Anisotropic partial regularity criteria for the Navier-Stokes - - PowerPoint PPT Presentation

Anisotropic partial regularity criteria for the Navier-Stokes equations

Walter Rusin

Department of Mathematics

Mathflows 2015

Porquerolles September 17, 2015

The question of regularity of the weak solutions of the 3D incompressible Navier-Stokes equations ∂tu − ∆u +

3

• j=1

∂j(uju) + ∇p = 0 (NSE) div u = 0 where u(x, t) = (u1(x, t), u2(x, t), u3(x, t)) and p(x, t) denote the unknown velocity and the pressure, has remained open since 1930’s (works of Leray ’34).

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For initial data u0(x) ∈ L2(R3), the energy estimate u(·, t)2

L2 + 2

t ∇u(·, s)2

L2 ds ≤ u02 L2

plays a crucial role in establishing the existence of global weak

• solutions. In particular, for any T > 0 we obtain

u ∈ L∞((0, T), L2(R3)) ∩ L2((0, T), ˙ H1(R3)).

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The Sobolev embedding and interpolation yield that for such u(x, t) we have u ∈ L10/3

t,x ((0, T) × R3)

• r any other Lebesgue space Lp

t Lq x((0, T) × R3) provided that

2 p + 3 q = 3 2.

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The available regularity criteria (e.g. Ladyzhenskaya-Prodi-Serrin) require however 2 p + 3 q ≤ 1. Note that not only we cannot draw any conclusion but there is also a substantial gap between what is at hand and what would be necessary to conclude regularity.

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The theory of partial regularity for the NSE aims at estimating the Husdorff dimension of the (possible) singular set and development of interior regularity criteria. Recall that a point is regular if there exists a neighborhood in which u is bounded (and thus H¨

• lder continuous); otherwise, the point is

called singular .

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The program has been started by Scheffer in the 70’s. In a classical paper, Caffarelli, Kohn, and Nirenberg proved that the

• ne-dimensional parabolic Hausdorff measure (parabolic Hausdorff

length) of the singular set equals zero. In this case, the interior regularity criterion is as follows: There exist two constants ǫCKN ∈ (0, 1] and α ∈ (0, 1) such that if

• Q1

(|u|3 + |p|3/2) dxdt ≤ ǫCKN then u(x, t)C α(Q1/2) < ∞ where Qr = {(x, t) : |x| < r, −r2 ≤ t ≤ 0}.

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Alternative proofs were given by Ladyzhenskaya and Ser¨ egin, Kukavica, Lin, Vasseur, and Wolf. The problem of partial regularity of the solutions of the Navier–Stokes equations has been since then addressed in various contexts and a variety of interior regularity criteria has been proposed.

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In particular Wolf proved the following: There exists ǫW > 0 such that if

• Q1

|u|3 dxdt ≤ ǫW then the solution u(x, t) is regular at the point (0, 0).

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Fairly recently, Wang and Zhang proved an anisotropic interior regularity criterion, which states: For every M > 0 there exists ǫWZ(M) > 0 such that if

• Q1

(|u|3 + |p|3/2) dxdt ≤ M and

• Q1

|uh|3 dxdt ≤ ǫWZ(M) where uh = (u1, u2), then the solution u(x, t) is regular at the point (0, 0).

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One component regularity/anisotropic results:

• Cao, Titi
• Chemin, Zhang
• Zhou
• Raugel, Sell
• He
• Mucha
• Nestupa, Novotny, Penel, Pokorny
• Kukavica, Ziane, R.
• ...

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The purpose of this talk is to present interior regularity criteria involving only one component of the velocity. We prove, with a different argument, the following statement: For every M > 0 there exists a constant ǫ(M) > 0 such that if

• Q1

(|u|3 + |p|3/2) dxdt ≤ M and

• Q1

|u3|3 ≤ ǫ(M) then u(x, t) is regular at the point (0, 0).

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Let D be an open, bounded, and connected subset of R3 × (0, ∞). We work in the class of suitable weak solutions, that is (u, p) satisfies (i) u ∈ L∞

t L2 x(D) ∩ L2 t H1 x (D) and p ∈ L3/2(D),

(ii) the Navier-Stokes equations (NSE) are satisfied in the weak sense (iii) the local energy inequality holds in D, i.e.,

• R3 |u|2φ dx
• T

+ 2

R3×(−∞,T]

|∇u|2φ dxdt ≤

R3×(−∞,T]

• |u|2(∂tφ + ∆φ) + (|u|2 + 2p)u · ∇φ
• dxdt

for all φ ∈ C ∞

0 (D) such that φ ≥ 0 in D and almost all T ∈ R.

