial regularity results for surface growth model
”First local regularity”, surface growth model
Let Q(x, t, r) := (t − r4 , t) × (x − r, x + r).
Theorem (J. Robinson, D. Blömker, 2016a )
There exists an ε > 0 such that whenever
ˆ
1
|ux |3 < ε
r2 Q(x,t,r)
then u is Hölder continuous in some neighbourhood of (x, t).
Let S denote the singular set of a weak solution u of the surface growth model,
that satisfies the local energy inequality.
Corollary
For any compact K ⊂ (0, T ) × (0, 2π) we have dB (S ∩ K) ≤ 5/3.
a J. C. Robinson, D. Blömker, Partial regularity for weak solutions of a surface growth
model, preprint (2016)
Wojciech Ożański
Singular sets in NSE
Cambridge, 18 March 2016
11 / 14
Partial regularity results for surface growth model
”Second local regularity”, surface growth model
Theorem (J. Robinson, D. Blömker, 2016a )
There exists a δ > 0 such that whenever
ˆ
1
lim sup
u2xx < δ
r→0 r Q(x,t,r)
then u is regular at (x, t).
Corollary
We have
dH (S) ≤ 1.
a J. C. Robinson, D. Blömker, Partial regularity for weak solutions of a surface growth
model, preprint (2016)
Wojciech Ożański
Singular sets in NSE
Cambridge, 18 March 2016
12 / 14
Nearly one dimensional singular set
Theorem (V. Scheffer, 1987a )
Let ξ ∈ (0, 1). There exists a Cantor set S ⊂ R3 × {1} and functions
u : R3 × [0, ∞) → R3 and p : R3 × [0, ∞) → R such that
1) dH (S) = ξ,
2) there exists a compact set K ⊂ R3 such that supp u(t) ⊂ K for all t,
3) u(x, t) is a C ∞ function for each t,
4) there exists M > 0 such that ku(t)kL2 ≤ M for all t,
5) ∇u ∈ L2 ((0, ∞) × R3 ), u ∈ L3 ((0, ∞) × R3 ), |u||p| ∈ L1 ((0, ∞) × R3 ),
6) the local energy inequality
ˆ
ˆ tˆ
2
|∇u|2 φ dx ds
|u(t)| φ dx + 2
3
3
0
R
R
ˆ tˆ
ˆ tˆ
2
≤
|u| (φt + ∆φ)dx ds +
(|u|2 + 2p)u · ∇φ dx ds
0
R3
holds for all t ≥ 0 and for all φ ∈
0
Cc∞ ((0, ∞)
R3
3
× R ) with φ ≥ 0,
7) the singular set of u contains S.
a V. Scheffer, Nearly One Dimensional Singularities of Solutions to the
Navier-Stokes Inequality, Comm. Math. Phys. 110, 525-551 (1987)
Wojciech Ożański
Singular sets in NSE
Cambridge, 18 March 2016
13 / 14
Nearly one dimensional singular set
Directions:
1) Construct an example of a solution to the surface growth inequality (or a
weak solution to the equation) with a given dH (S) ∈ (0, 1),
2) Construct a shorter proof of the local regularity results for the surface growth
model and try showing boundedness in the half-cylinder (modifying the
approach of Caffarelli, Kohn, Nirenberg for Navier-Stokes equations),
3) Construct an example of a solution to the Navier-Stokes inequality with
dH (S) ≤ 1 and dB (S) > 1 (add concentration of blow-up times),
4) Establish any bounds on dA of the singular set S of a solution to the
Navier-Stokes equations.
Wojciech Ożański
Singular sets in NSE
Cambridge, 18 March 2016
14 / 14