Remarks on the blow-up criterion of the 3D Euler equations Dongho Chae Department of Mathematics Sungkyunkwan University Suwon 440-746, Korea e-mail :
[email protected] Abstract In this note we prove that the finite time blow-up of classical solutions of the 3-D homogeneous incompressible Euler equations is controlled by the Besov 0 , norm of the two components of the vorticity. For the axisymmetspace, B˙ ∞,1 ric flows with swirl we deduce that the blow-up of solution is controlled by the same Besov space norm of the angular component of the vorticity. For the proof of these results we use the Beale-Kato-Majda criterion, and the special structure of the vortex stretching term in the vorticity formulation of the Euler equation. Keywords: 3-D Euler equations, finite time blow-up AMS Subject Classification Number: 35Q35, 76B03
1
Introduction
We are concerned on the Euler equations for the homogeneous incompressible fluid flows in R3 × (0, ∞). ∂v + (v · ∇)v = −∇p, (1.1) ∂t div v = 0, (1.2) v(x, 0) = v0 (x),
x ∈ R3
(1.3)
where v = (v 1 , v 2 , v 3 ), v j = v j (x, t), j = 1, 2, 3 is the velocity of the fluid flows, p = p(x, t) is the scalar pressure, and v0 is the given initial velocity satisfying div v0 = 0. Taking curl of (1.1), we obtain the following vorticity formulation for the vorticity field ω =curl v. ∂ω + (v · ∇)ω = ω · ∇v, (1.4) ∂t 1
div v = 0,
curl v = ω,
(1.5)
x ∈ R3
(1.6)
ω(x, 0) = ω0 (x),
The local in time solution of the Euler equations in the Sobolev space H m (Rn ) for m > n/2 + 1, n = 2, 3 was obtained by Kato in [13], and there are several other local well-posedness results after that, using various function spaces([14, 15, 8, 2, 3]). The most outstanding open problem for the Euler equations is whether or not there exists any smooth initial data, say v0 ∈ C0∞ (R3 ), which evolves in finite time into a blowing up solution(breakdown of the initial data regularity). In this direction there is a celebrated criterion of the blow-up due to Beale, Kato and Majda(called the BKM criterion)[1], which states for m > 52 Z T∗ lim sup kv(t)kH m = ∞ if and only if kω(t)kL∞ dt = ∞. (1.7) t%T∗
0
(See [16, 2, 3, 17] for the refinements of this result by replacing the L∞ norm of the vorticity by weaker norms close to the L∞ norm.) In this note we are concerned on refining the BKM criterion by reducing the number of components of the vorticity vector field. For the statement of our main results we introduce a particular Besov 0 space, B˙ ∞,1 . Given f ∈ S, the Schwartz class of rapidly deceasing functions. Its Fourier transform fˆ is defined by Z 1 ˆ F(f ) = f (ξ) = e−ix·ξ f (x)dx. n/2 (2π) Rn We consider ϕ ∈ S satisfying the following three conditions: (i) Supp ϕˆ ⊂ {ξ ∈ Rn | 12 ≤ |ξ| ≤ 2}, (ii) ϕ(ξ) ˆ ≥ C > 0 if 23 < |ξ| < 32 , P (iii) ˆj (ξ) = 1, where ϕˆj = ϕ(2 ˆ −j ξ). j∈Z ϕ Construction of such sequence of functions {ϕj }j∈Z is well-known(See e.g. [21]). Note that ϕˆj is supported on the annulus of radius about 2j . 0 Then, B˙ ∞,1 is defined by X 0 f ∈ B˙ ∞,1 ⇐⇒ kf kB˙ ∞,1 = kϕj ∗ f kL∞ < ∞, 0
Z
j∈
R where ∗ is the standard notation for convolution, (f ∗g)(x) = Rn f (x−y)g(y)dy. Note 0 ,→ that the condition (iii)(partition of unity) above implies immediately that B˙ ∞,1 ∞ 0 ˙ L . The space B∞,1 can be embedded into the class of continuous bounded functions, thus having slightly better regularity than L∞ , but containing as a subspace the 0 , compared with L∞ H¨older space C γ , for any γ > 0. One distinct feature of B˙ ∞,1 0 into is that the singular integral operators of the Calderon-Zygmund type map B˙ ∞,1 ∞ itself boundedly, the property which L does not have. We now state our main theorems. 2