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Theorem 1

Let (u, p) be a suitable weak solution of (NSE) in a neighborhood of Qr(x0, t0) ⊂ D which satisfies 1 r2

• Qr(x0,t0)
• |u|3 + |p|3/2

dxdt ≤ M. (1) Then there exists ǫ > 0 depending on M such that if 1 r2

• Qr(x0,t0)

|u3|3 dxdt ≤ ǫ (2) then u is regular at (x0, t0).

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Corollary 1

Let (u, p) be Leray’s weak solution defined in a neighborhood of [0, T]. If T is an epoch of irregularity, then for every q ≥ 3 and M > 0, we have (u1, u2)(·, t)Lq ≥ M (T − t)(1−3/q)/2 and u3(·, t)Lq ≥ ǫ(M) (T − t)(1−3/q)/2 , for t < T sufficiently close to T.

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Theorem 2

For every M > 0, there exists a constant ǫ(M) > 0 with the following property: If (u, p) is a suitable weak solution of (NSE) in a neighborhood of Qr(x0, t0) ⊂ D which satisfies 1 r2

• Qr(x0,t0)

(|u|3 + |p|3/2) dxdt ≤ M (3) and either 1 r2

• Qr(x0,t0)

|u|3 dxdt ≤ ǫ (4)

• r

1 r2

• Qr(x0,t0)

|p|3/2 dxdt ≤ ǫ, (5) then u is regular at (x0, t0).

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Corollary 2

Let (u, p) be Leray’s weak solution defined in a neighborhood of [0, T]. If T is an epoch of irregularity, then for every q ≥ 3 and M > 0, we have u(·, t)Lq ≥ M (T − t)(1−3/q)/2 and p(·, t)Lq/2 ≥ ǫ(M) (T − t)(1−3/q)/2 , for t < T sufficiently close to T.

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Sketch of the proof of Theorem 1. We argue by contradiction. Fix r > 0 and assume that there exists a sequence of suitable weak solutions (u(n), p(n)), with (x0, t0) a singular point for each of them with 1 r2

• Qr(x0,t0)

|u(n)|3 + |p(n)|3/2 dxdt ≤ M and 1 r2

• Qr(x0,t0)

|u(n)

3 |3 dxdt → 0.

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Lemma

Let (u(n), p(n)) be a sequence of suitable weak solutions such that 1 r2

• Qr(x0,t0)
• |u(n)|3 + |p(n)|3/2

dxdt ≤ M, (6) and let 0 < ρ < r. Then there exists a subsequence (u(nk), p(nk)) such that u(nk) → u strongly in Lq(Qρ(x0, t0)) for all 1 ≤ q < 10/3 and p(nk) ⇀ p weakly in L3/2(Qρ(x0, t0)).

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Note that the pair (u, p) solves the system ∂tui − ∆ui +

2

• j=1

uj∂jui + ∂ip = 0 in D i = 1, 2 (LS) ∂3p = 0 in D ∂1u1 + ∂2u2 = 0 in D where u(x1, x2, x3, t) and p(x1, x2, x3, t) are unknown.

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Theorem 3

Let (u, p) be a solution of (LS). Then u is regular.

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Theorem 3 implies that we may reduce r and assume M0 = uL6(Qr(x0,t0)) < ∞. Using H¨

• lder’s inequality we obtain

uL3(Qρ(x0,t0)) ≤ Cρ5/2uL6(Qρ(x0,t0)) ≤ Cρ5/2M0 0 < ρ ≤ r from where 1 ρ2/3 uL3(Qρ(x0,t0)) ≤ Cρ11/6uL6(Qρ(x0,t0)) ≤ Cρ11/6M0 0 < ρ ≤ r.

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Let r0 ≤ r be sufficiently small so that Cr11/6 M0 ≤ 1 2ǫ1/3

W .

Then 1 r2/3 uL3(Qr0(x0,t0)) ≤ 1 2ǫ1/3

W .

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Since u(n) → u strongly in L3

loc(Qr0(x0, t0)), we may choose n large

enough so that 1 r2/3 u(n) − uL3(Qr0(x0,t0)) ≤ 1 2ǫ1/3

W .

Thus we obtain 1 r2

• Qr(x0,t0)

|u(n)|3 dxdt ≤ ǫW. This leads to a contradiction as the above inequality implies that the suitable weak solution u(n) is regular at (x0, t0). Note that instead of Wolf’s result we may use Theorem 2.

