A LOWER BOUND ON BLOWUP RATES FOR THE ... - UT Mathematics
A LOWER BOUND ON BLOWUP RATES FOR THE 3D INCOMPRESSIBLE EULER EQUATION AND A SINGLE EXPONENTIAL BEALE-KATO-MAJDA ESTIMATE ˇ PAVLOVIC ´ THOMAS CHEN AND NATASA
Abstract. We prove a Beale-Kato-Majda criterion for the loss of regularity for solutions of the incompressible Euler equations in H s (R3 ), for s > 52 . Instead of double exponential estimates of Beale-Kato-Majda type, we obtain a single exponential bound on ku(t)kH s involving the dimensionless parameter introduced by P. Constantin in [2]. In particular, we derive lower bounds on the blowup rate of such solutions.
1. Introduction In this paper, we revisit the Beale-Kato-Majda criterion for the breakdown of smooth solutions to the 3D Euler equations. More precisely, we consider the incompressible Euler equations ∂t u + (u · ∇)u + ∇p = 0
(1.1)
∇·u = 0
(1.2)
u(x, 0) = u0
(1.3) 3
for an unknown velocity vector u(x, t) = (ui (x, t))1≤i≤3 ∈ R and pressure p = p(x, t) ∈ R, for position x ∈ R3 and time t ∈ [0, ∞). Existence and uniqueness of local in time solutions to (1.1) – (1.3) in the space C([0, T ], H s ) ∩ C 1 ([0, T ]; H s−1 ) ,
(1.4)
5 2,
has long been known for s > see for instance [6]. However, it is an open problem to determine whether such solutions can lose their regularity in finite time. An important result that addresses the question of a possible loss of regularity of solutions to Euler equations (1.1) – (1.3) is the criterion formulated by Beale-KatoMajda [1] in terms of the L∞ norm of the vorticity ω = ∇ ∧ u. More precisely, Beale-Kato-Majda in [1] proved the following theorem: Theorem 1.1. Let u be a solution to (1.1) – (1.3) in the class (1.4) for s ≥ 3 integer. Suppose that there exists a time T ∗ such that the solution cannot be continued in the class (1.4) to T = T ∗ . If T ∗ is the first such time, then Z T∗ kω(·, t)kL∞ dt = ∞. (1.5) 0
Date: June 17, 2011. 1
2
´ T. CHEN AND N. PAVLOVIC
The theorem is proved with a contradiction argument. Under the assumption Z T∗ kω(·, t)kL∞ dt < ∞ , 0
the authors of [1] show that ku(·, t)kH s ≤ C0 , for all t < T ∗ contradicting the hypothesis that T ∗ is the first time such that the solution cannot be continued to T = T ∗ . In particular, Beale-Kato-Majda obtain a double exponential bound for ku(·, t)kH s , which follows from the following estimates: Step 1 An energy-type bound on kukH s in terms of kDukL∞ , where Du = [∂i uj ]ij is a 3×3-matrix valued function. More specifically, one applies the operator Dα to equations (1.1)-(1.2), where α is an integer-valued multi-index with |α| ≤ s and uses a certain commutator estimate to derive d ku(·, tk2H s ≤ 2CkDukL∞ ku(·, t)k2H s , dt which via Gronwall’s inequality gives the bound: Z t ku(·, t)kH s ≤ ku0 kH s exp C kDu(·, τ )kL∞ dτ .
(1.6)
(1.7)
0
Step 2 An estimate on kDu(·, t)kL∞ based on the quantities kω(·, t)kL∞ , kω(·, t)kL2 , and log+ ku(·, t)kH 3 , given by kDu(·, t)kL∞ ≤ C 1 + 1 + log+ ku(·, t)kH 3 kω(·, t)kL∞ + kω(·, t)kL2 , (1.8) where C is a universal constant. Step 3 The bound on kω(·, t)kL2 in terms of kω(·, t)kL∞ given by d kω(·, t)k2L2 ≤ 2 D kω(·, t)kL∞ kω(·, t)k2L2 , dt which follows from taking the L2 (R3 )-inner product of ω with the equation for vorticity. Then, Gronwall’s inequality yields Z t kω(·, t)kL2 ≤ kω(·, 0)kL2 exp D kω( · , τ )kL∞ dτ . (1.9) 0
Consequently, one obtains the double exponential bound Z t ku(·, t)kH s ≤ ku0 kH s exp exp C kω( · , τ )kL∞ dτ .
(1.10)
0
from combining (1.7), (1.8) and (1.9). It is an open question whether (1.10) is sharp1. While we do not attempt to answer that question itself in this paper, we obtain a single exponential bound on the H s -norm of solution to Euler equations (1.1) - (1.3) in terms of the quantity 2 o n kω(t)k δ − 2δ+5 C , (1.11) `δ (t) = min L , ku0 kL2 1Single exponential bounds have been obtained in other solution spaces than those displayed above, see for instance [7] for such a result in BMO.
