Singularities in the three-dimensional incompressible Navier ......Navier-Stokes equations,...
Transcript of Singularities in the three-dimensional incompressible Navier ......Navier-Stokes equations,...
Singularities in the three-dimensional incompressibleNavier-Stokes equations
Wojciech Ozanski
Young Researchers in Mathematics 2016,
St. Andrews, 2nd August 2016
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Let K ⊂ Rn. What is the dimension of K?
1) Hausdorff dimensionFor δ > 0 let
H sδ (K) := inf
{ ∞∑i=1
(diamUi)s : K ⊂
∞⋃i=1
Ui, diamUi ≤ δ
}.
and
H s(K) := limδ→0
H sδ (K) ← this is s-dimensional Hausdorff measure
The Hausdorff dimension is
dH(K) := inf {s ≥ 0 : H s(K) = 0} .
2) Box-counting dimension (also fractal dimension or Minkowski dimension)
dB(K) := lim supε→0
logM(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−dj .
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Examples:
1) If K = [0, 1]m then dH(K) = dB(K) = m,
2) If K is countable then dH(K) = 0,
3) For any compact K ⊂ Rn we have
dH(K) ≤ dB(K),
4) If K = {0} ∪ {1/m}∞m=1 then dH(K) = 0, dB(K) = 1/2,
5) For any ξ ∈ (0, 1) there exists a ”Cantor-like” set C ⊂ R such thatdH(C) = dB(C) = ξ.
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The 3D incompressible Navier-Stokes equations are
ut −∆u+ (u · ∇)u+∇p = 0, div u = 0, u(0) = u0,
where (typically) u0 ∈ L2 or H1 (on some domain, we will take T3 - the torus inR3) and div u0 = 0.The essentials:
1) u0 ∈ H1 ⇒ a unique local strong solution until at least T0 := M‖∇u0‖−4,
2) u0 ∈ L2 ⇒ a (nonunique) global weak solution (understood as a functionu(x, t) such that u ∈ L∞((0, T );L2) ∩ L2((0, T );H1) and u satisfies theweak form of the Navier-Stokes equations (tested against divergence free testfunctions)) satisfying the strong energy inequality
1
2‖u(t)‖2 +
ˆ t
s
‖∇u‖2 ≤ 1
2‖u(s)‖2 for a.e. s ≥ 0 and every t > s.
Also, u is a weak solution ⇒ u ∈ L10/3((0, T )× T3).
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Some other results:
1) ‖u0‖H1 small enough ⇒ the global weak solution is strong,
2) u0 ∈ H1/2 or u0 ∈ L3 ⇒ local strong solution (H1/2, L3 are the criticalspaces for 3D NSE - i.e. the spaces with norms invariant under the rescalingu0(x) 7→ λu0(λx)),
3) A weak solution u is strong on (s, t) if either
(A) u ∈ Lq((s, t), Lr(T3)) for some q, r satisfying 2q
+ 3r≤ 1 (the Serrin
condition),
(B)´ t
s‖∇u‖L∞ <∞ or
(C)´ t
s‖curlu‖L∞ <∞ (the Beale-Kato-Majda condition),
4) If blow-up occurs at time t0 then ‖∇u(t)‖2 ≥ C√t0−t
.
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Let
S :={
(x, t) ∈ T3 × (0, T ) : u(x, t) is unbounded in any neighbourhood
of (x, t)} , the set of singular points,
St :={τ > 0 : ‖∇u(t)‖ → ∞ as t→ τ−
}, the set of singular times.
ThenSt = {τ > 0 : (x, τ) ∈ S for some x ∈ T3}.
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Theorem
dH(St) ≤ dB(St) ≤ 1/2.