Theorem 1.1 Let m > 5/2. Suppose v ∈ C([0, T1 ); H m (R3 )) is the local classical solution of (1.1)-(1.3) for some T1 > 0, corresponding to the initial data v0 ∈ H m (R3 ), and ω = curl v is its vorticity. We decompose ω = ω ˜ + ω 3 e3 , where ω ˜ = ω 1 e1 + ω 2 e2 , and {e1 , e2 , e3 } is the canonical basis of R3 . Then, Z T lim sup kv(t)kH m = ∞ if and only if k˜ ω (t)k2B˙ 0 dt = ∞. (1.8) t%T
∞,1
0
Remark 1.1. Actually ω ˜ could be the projected component of ω onto any plane in R3 . For the solution v = (v 1 , v 2 , 0) of the Euler equations on the x1 − x2 plane, the vorticity is ω = ω 3 e3 with ω3 = ∂x1 v 2 − ∂x2 v 1 , and ω ˜ ≡ 0. Hence, as a trivial application of the above theorem we obtain the global in time existence of classical solutions for the 2-D Euler equations. Remark 1.2. For the 3-D Navier-Stokes equations it is possible to control the regularity also by ω ˜ , but using the same scale invariant norm as for the whole components of the vorticity field, ω as obtained in [4]. Next, we consider the axisymmetric solution of the Euler equations, which means velocity field v(r, x3 , t), solving the Euler equations, and having the representation v(r, x3 , t) = v r (r, x3 , t)er + v θ (r, x3 , t)eθ + v 3 (r, x3 , t)e3 in the cylindrical coordinate system, where x1 x2 er = ( , , 0), r r
x2 x1 eθ = (− , , 0), r r
e3 = (0, 0, 1),
q r = x21 + x22 .
In this case also the question of finite time blow-up of solution is widely open(See [11],[5],[6] for previous studies in such case). The vorticity ω = curl v is computed as ω = ω r er + ω θ eθ + ω 3 e3 , where ω r = −∂x3 v θ ,
ω θ = ∂x3 v r − ∂r v 3 ,
1 ω 3 = ∂r (rv θ ). r
We denote v˜ = v r er + v 3 e3 ,
ω ˜ = ω r er + ω 3 e3 .
Hence, ω = ω ˜+ω ~ θ , where ω ~ θ = ω θ eθ . The Euler equations for the axisymmetric solution are ∂v r ˜ r = − ∂p , + (˜ v · ∇)v (1.9) ∂t ∂r r θ ∂v θ ˜ θ = −v v , + (˜ v · ∇)v (1.10) ∂t r ∂v 3 ˜ 3 = − ∂p , + (˜ v · ∇)v (1.11) ∂t ∂x3 3
div v˜ = 0,
(1.12)
v(r, x3 , 0) = v0 (r, x3 ),
(1.13)
˜ = er ∂ + e3 ∂ . In the axisymmetry the Euler equations in the vorticity where ∇ ∂r ∂x3 formulation becomes ∂ω r ˜ r (˜ ˜ r + (˜ v · ∇)ω ω · ∇)v ∂t ∂ω 3 ˜ 3 (˜ ˜ 3 + (˜ v · ∇)ω ω · ∇)v ∂t µ θ¶ · ¸µ θ¶ v ∂ ω ˜ ˜ = (˜ ω · ∇) + v˜ · ∇ ∂t r r div v˜ = 0, curl v˜ = ω ~ θ.
(1.14) (1.15) (1.16) (1.17)
We now state our main theorem for the axisymmetric solutions of the Euler equations. Theorem 1.2 Let v be the local classical axisymmetric solution of the 3-D Euler equations considered in Theorem 1.1, corresponding to an axisymmetric initial data v0 ∈ H m (R3 ). As in the above we decompose ω = ω ˜ +ω ~ θ , where ω ˜ = ω r er + ω 3 e3 and θ ω ~ θ = ω eθ . Then, Z T lim sup kv(t)kH m = ∞ if and only if k~ωθ (t)kB˙ ∞,1 dt = ∞. (1.18) 0 t%T
0
Remark 1.3. We note that for the axisymmetric 3-D Navier-Stokes equations with swirl it is possible to control the regularity also by ω ~ θ without strengthening its norm as obtained in [4]. Remark 1.4. We compare this result with that of [6], where we proved that the RT regularity/singularity is controlled by the integral 0 k~ωθ (t)kL∞ (1+log+ k~ωθ (t)kC γ )dt. 0 We note that this integral contains C γ norm of ω ~ θ , higher than B˙ ∞,1 norm.