BLOWUP RATE FOR EULER EQUATION
3
where kωkC δ =
|ω(x) − ω(y)| |x − y|δ |x−y| 0 fixed, and δ > 0. More precisely, we prove the following theorem: Theorem 1.2. Let u be a solution to (1.1) - (1.3) in the class (1.4), for s = 52 + δ. Assume that `δ (t) is defined as above, and that Z T −5 (`δ (τ )) 2 dτ < ∞ . (1.13) 0
Then, there exists a finite positive constant Cδ = O(δ −1 ) independent of u and t such that Z t i h −5 ku(·, t)kH s ≤ ku0 kH s exp Cδ ku0 kL2 (`δ (τ )) 2 dτ 0
holds for 0 ≤ t ≤ T . The quantity `δ (t) has the dimension of length, and was introduced by Constantin in [2] (see also the work of Constantin, Fefferman and Majda [4] where a criterion for loss of regularity in terms of the direction of vorticity was obtained), where it was observed that Z T −5 (`δ (t)) 2 dt = ∞ (1.14) 0
is a necessary and sufficient condition for blow-up of Euler equations. In particular, the necessity of the condition follows from the inequality obtained in [2] kω(·, t)kL∞ ≤ ku(·, t)kL2 (`δ (t))
− 52
,
(1.15)
and Theorem 1.1 of Beale-Kato-Majda. This is so because Theorem 1.1 implies that RT if the solution cannot be continued to some time T , then 0 kω(·, t)kL∞ dt = ∞. As a consequence of (1.15), and conservation of energy ku(·, t)kL2 = ku0 kL2 ,
(1.16)
this in turn implies (1.14). However, by invoking the result of Beale-Kato-Majda in this argument, one again obtains a double exponential bound on ku(·, t)kH s in RT −5 terms of 0 (`δ (t)) 2 dt. We refer to [3, 5] for recent developments in this and related areas. In this paper, we observe that one can actually obtain a single exponential bound RT −5 on the H s -norm of the solution u(t) in terms of 0 (`δ (t)) 2 dt, as stated in Theorem 1.2. This is achieved by avoiding the use of the logarithmic inequality (1.8) from [1]. More precisely, we combine the energy bound (1.6) with a Calderon-Zygmund type bound on the symmetric and antisymmetric parts of Du. 5
Also, we obtain a lower bound on the blowup rate of solutions in H 2 +δ . Specifically, we prove:
4
´ T. CHEN AND N. PAVLOVIC
Theorem 1.3. Let u be a solution to (1.1) – (1.3) in the class 5
3
C([0, T ]; H 2 +δ ) ∩ C 1 ([0, T ]; H 2 +δ ).
(1.17)
Suppose that there exists a time T ∗ such that the solution cannot be continued in the class (1.17) to T = T ∗ . If T ∗ is the first such time then there exists a finite, positive constant C(δ, ku0 kL2 ) such that 1 1+ 25 δ ku(·, t)k 25 +δ ≥ C(δ, ku0 kL2 ) , (1.18) H T∗ − t under the condition that t is sufficiently close to T ∗ (see the conditions (3.22) and (3.23) below, with t0 = t). The proof of Theorem 1.3 can be outlined as follows. We assume that u is a solution in the class (1.17) that cannot be continued to T = T ∗ , and that T ∗ is the first such time. Invoking the local in time existence result, we derive a lower bound Tloc,t1 > 0 on the time of existence of solutions to Euler equations in (1.17) 5 for initial data u(t1 ) ∈ H 2 +δ at an arbitrary time t1 < T ∗ . By definition of T ∗ , we thus have t1 + Tloc,t1 < T ∗ .
(1.19)
5 2 +δ
-norm of the solution, we obtain in Section Based on an energy bound on the H 3 an expression for Tloc,t1 of the form Cku(·,t11)k 5 , which together with (1.19) H2
+δ
implies that 1 , (1.20) − t1 ) for all t1 < T ∗ . This is an “a priori” lower bound on the blowup rate. Subsequently, we improve (1.20) by a recursion argument in Theorem 1.3 for times t close to T ∗ , to yield the stronger bound (1.18). ku(·, t1 )k
5
H 2 +δ
>
C(T ∗
2. Proof of theorem 1.2 First we recall that the full gradient of velocity Du can be decomposed into symmetric and antisymmetric parts, Du = Du+ + Du−
(2.1)
where 1 Du ± DuT . 2 Du+ is called the deformation tensor. Du± =
(2.2)
In the following lemma we recall important properties of Du+ and Du− . For the convenience of the reader, we give proofs of those properties, although some of them are available in the literature, see e.g. [2]. Lemma 2.1. For both the symmetric and antisymmetric parts Du+ , Du− of Du, the L2 bound kDu± kL2 ≤ CkωkL2 . holds.
(2.3)
BLOWUP RATE FOR EULER EQUATION
5
The antisymmetric part Du− satisfies 1 Du− v = ω ∧ v 2 for any vector v ∈ R3 . The vorticity ω satisfies the identity Z 1 dy ω(ξ) = P.V. σ(b y ) ω(x + y) 3 , 4π |y| (”P.V.” denotes principal value) where σ(b y ) = 3 yb ⊗ yb − 1, with yb = Z σ(b y ) dµS 2 (y) = 0 ,
(2.4)
(2.5) y |y| .