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For u0 ∈ L2 there exists a global weak solution u such that the pair (u, p) satisfies
a) −∆p = ∂xi∂xj
(uiuj) for every t ∈ (0, T ),
b) p ∈ L5/3((0, T )× T3),
c) the strong energy inequality
1
2‖u(t)‖L2 +
ˆ t
s
‖∇u‖2 ≤ 1
2‖u(s)‖L2 for a.e. s ≥ 0 and all t > s,
d) the local energy inequality
ˆT3
|u(t)|2φdx+ 2
ˆ t
0
ˆT3
|∇u|2φ dxds
≤ˆ t
0
ˆT3
|u|2(φt + ∆φ)dx ds+
ˆ t
0
ˆT3
(|u|2 + 2p)u · ∇φ dx ds
for all t ∈ (0, T ) and for all φ ∈ C∞c ((0, T )× T3) with φ ≥ 0.
This is called a suitable weak solution.
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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)
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Theorem (Local Regularity of NSEa)
Let (u, p) be a suitable Leray-Hopf solution of the Navier-Stokes equations.
1) There exists an ε0 > 0 such that
1
r2
ˆQr
(|u|3 + |p|3/2
)< ε0 ⇒ u ∈ L∞(Qr/2),
2) There exists an ε1 > 0 such that
lim supr→0
1
r
ˆQr
|∇u|2 < ε1 ⇒ u ∈ L∞(Qρ) for some ρ ∈ (0, R).
Corollary (Partial Regularity of NSE)
For any compact K ⊂ (0, T )× T3 we have
dB(S ∩K) ≤ 5/3,
dH(S ∩K) ≤ 1.
a L. Caffarelli, R. Kohn, L. Nirenberg, Partial Regularity of Suitable Weak Solutions of theNavier-Stokes Equations, Comm. Pure Appl. Math., Vol XXXV, 771-831 (1982)
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Recent improvements of the Partial Regularity
Let (u, p) be a suitable weak solution on some time interval (0, T ) and let Sdenote its singular set.
Theorem
(Kukavica, (2009)a) dB(S) ≤ 135/82(≈ 1.65),
(Kukavica & Pei (2012)b) dB(S) ≤ 45/29(≈ 1.55),
(Koh & Yang (March 2016)c) dB(S) ≤ 95/63(≈ 1.51),
(Wang & Wu (April 2016)d) dB(S) ≤ 180/131(≈ 1.37).
a I. Kukavica, The fractal dimension of the singular set for solutions of the Navier-Stokessystem, Nonlinearity 22(12):2889-2900 (2009)
b I. Kukavica, Y. Pei An estimate on the parabolic fractal dimension of the singular set forsolutions of the Navier-Stokes system, Nonlinearity 25(9):2775-2783 (2012)
c Y. Koh, M. Yang The Minkowski dimension of interior singular points in the incompressibleNavier-Stokes equations, arXiv:1603.01007 (2016)
d Y. Wang, G. Wu On the box-counting dimension of potential singular set for suitable weaksolutions to the 3D Navier-Stokes equations, arXiv:1604.05032 (2016)
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Theorem (Scheffer (1985)a)
There exist functions u : R3 × [0,∞)→ R3 and p : R3 × [0,∞)→ R such that
1) S 6= ∅,2) there exists a compact set K ⊂ R3 such that suppu(t) ⊂ K for all t,
3) u(x, t) is a C∞ function for each t ≥ 0,
4) there exists M > 0 such that ‖u(t)‖ ≤M for all t ≥ 0,
5) ∇u ∈ L2((0,∞)× R3), u ∈ L3((0,∞)× R3), |u||p| ∈ L1((0,∞)× R3),
6) the local energy inequality
ˆR3
|u(t)|2φ(t)−ˆR3
|u(s)|2φ(s) + 2
ˆ t
s
ˆR3
|∇u|2φ
≤ˆ t
s
ˆR3
|u|2(φt + ∆φ) +
ˆ t
s
ˆR3
(|u|2 + 2p)u · ∇φ
holds for all 0 ≤ s < t and for all φ ∈ C∞c ((0,∞)× R3) with φ ≥ 0,
a V. Scheffer, A solution to the Navier-Stokes inequality with an internalsingularity, Comm. Math. Phys. 101, 47-85 (1985)
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Scheffer’s construction, sketch
Define u(0) on (0, t1) by
1)u(0)(x1, x2, 0, 0) := u[v, f ](x1, x2, 0) := (v1, v2, (f
2 − |v|2)1/2),
2)u(0)(Rc(x1, x2, 0), 0) := Rc(u
(0)(x1, x2, 0, 0)),
where Rc(x1, x2, x3) := (x1, c1x2 − c2x3, c1x3 + c2x2) with c21 + c22 = 1denotes the rotation around the Ox1 axis, v = (v1, v2) : R2 → R2 satisfiesthe PDE
x2div v + v2 = 0
and f : R2 → R satisfies f > |v| everywhere,
3)|u(0)(x, 0)| = f1(x), |u(0)(x, t1)| = f2(x)
for all x, where f1, f2 : R3 → R satisfy
f2(τx+ (a1, a2, 0)) ≥ τ−1f1(x)
for all x for some fixed τ ∈ (0, 1), and a := (a1, a2, 0) ∈ R3.