2
Proof of the Main Results
Multiplying (1.5) by e3 , we obtain ∂ω 3 + (v · ∇)ω 3 = (ω · ∇)v · e3 . ∂t
(2.1)
Given a vector field v(x, t), we consider the particle trajectory mapping X(α, t) defined by the system of ordinary differential equation, ∂X(α, t) = v(X(α, t), t), ∂t
4
X(α, 0) = α ∈ R3 .
Integrating (2.1) along X(α, t), we have Z t 3 3 ω (X(α, t), t) = ω0 (α) + [(ω · ∇)v · e3 ](X(α, s), s)ds. 0
Hence, taking supremum over α ∈ R3 yields Z t 3 3 kω (t)kL∞ ≤ kω0 kL∞ + k(ω · ∇)v · e3 ](s)kL∞ ds.
(2.2)
0
We now estimate the vortex stretching term (ω · ∇)v · e3 pointwise. Using the BiotSavart law, which follows from (1.5), Z 1 y × ω(x + y, t) v(x, t) = dy, 4π R3 |y|3 we can compute(See e.g. [19]) ∂v i (x, t) = ∂xj
¾ Z ½ 3 3 δil 1 X yi yl 1X ²jlm P V ²ijl ωl (x, t) − 3 5 ωm (x + y, t) dy − 4π l,m=1 |y| 3 l=1 R3 |y|3
:= Pij (ω)(x, t), where P V denotes the principal value of the integrals, and ²jlm is the skew symmetric tensor with the normalization ²123 = 1. We note that Pij (·) is a matrix valued singular integral operator of the Calderon-Zygmund type. Hence, we compute explicitly the vortex stretching term as [(ω · ∇)v · e3 ](x, t) =
3 X i,j=1
ωi (x, t)
∂v i (x, t)(e3 )j ∂xj
¾ ω(x, t) × ω(x + y, t) y × ω(x + y, t) 1 PV · e3 − 3 · e3 (y · ω(x, t)) = 4π |y|3 |y|5 R3 = ( Decomposing the vorticity into ω = ω ˜ + ω 3 e3 ,) ½ Z y×ω ˜ (x + y, t) 1 ω ˜ (x, t) × ω ˜ (x + y, t) · e3 − 3 · e3 y3 ω3 (x, t) = PV 3 4π |y| |y|5 R3 ¾ y×ω ˜ (x + y, t) −3 · e3 (y · ω ˜ (x, t)) dy |y|5 3 3 X X ω 3 (x, t)(e3 )i Pij (˜ ω )(x, t)(e3 )j . ω ˜ i (x, t)Pij (˜ ω )(x, t)(e3 )j + = Z
½
i,j=1
i,j=1
We thus have the pointwise estimate |[(ω · ∇)v · e3 ](x, t)| ≤ C|˜ ω (x, t)||P(˜ ω )(x, t)| + C|ω 3 (x, t)||P(˜ ω )(x, t)|,
5
0 and from the embedding, B˙ ∞,1 ,→ L∞ , we obtain
ω )kL∞ + Ck˜ ω kL∞ kP(˜ ω )kL∞ k[Dω · e3 ]kL∞ ≤ Ckω 3 kL∞ kP(˜ 3 ≤ Ckω kL∞ kP(˜ ω )kB˙ ∞,1 + Ck˜ ω kL∞ kP(˜ ω )kB˙ ∞,1 0 0 ≤ Ckω 3 kL∞ k˜ ω kB˙ ∞,1 + Ck˜ ω k2B˙ 0 , 0
(2.3)
∞,1
where we used the fact that the Calderon-Zygmund singular integral operator maps 0 into itself boundedly. Substituting (2.3) into (2.2), we have the estimate B˙ ∞,1 Z 3
kω (t)kL∞ ≤
t
ds ω (s)kB˙ ∞,1 +C kω 3 (s)kL∞ k˜ 0 0 Z t +C k˜ ω (s)k2B˙ 0 ds.