Notably, (2.6)
S2
where dµS 2 denotes the standard measure on the sphere S 2 . The matrix components of the symmetric part have the form X X ` Du+ Tij` (ω` ) = Kij ∗ ω` , ij = `
(2.7)
`
` where ω` are the vector components of ω, and where the integral kernels Kij have the properties ` Kij (y) ` kσij kC 1 (S 2 )
Z S2
` σij (b y ) dµS 2 (y)
` = σij (b y ) |y|−3
(2.8)
≤ C
(2.9)
=
0.
(2.10)
Thus in particular, Tij` is a Calderon-Zygmund operator, for every i, j, ` ∈ {1, 2, 3}. Proof. An explicit calculation shows that the Fourier transform of Du as a function of ω b is given by c b b Du(ξ) = −[(∂i (∆−1 ∇ ∧ ω)j )b (ξ)]i,j = G(ξ) + H(ξ)
(2.11)
where ξ1 ξ2 ω b 3 − ξ1 ξ3 ω b2 1 b ξ ξ ω b G(ξ) := 1 3 1 2|ξ|2 −ξ1 ξ2 ω b1
−ξ2 ξ3 ω b2 ξ2 ξ3 ω b 1 − ξ1 ξ2 ω b3 ξ1 ξ2 ω b2
ξ2 ξ3 ω b3 (2.12) −ξ1 ξ3 ω b3 ξ1 ξ3 ω b2 − ξ2 ξ3 ω b1
and 0 1 b −ξ12 ω b3 H(ξ) := 2|ξ|2 ξ12 ω b2
ξ22 ω b3 0 −ξ22 ω b1
−ξ32 ω b2 ξ32 ω b1 , 0
(2.13)
using the notation ω bj ≡ ω bj (ξ) for brevity. Clearly, every component of G is given by a sum of Fourier multiplication opξ i ξj erators with symbols of the form |ξ| 2 , i 6= j, applied to a component of ω. For instance, Z dy G21 (x) = const. P.V. yb1 yb3 ω1 (x + y) 3 (2.14) |y| corresponds to the component G21 . It is easy to see that every component Gij is a sum of Calderon-Zygmund operators applied to components of ω, with kernel
´ T. CHEN AND N. PAVLOVIC
6
satisfying the asserted properties (2.8) ∼ (2.10). The same is true for the symmetric part, G+ = 12 (G + GT ). b The symmetric part of H(ξ) is given by ω3 0 (ξ22 − ξ12 )b 1 2 + 2 b (ξ − ξ )b ω 0 H (ξ) = 3 2 1 2|ξ|2 (ξ12 − ξ32 )b ω2 (ξ32 − ξ22 )b ω1
(ξ12 − ξ32 )b ω2 (ξ32 − ξ22 )b ω1 0
(2.15)
so that each component defines a Fourier multiplication operator with symbol of ξi2 −ξj2 |ξ|2 , i 6= j, acting on a x2i −x2j |x|n+2 ). That is, for instance,
the form form
component of ω (with associated kernel of the
+ H12 (x) = const P.V.
Z
(b y22 − yb12 ) ω3 (x + y)
dy . |y|3
(2.16)
The properties (2.8) ∼ (2.10) follow immediately. ` The Fourier transforms of the integral kernels Kij can be read off from the + + ` b b components Gij + Hij . In position space, one finds that σij (b y ) is obtained from a sum of terms proportional to terms of the form ybi1 ybj1 and (b yi22 − ybj22 ).
For the antisymmetric part Du− , one generally has Du− v = 21 (∇ ∧ u) ∧ v for any v ∈ R3 , and from u = −∆−1 ∇ ∧ ω, we get Du− v = 12 ω ∧ v, using that ∇ · u = 0. As a side remark, we note that while H − does not by itself exhibit the properties (2.8) ∼ (2.10), it combines with G− in a suitable manner to yield the stated properties of Du− , thanks to the condition ∇ · ω = 0. Next, Lemma 2.2 below provides an upper bound in terms of the quantity `δ (t) on singular integral operators applied to ω of the type appearing in (2.7). We note that similar bounds were used in [2] and [4] for the antisymmetric part Du− . Here, we observe that they also hold for the symmetric part Du+ . As shown in [4] for Du− , the proof of such a bound follows standard steps based on decomposing the singular integral into an inner and outer contribution. The inner contribution can be bounded based on a certain mean zero property, while the outer part is controlled via integration by parts. Lemma 2.2. For L > 0 fixed, and δ > 0, let `δ (t) be defined as above. Moreover, let ω` , ` = 1, 2, 3, denote the components of the vorticity vector ω(t). Then, any singular integral operator Z 1 dy T ω` (x) = P.V. σT (b y ) ω` (x + y) 3 (2.17) 4π |y| with Z σT (b y )dµS 2 (y) = 0
,
kσT kC 1 (S 2 ) < C ,
(2.18)
S2
satisfies 5
kT ω` kL∞ ≤ C(δ) ku0 kL2 `δ (t)− 2 for ` ∈ {1, 2, 3}, for a constant C(δ) = O(δ
−1
) independent of u and t.