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Scheffer’s construction, sketch
We construct u(1) on the time interval (t1, t2) by the rescaling
u(1)(x, t) := τ−1u(0)(τ−1(x− a), τ−2(t− t1)).
This way
|u(1)(x, t1)| = |τ−1u(0)(τ−1(x− a), 0)|= τ−1f1(τ−1(x− a))
≤ f2(x)
= |u(0)(x, t1)|.
Hence, by local energy inequality, we can combine u(0) and u(1) (one afteranother).
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Theorem (V. Scheffer, 1987a)
Let ξ ∈ (0, 1). There exists a Cantor set S ⊂ R3 × {1} and functionsu : R3 × [0,∞)→ R3 and p : R3 × [0,∞)→ R such that
1) dH(S) = ξ,
2) there exists a compact set K ⊂ R3 such that suppu(t) ⊂ K for all t,
3) u(x, t) is a C∞ function for each t,
4) there exists M > 0 such that ‖u(t)‖L2 ≤M for all t,
5) ∇u ∈ L2((0,∞)× R3), u ∈ L3((0,∞)× R3), |u||p| ∈ L1((0,∞)× R3),
6) the local energy inequality
ˆR3
|u(t)|2φ dx+ 2
ˆ t
0
ˆR3
|∇u|2φ dx ds
≤ˆ t
0
ˆR3
|u|2(φt + ∆φ)dxds+
ˆ t
0
ˆR3
(|u|2 + 2p)u · ∇φ dxds
holds for all t ≥ 0 and for all φ ∈ C∞c ((0,∞)× R3) with φ ≥ 0,
7) the singular set of u contains S.
a V. Scheffer, Nearly One Dimensional Singularities of Solutions to theNavier-Stokes Inequality, Comm. Math. Phys. 110, 525-551 (1987)
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Directions:
1) Understand the solutions constructed by Scheffer and try to construct anexample of a solution to the Navier-Stokes inequality with dH(S) ≤ 1 anddB(S) > 1 (add concentration of blow-up times),
2) Consider the 1D surface growth model
ut + uxxxx +(|ux|2
)xx
= 0, u(0) = u0
for x ∈ (0, 2π), t ≥ 0. This model shares many striking similarities with the3D incompressible Navier-Stokes equations, including the existence anduniqueness results.The local energy inequality:
1
2
ˆ 2π
0
|u(t)|2φ(t) +
ˆ t
0
ˆ 2π
0
u2xxφ
≤ˆ t
0
ˆ 2π
0
(1
2(φt + φxxxx)|u|2 + 2u2xφxx −
5
3u3xφx − u2xuφxx
).
2a) Prove the partial regularity theory for the surface growth model in the similarway as in the case of 3D NSE,
2b) Construct an example of a solution to the surface growth inequality (or a weaksolution to the equation) with a given dH(S) ∈ (0, 1).
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