kω03 kL∞
∞,1
0
The Gronwall lemma yields 3
kω (t)kL∞
¶ µ Z t ≤ k˜ ω (s)kB˙ ∞,1 ds exp C 0 0 µ Z t ¶ Z t 2 k˜ ω (τ )kB˙ ∞,1 +C k˜ ω (s)kB˙ 0 exp C dτ ds 0 ∞,1 s 0 µ ¶ µ Z t ¶ Z t 3 2 ≤ kω0 kL∞ + k˜ ω (s)kB˙ 0 ds exp C k˜ ω (s)kB˙ ∞,1 ds . 0 kω03 kL∞
∞,1
0
Hence, denoting Z
³R
T 0
´ 21 k˜ ω (t)k2B˙ 0 dt = AT , we deduce that ∞,1
Z
T
kω(t)kL∞ dt ≤ 0
0
Z
T
k˜ ω (t)kL∞ dt + 0
√ £ ≤ T AT + kω03 kL∞
T
kω 3 (t)kL∞ dt 0 ³ √ ´ ¤ + CA2T T exp C T AT .
Combining this with (1.7) implies the necessity part of (1.8). The sufficiency part 0 easily follows by trivial application of the imbedding, H m (R3 ) ,→ B˙ ∞,1 (R3 ) for m > 25 . This completes the proof of Theorem 1.1.¤ Remark after Proof: The special structure of the vortex stretching term used in the above proof was emphasized and used previously in [9, 10]. Proof of Theorem 1.2: We will use the notations, r ∂v ∂v r µ ¶3 ∂r ∂x3 ∂˜ vj ˜ . , ∇˜ v= ∇˜ v= 3 ∂v ∂v 3 ∂xk j,k=1 ∂r ∂x3 6
One can check easily(or, may see [6] for detailed computations.) ˜ v (x)| ≤ |∇˜ |∇˜ v (x)|
∀x ∈ R3 .
(2.4)
As in the proof of Theorem 1.1 the elliptic system, (1.17) implies ∇˜ v (x) = P(~ωθ )(x) + C0 ω ~ θ (x), where P(·) is a matrix valued singular integral operator of the Calderon-Zygmund type, and C0 is a constant matrix. Given v˜(x, t), we consider the particle trajectory ˜ mapping X(α, t) defined by the system of ordinary differential equation, ˜ ∂ X(α, t) ˜ ˜ = v˜(X(α, t), t), X(α, 0) = α. ∂t ˜ Then, integrating (1.14)-(1.15) along X(α, t) , we find that Z t r ˜ r ˜ r (X(α, ˜ ω (X(α, t), t) = ω0 (α) + (˜ ω · ∇)v s), s)ds, 0
Z ˜ ω 3 (X(α, t), t) = ω03 (α) +
t
˜ 3 (X(α, ˜ (˜ ω · ∇)v s), s)ds.
0
3
Thus, taking supremum over α ∈ R , we infer Z t ˜ v (s)kL∞ ds k˜ ω (t)kL∞ ≤ k˜ ω0 kL∞ + k˜ ω (s)kL∞ k∇˜ 0 Z t ≤ k˜ ω0 kL∞ + k˜ ω (s)kL∞ k∇˜ v (s)kL∞ ds, 0
where we used (2.4). By Gronwall’s lemma we obtain µZ t ¶ k˜ ω (t)kL∞ ≤ k˜ ω0 kL∞ exp k∇˜ v (s)kL∞ ds 0 ¶ µ Z t ≤ k˜ ω0 kL∞ exp C k∇˜ v (s)kB˙ ∞,1 ds 0 0 µ Z t ¶ ≤ k˜ ω0 kL∞ exp C k~ωθ (s)kB˙ ∞,1 ds , 0 0
where we used the fact that the Calderon-Zygmund singular integral operator maps 0 0 ,→ L∞ , into itself boundedly. Combining this estimate with the embeddig, B˙ ∞,1 B˙ ∞,1 we find Z T Z T Z T kω(t)kL∞ dt ≤ k˜ ω (t)kL∞ dt + k~ωθ (t)kL∞ dt 0 0 0 µ Z T ¶ ≤ T k˜ ω0 kL∞ exp C k~ωθ (t)kB˙ ∞,1 dt 0 0 Z T +C k~ωθ (t)kB˙ ∞,1 dt. 0 0
7
Thus, the BKM criterion, (1.7) implies the necessity part of Theorem 1.2. Similarly to the proof of Theorem 1.1 the sufficiency part easily follows from the imbedding, H m (R3 ) ,→ B˙ 0 (R3 ) for m > 5 . This completes the proof of Theorem 1.2. ¤ ∞,1
2
Acknowledgements This research is supported partially by the grant no.2000-2-10200-002-5 from the basic research program of the KOSEF.
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