(2.19)
BLOWUP RATE FOR EULER EQUATION
7
Proof. Let χ1 (x) be a smooth cutoff function which is identical to 1 on [0, 1], and identically 0 for x > 2. Moreover, let χR (x) = χ1 (x/R), and χcR = 1 − χR . We consider Z σT (b y ) ω` (x + y) |y|>
for > 0 arbitrary, where Z (I) :=
dy = (I) + (II) |y|3 dy |y|3
(2.21)
dy . |y|3
(2.22)
σT (b y ) ω` (x + y) χ`δ (t) (|y|)
|y|>
(2.20)
and Z (II) :=
σT (b y ) ω` (x + y) χc`δ (t) (|y|)
From the zero average property (2.18), we find Z dy σT (b y ) (ω` (x + y) − ω` (x)) χ`δ (t) (|y|) k(I)kL∞ = |y|3 |y|> Z dy ≤ kω` kC δ 3−δ |y| 52 , we recall the definitions of the homogenous and inhomogenous Besov norms for 1 ≤ p, q ≤ ∞, X q1 kukB˙ s = 2jqs kuj kqLp , (2.34) p,q
j∈Z
BLOWUP RATE FOR EULER EQUATION
9
respectively, s kukBp,q =
kukqLp + kukqB˙ s
q1
,
(2.35)
p,q
where uj = Pj u is the Paley-Littlewood projection of u of scale j. In analogy to s (1.6), we obtain the bound on the B2,2 Besov norm of u(t) given by 1 ∂t ku(t)k2B2,2 ≤ kDu(t)kL∞ ku(t)k2B2,2 , (2.36) s s 2 from a straightforward application of estimates obtained in [8]; details are given in the Appendix. Accordingly, since the left hand side yields s ∂t ku(t)kB s ∂t ku(t)k2B2,2 = 2ku(t)kB2,2 , s 2,2
(2.37)
s s ∂t ku(t)kB2,2 ≤ kDu(t)kL∞ ku(t)kB2,2 .
(2.38)
we get
However, Corollary 2.3 implies that kDu(t)kL∞
≤
kDu+ (t)kL∞ + kDu− (t)kL∞
≤
Cδ ku0 kL2 (`δ (t))− 2 .
5
(2.39)
Therefore, by combining (2.38) and (2.39) we obtain s ∂t ku(t)kB2,2
≤
5
s Cδ ku0 kL2 (`δ (t))− 2 ku(t)kB2,2 ,
which implies that ku(t)kH s
∼ ≤ ∼
s ku(t)kB2,2
t i 5 s ku0 kB2,2 exp Cδ ku0 kL2 `δ (s)− 2 ds 0 Z t h i 5 ku0 kH s exp Cδ ku0 kL2 `δ (s)− 2 ds ,
Z
h
0
for s ≥ 0, where we recall from (2.23) that Cδ = O(δ
−1
).
This completes the proof of Theorem 1.2.
3. Lower bounds on the blowup rate In this section, we prove Theorem 1.3. Recalling the energy bound (2.38), s s ∂t ku(t)kB2,2 ≤ kDu(t)kL∞ ku(t)kB2,2 ,
(3.1)
we invoke the Sobolev embedding kDukL∞
≤
c L1 kDuk Z 21 ≤ dξ hξi−3−2δ kDuk
3
H 2 +δ
≤ Cδ kuk
5
∼ Cδ kuk
5 +δ
H 2 +δ 2 B2,2
,
(3.2)
´ T. CHEN AND N. PAVLOVIC
10 1
with Cδ = O(δ − 2 ), to get, for s =
5 2
+ δ,
2 s s ) . ∂t ku(t)kB2,2 ≤ Cδ (ku(t)kB2,2
(3.3)
Straightforward integration implies 1 1 − ≤ Cδ (t − t0 ) . − s s ku(t)kB2,2 ku(t0 )kB2,2
(3.4)
Hence, ku(t)kH s
∼ ≤
s ku(t)kB2,2 s ku(t0 )kB2,2 s 1 − (t − t0 )Cδ ku(t0 )kB2,2
ku(t0 )kH s , (3.5) 1 − (t − t0 )Cδ ku(t0 )kH s where a possible trivial modification of Cδ is implicit in passing to the last line. This implies that the solution u(t) is locally well-posed in H s , with s = 25 + δ, for ∼
1 . (3.6) Cδ ku(t0 )kH s In particular, this infers that if T ∗ is the first time beyond which the solution u cannot be continued, one necessarily has that 1 T ∗ > t0 + . (3.7) Cδ ku(t0 )kH s This in turn implies an a priori lower bound on the blowup rate given by 1 (3.8) ku(t)kH s > Cδ (T ∗ − t) for all 0 ≤ t < T ∗ . The lower bound on the blowup rate stated in Theorem 1.3 is stronger than this estimate, and we shall prove it in the sequel. t0 ≤ t < t0 +
To begin with, we note that kω(t)kC δ
≤ Cδ kω(t)k ≤ Cδ ku(t)k
3
H 2 +δ 5
H 2 +δ
≤
Cδ ku(t0 )k
5
H 2 +δ
1 − (t − t0 )Cδ ku(t0 )k
.