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Sketch of the proof of Theorem 2. First, we prove the assertion under the condition on the pressure. We argue by contradiction. Fix r > 0 and assume that there exists a sequence of suitable weak solutions (u(n), p(n)), with (x0, t0) a singular point for each of them with 1 r2

• Qr(x0,t0)

|u(n)|3 dxdt ≤ M and 1 r2

• Qr(x0,t0)

|p(n)|3/2 dxdt → 0.

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By possibly reducing r > 0 we may assume that u(n) → u strongly in L3(Qr(x0, t0)). Note that u solves the Burgers-type system ∂tui − ∆ui +

3

• j=1

ui∂juj = 0 in D i = 1, 2, 3. with ∂1u1 + ∂2u2 + ∂3u3 = 0 in D. Therefore u is regular.

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Thus, there exists a constant M0 > 0 (which depends on M) such that uL6(Qr(x0,t0)) = M0 < ∞. We conclude as in the proof of Theorem 1. Namely, H¨

• lder’s

inequality yields uL3(Qρ(x0,t0)) ≤ Cρ5/2uL6(Qρ(x0,t0)) ≤ Cρ5/2M0 0 < ρ ≤ r which implies 1 ρ2/3 uL3(Qρ(x0,t0)) ≤ Cρ11/6uL6(Qρ(x0,t0)) ≤ Cρ11/6M0 0 < ρ ≤ r.

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Let r0 ≤ r be sufficiently small so that Cr11/6 M0 ≤ 1 6ǫ1/3

CKN.

Thus 1 r2/3 uL3(Qr0(x0,t0)) ≤ 1 6ǫ1/3

CKN.

Since u(n) → u strongly in L3

loc(Qr0(x0, t0)), we may choose n large

enough so that 1 r2/3 u(n) − uL3(Qr0(x0,t0)) ≤ 1 6ǫ1/3

CKN.

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Therefore 1 r2/3 u(n)L3(Qr(x0,t0)) ≤ 1 2ǫ1/3

CKN.

Moreover, if n is sufficiently large, we also have 1 r2

• Qr(x0,t0)

|p(n)|3/2 dxdt ≤ 1 2ǫCKN. Hence 1 r2

• Qr0(x0,t0)

(|u|3 + |p|3/2) dxdt ≤ ǫCKN. This leads to a contradiction as the above inequality implies that the suitable weak solution u(n) is regular at (x0, t0).

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Now, we sketch the proof of the assertion under the condition (4). Without loss of generality, we may assume that (x0, t0) = (0, 0). Denote Qr = Qr(x0, t0) Assume that uL3(Qr) ≤ ǫ0 and pL3/2(Qr) ≤ M2

0.

We shall prove that there exists ǫ0 > 0 sufficiently small, depending

• n M0, such that (0, 0) is regular.

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We may rewrite the pressure equation as ∆(ηp) = − ∂ij(ηuiuj) − uiuj∂ijη + ∂j(uiuj∂iη) + ∂i(uiuj∂jη) − p∆η + 2∂j((∂jη)p). With N = −1/4π|x|, the Newtonian potential, we obtain ηp = RiRj(ηuiuj) − N ∗ (uiuj∂ijη) + ∂jN ∗ (uiuj∂iη) + ∂iN ∗ (uiuj∂jη) − N ∗ (p∆η) + 2∂jN ∗ ((∂jη)p) = p1 + p2 + p3 + p4 + p5 + p6.

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For p1, we have by the Calder´

• n-Zygmund theorem

p1L3/2(Qr0) ≤ Cu2

L3(Qr).

For the rest of the terms, we use the fact that they all contain derivatives of η. This makes all the convolutions nonsingular when |x| ≤ r0

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Summarizing, we obtain 1 r2/3 p1/2

L3/2(Qr0) ≤

C r2/3 uL3(Qr)+C r1/3 r M0 ≤ C0 r2/3 r2/3 ǫ0+C0 r1/3 r1/3 M0. where C0 is a constant.

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Let ǫ > 0 be arbitrary. Then, fix r0 = ǫ3r 8C 3

0 M3

and ǫ0 = ǫ 2C0 r0 r 2/3 = ǫ3 8C 3

0 M2

and we get uL3(Qr0) + p1/2

L3/2(Qr0) ≤ ǫ, which implies that (0, 0) is

regular if ǫ = ǫ1/3

CKN/C with C a sufficiently large constant.

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Questions?

Thank you for your attention!

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