(3.9)
5
H 2 +δ
5
That is, local well-posedness of u in H 2 +δ implies δ-Holder continuity of the vorticity. The parameter L in the definition (1.11) of `δ (t) is arbitrary. Thus, in view of (3.9), we may now let L → ∞ for convenience. Then, 2 kω(t)k δ 2δ+5 · 52 5 C `δ (t)− 2 = ku0 kL2 Cδ ku(t)k 5 +δ 1−δ˜ H2 ≤ ku0 kL2 C 1−δ˜ 1−δ˜ ku(t0 )kH s δ , (3.10) ≤ ku0 kL2 1 − (t − t0 )Cδ ku(t0 )kH s
BLOWUP RATE FOR EULER EQUATION
11
where 2δ 5 and s = + δ . 5 + 2δ 2 We note that while the right hand side of (3.10) diverges as t approaches δ˜ :=
1 , Cδ ku(t0 )kH s
t1 := t0 + the integral Z t1 5 `δ (t)− 2 dt
Cδ 1−δ˜ ku0 kL2 =: B0 (δ)
≤
t0
Z
t1
t0
(3.11)
(3.12)
1−δ˜ ku(t0 )kH s dt 1 − (t − t0 )Cδ ku(t0 )kH s (3.13)
converges for δ > 0 (⇔ δe > 0). This implies that the solution u(t) for t ∈ [t0 , t1 ) can be extended to t > t1 . In particular, we obtain that ku(t1 )k
H
5 +δ 2
≤
ku(t0 )k
5 +δ 2
≤
ku(t0 )k
5
H
H 2 +δ
Z t1 5 exp Cδ ku0 kL2 (`δ (t))− 2 dt t0 exp Cδ ku0 kL2 B0 (δ)
(3.14)
from Theorem 1.2. 5
We may now repeat the above estimates with initial data u(t1 ) in H 2 +δ , thus obtaining a local well-posedness interval [t1 , t2 ]. Accordingly, we may set t2 to be given by 1 t2 := t1 + . (3.15) Cδ ku(t1 )kH s More generally, we define the discrete times tj by 1 . Cδ ku(tj )kH s
tj+1 := tj +
(3.16)
We then have ku(tj+1 )kH s ≤ exp Cδ ku0 kL2 Bj (δ) ku(tj )kH s ,
(3.17)
where Bj (δ) is defined by Cδ ku0 kL2 Bj (δ) Cδ 1−δ˜ ku0 kL2 ku k 2 δ˜ ˜
:= Cδ ku0 kL2
Z
tj+1
tj
1−δ˜ ku(tj )kH s dt 1 − (t − tj )Cδ ku(tj )kH s
1 1−δ 0 L Cδ ˜ ku(t δ j )kH s ku k 2 δ˜ 0 L =: bδ . ku(tj )kH s =
(3.18)
Letting ku k 2 δ˜ 0 L ρj := exp bδ , ku(tj )kH s
(3.19)
´ T. CHEN AND N. PAVLOVIC
12
we have ku(tj )kH s ≤ ρj−1 ku(tj−1 )kH s ,
(3.20)
and we remark that (ρj )j satisfy the recursive estimates ρj
δ˜ ku0 kL2 ρj−1 ku(tj−1 )kH s
≥
exp bδ
=
( ρj−1 )ρj−1 ˜ δ exp ρ− j−1 ln ρj−1 .
=
˜ −δ
(3.21)
We note that from its definition, ρj > 1 for all j. We shall now assume that T ∗ > 0 is the first time beyond which the solution u(t) cannot be continued. Thus, by choosing t0 close enough to T ∗ , (3.8) implies that ku(t0 )kH s can be made sufficiently large that the following hold: (1) The quantity bδ
ku k 2 δ˜ 0 L 1 ku(t0 )kH s
(3.22)
is small. e independent of j such that (2) There is a positive, finite constant C e ku(t0 )kH s ku(tj )kH s ≥ C
(3.23)
holds for all j ∈ N. Without any loss of generality (by a redefinition of the e = 1. constant bδ if necessary), we can assume that C e = 1 implies that ρj ≤ ρ0 for all j. Then, for any Accordingly, (3.23) with C N ∈ N, T ∗ − t0
≥
N X (tj+1 − tj ) j=0
= = ≥ ≥
1 1 1 + ··· + Cδ ku(t0 )kH s ku(tN )kH s 1 ku(t0 )kH s ku(t0 )kH s 1+ + ··· + Cδ ku(t0 )kH s ku(t1 )kH s ku(tN )kH s 1 1 1 1+ + ··· + Cδ ku(t0 )kH s ρ0 ρ0 · · · ρN 1 1 1 1+ + ··· + N Cδ ku(t0 )kH s ρ0 ρ0
(3.24)
from ρ1j ≥ ρ10 for all j, and the fact that ρ0 > 1 since the argument in the exponent (3.19) is positive.
BLOWUP RATE FOR EULER EQUATION
13
Then, letting N → ∞, we obtain 1 ∗ T − t0
1 ≤ Cδ ku(t0 )kH s 1 − ρ0 ku k 2 δ˜ 0 L = Cδ ku(t0 )kH s 1 − exp − bδ . ku(t0 )kH s
(3.25)
Next, we deduce a lower bound on the blowup rate. Invoking (3.22), we obtain
T∗
1 − t0
ku k 2 δ˜ 0 L ≤ Cδ ku(t0 )kH s 1 − exp − bδ ku(t0 )kH s ku k 2 δ˜ 0 L ≈ Cδ ku(t0 )kH s bδ ku(t0 )kH s =
˜
˜
δ Cδ bδ ku0 kδL2 ku(t0 )k1− Hs .
(3.26)
This implies a lower bound on the blowup rate of the form ku(t0 )k
H
5 +δ 2
≥ =
1 1−1 δ˜ T ∗ − t0 1 2δ+5 5 C(δ, ku0 kL2 ) , T ∗ − t0 C(δ, ku0 kL2 )
(3.27)
under the condition that (3.22) and (3.23) hold. This concludes our proof of Theorem 1.3.
Appendix A. Proof of inequality (2.38) for s >
5 2
In this Appendix, we prove (2.38) which follows from (2.36), 1 ∂t ku(t)k2B2,2 . kDu(t)kL∞ ku(t)k2B2,2 , s s 2
(A.1)
for s > 25 . We invoke Eq. (26) in the work [8] of F. Planchon, which is valid for s > 1 + n2 in n dimensions (thus, s > 25 in our case of n = 3), for parameter values p = q = 2 in the notation of that paper. It yields 1 ∂t 22js kuj k2L2 2
. 22js
X
kSj+1 DukL∞ kuk kL2 kuj kL2
k∼j
+ 22js
X
kuk kL2 kuk0 kL2 kDuj kL∞
(A.2)
j.k∼k0
where uk = Pk u is the Paley-Littlewood projection of u at scale k, and Sj = P 0 P 0 j ≤j j is the Paley-Littlewood projection to scales ≤ j.
´ T. CHEN AND N. PAVLOVIC
14
Summing over j, 1 X 2js ∂t 2 kuj k2L2 2 j
. sup kSj+1 DukL∞
X
j
22js
j
+
kuk kL2 kuj kL2
k∼j 0
X X
22s(j−k) 2ks kuk kL2 2k s kuk0 kL2
k∼k0 &j
j
. kDukL∞
X
X
22js kuj k2L2
j
+
XX k
. kDukL∞
X
22s(j−k) 2ks kuk k2L2
j.k
22js kuj k2L2 .
(A.3)
j
To pass to the second inequality, we used that kSj+1 DukL∞ = kmj+1 ∗ DukL∞ . kDukL∞ kmj+1 kL1 ,
(A.4)
where m cj is the symbol of the Fourier multiplication operator Sj , and the fact that kmj kL1 ∼ 1 uniformly in j. Accordingly, we get 1 ∂t ku(t)k2B˙ s . kDu(t)kL∞ ku(t)k2B˙ s . 2,2 2,2 2
(A.5)
ku(t)k2B2,2 = ku(t)k2L2 + ku(t)k2B˙ s , s
(A.6)
From 2,2
and energy conservation, ∂t ku(t)k2L2 = 0, we obtain 1 ∂t ku(t)k2B2,2 s 2
1 ∂t ku(t)k2B˙ s 2,2 2 . kDu(t)kL∞ ku(t)k2B˙ s
=
2,2
. kDu(t)k
L∞
This proves (A.1).
ku(t)k2B2,2 s
.
(A.7)
Acknowledgments. The work of T.C. was supported by NSF grant DMS 1009448. The work of N.P. was supported NSF grant number DMS 0758247 and an Alfred P. Sloan Research Fellowship.
References [1] J.T. Beale, T. Kato, A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys. 94 (1), 61–66, 1984. [2] P. Constantin, Geometric statistics in turbulence, SIAM Rev. 36 (1), 73 – 98, 1994. [3] P. Constantin, On the Euler equations of incompressible fluids, Bull. Amer. Math. Soc. (N.S.) 44 (4), 603 – 621, 2007. [4] P. Constantin, C. Fefferman, A. Majda, Geometric constraints on potentially singular solutions for the 3-D Euler equations, Comm. Part. Diff. Eq., 21 (3-4), 559 – 571 [5] J. Deng, T.Y. Hou, X. Yu, Improved Geometric Conditions for Non-Blowup of the 3D Incompressible Euler Equation, Commun. Part. Diff. Eq. 31, 293 – 306, 2006.
BLOWUP RATE FOR EULER EQUATION
15
[6] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, Spectral theory and differential equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad J¨ orgens), 25–70. Lecture Notes in Mathematics 448. Springer, Berlin, 1975. [7] H. Kozono,Y. Taniuchi, Limiting case of the Sobolev inequality in BMO, with application to the Euler equations. Commun. Math. Phys. 214 (1), 191 – 200, 2000. [8] F. Planchon, An Extension of the Beale-Kato-Majda Criterion for the Euler Equations, Commun. Math. Phys. 232, 319 – 326, 2003.
T. Chen, Department of Mathematics, University of Texas at Austin. E-mail address: [email protected] ´, Department of Mathematics, University of Texas at Austin. N. Pavlovic E-mail address: [email protected]
Abstract. We prove a Beale-Kato-Majda criterion for the loss of regularity for solutions of the incompressible Euler equations in H s (R3 ), for s > 52 . Instead of double exponential estimates of Beale-Kato-Majda type, we obtain a single exponential bound on ku(t)kH s involving the dimensionless parameter introduced by P. Constantin in [2]. In particular, we derive lower bounds on the blowup rate of such solutions.
1. Introduction In this paper, we revisit the Beale-Kato-Majda criterion for the breakdown of smooth solutions to the 3D Euler equations. More precisely, we consider the incompressible Euler equations ∂t u + (u · ∇)u + ∇p = 0
(1.1)
∇·u = 0
(1.2)
u(x, 0) = u0
(1.3) 3
for an unknown velocity vector u(x, t) = (ui (x, t))1≤i≤3 ∈ R and pressure p = p(x, t) ∈ R, for position x ∈ R3 and time t ∈ [0, ∞). Existence and uniqueness of local in time solutions to (1.1) – (1.3) in the space C([0, T ], H s ) ∩ C 1 ([0, T ]; H s−1 ) ,
(1.4)
5 2,
has long been known for s > see for instance [6]. However, it is an open problem to determine whether such solutions can lose their regularity in finite time. An important result that addresses the question of a possible loss of regularity of solutions to Euler equations (1.1) – (1.3) is the criterion formulated by Beale-KatoMajda [1] in terms of the L∞ norm of the vorticity ω = ∇ ∧ u. More precisely, Beale-Kato-Majda in [1] proved the following theorem: Theorem 1.1. Let u be a solution to (1.1) – (1.3) in the class (1.4) for s ≥ 3 integer. Suppose that there exists a time T ∗ such that the solution cannot be continued in the class (1.4) to T = T ∗ . If T ∗ is the first such time, then Z T∗ kω(·, t)kL∞ dt = ∞. (1.5) 0
Date: June 17, 2011. 1
2
´ T. CHEN AND N. PAVLOVIC
The theorem is proved with a contradiction argument. Under the assumption Z T∗ kω(·, t)kL∞ dt < ∞ , 0
the authors of [1] show that ku(·, t)kH s ≤ C0 , for all t < T ∗ contradicting the hypothesis that T ∗ is the first time such that the solution cannot be continued to T = T ∗ . In particular, Beale-Kato-Majda obtain a double exponential bound for ku(·, t)kH s , which follows from the following estimates: Step 1 An energy-type bound on kukH s in terms of kDukL∞ , where Du = [∂i uj ]ij is a 3×3-matrix valued function. More specifically, one applies the operator Dα to equations (1.1)-(1.2), where α is an integer-valued multi-index with |α| ≤ s and uses a certain commutator estimate to derive d ku(·, tk2H s ≤ 2CkDukL∞ ku(·, t)k2H s , dt which via Gronwall’s inequality gives the bound: Z t ku(·, t)kH s ≤ ku0 kH s exp C kDu(·, τ )kL∞ dτ .
(1.6)
(1.7)
0
Step 2 An estimate on kDu(·, t)kL∞ based on the quantities kω(·, t)kL∞ , kω(·, t)kL2 , and log+ ku(·, t)kH 3 , given by kDu(·, t)kL∞ ≤ C 1 + 1 + log+ ku(·, t)kH 3 kω(·, t)kL∞ + kω(·, t)kL2 , (1.8) where C is a universal constant. Step 3 The bound on kω(·, t)kL2 in terms of kω(·, t)kL∞ given by d kω(·, t)k2L2 ≤ 2 D kω(·, t)kL∞ kω(·, t)k2L2 , dt which follows from taking the L2 (R3 )-inner product of ω with the equation for vorticity. Then, Gronwall’s inequality yields Z t kω(·, t)kL2 ≤ kω(·, 0)kL2 exp D kω( · , τ )kL∞ dτ . (1.9) 0
Consequently, one obtains the double exponential bound Z t ku(·, t)kH s ≤ ku0 kH s exp exp C kω( · , τ )kL∞ dτ .
(1.10)
0
from combining (1.7), (1.8) and (1.9). It is an open question whether (1.10) is sharp1. While we do not attempt to answer that question itself in this paper, we obtain a single exponential bound on the H s -norm of solution to Euler equations (1.1) - (1.3) in terms of the quantity 2 o n kω(t)k δ − 2δ+5 C , (1.11) `δ (t) = min L , ku0 kL2 1Single exponential bounds have been obtained in other solution spaces than those displayed above, see for instance [7] for such a result in BMO.
BLOWUP RATE FOR EULER EQUATION
3
where kωkC δ =
|ω(x) − ω(y)| |x − y|δ |x−y| 0 fixed, and δ > 0. More precisely, we prove the following theorem: Theorem 1.2. Let u be a solution to (1.1) - (1.3) in the class (1.4), for s = 52 + δ. Assume that `δ (t) is defined as above, and that Z T −5 (`δ (τ )) 2 dτ < ∞ . (1.13) 0
Then, there exists a finite positive constant Cδ = O(δ −1 ) independent of u and t such that Z t i h −5 ku(·, t)kH s ≤ ku0 kH s exp Cδ ku0 kL2 (`δ (τ )) 2 dτ 0
holds for 0 ≤ t ≤ T . The quantity `δ (t) has the dimension of length, and was introduced by Constantin in [2] (see also the work of Constantin, Fefferman and Majda [4] where a criterion for loss of regularity in terms of the direction of vorticity was obtained), where it was observed that Z T −5 (`δ (t)) 2 dt = ∞ (1.14) 0
is a necessary and sufficient condition for blow-up of Euler equations. In particular, the necessity of the condition follows from the inequality obtained in [2] kω(·, t)kL∞ ≤ ku(·, t)kL2 (`δ (t))
− 52
,
(1.15)
and Theorem 1.1 of Beale-Kato-Majda. This is so because Theorem 1.1 implies that RT if the solution cannot be continued to some time T , then 0 kω(·, t)kL∞ dt = ∞. As a consequence of (1.15), and conservation of energy ku(·, t)kL2 = ku0 kL2 ,
(1.16)
this in turn implies (1.14). However, by invoking the result of Beale-Kato-Majda in this argument, one again obtains a double exponential bound on ku(·, t)kH s in RT −5 terms of 0 (`δ (t)) 2 dt. We refer to [3, 5] for recent developments in this and related areas. In this paper, we observe that one can actually obtain a single exponential bound RT −5 on the H s -norm of the solution u(t) in terms of 0 (`δ (t)) 2 dt, as stated in Theorem 1.2. This is achieved by avoiding the use of the logarithmic inequality (1.8) from [1]. More precisely, we combine the energy bound (1.6) with a Calderon-Zygmund type bound on the symmetric and antisymmetric parts of Du. 5
Also, we obtain a lower bound on the blowup rate of solutions in H 2 +δ . Specifically, we prove:
4
´ T. CHEN AND N. PAVLOVIC
Theorem 1.3. Let u be a solution to (1.1) – (1.3) in the class 5
3
C([0, T ]; H 2 +δ ) ∩ C 1 ([0, T ]; H 2 +δ ).
(1.17)
Suppose that there exists a time T ∗ such that the solution cannot be continued in the class (1.17) to T = T ∗ . If T ∗ is the first such time then there exists a finite, positive constant C(δ, ku0 kL2 ) such that 1 1+ 25 δ ku(·, t)k 25 +δ ≥ C(δ, ku0 kL2 ) , (1.18) H T∗ − t under the condition that t is sufficiently close to T ∗ (see the conditions (3.22) and (3.23) below, with t0 = t). The proof of Theorem 1.3 can be outlined as follows. We assume that u is a solution in the class (1.17) that cannot be continued to T = T ∗ , and that T ∗ is the first such time. Invoking the local in time existence result, we derive a lower bound Tloc,t1 > 0 on the time of existence of solutions to Euler equations in (1.17) 5 for initial data u(t1 ) ∈ H 2 +δ at an arbitrary time t1 < T ∗ . By definition of T ∗ , we thus have t1 + Tloc,t1 < T ∗ .
(1.19)
5 2 +δ
-norm of the solution, we obtain in Section Based on an energy bound on the H 3 an expression for Tloc,t1 of the form Cku(·,t11)k 5 , which together with (1.19) H2
+δ
implies that 1 , (1.20) − t1 ) for all t1 < T ∗ . This is an “a priori” lower bound on the blowup rate. Subsequently, we improve (1.20) by a recursion argument in Theorem 1.3 for times t close to T ∗ , to yield the stronger bound (1.18). ku(·, t1 )k
5
H 2 +δ
>
C(T ∗
2. Proof of theorem 1.2 First we recall that the full gradient of velocity Du can be decomposed into symmetric and antisymmetric parts, Du = Du+ + Du−
(2.1)
where 1 Du ± DuT . 2 Du+ is called the deformation tensor. Du± =
(2.2)
In the following lemma we recall important properties of Du+ and Du− . For the convenience of the reader, we give proofs of those properties, although some of them are available in the literature, see e.g. [2]. Lemma 2.1. For both the symmetric and antisymmetric parts Du+ , Du− of Du, the L2 bound kDu± kL2 ≤ CkωkL2 . holds.
(2.3)
BLOWUP RATE FOR EULER EQUATION
5
The antisymmetric part Du− satisfies 1 Du− v = ω ∧ v 2 for any vector v ∈ R3 . The vorticity ω satisfies the identity Z 1 dy ω(ξ) = P.V. σ(b y ) ω(x + y) 3 , 4π |y| (”P.V.” denotes principal value) where σ(b y ) = 3 yb ⊗ yb − 1, with yb = Z σ(b y ) dµS 2 (y) = 0 ,
(2.4)
(2.5) y |y| .
Notably, (2.6)
S2
where dµS 2 denotes the standard measure on the sphere S 2 . The matrix components of the symmetric part have the form X X ` Du+ Tij` (ω` ) = Kij ∗ ω` , ij = `
(2.7)
`
` where ω` are the vector components of ω, and where the integral kernels Kij have the properties ` Kij (y) ` kσij kC 1 (S 2 )
Z S2
` σij (b y ) dµS 2 (y)
` = σij (b y ) |y|−3
(2.8)
≤ C
(2.9)
=
0.
(2.10)
Thus in particular, Tij` is a Calderon-Zygmund operator, for every i, j, ` ∈ {1, 2, 3}. Proof. An explicit calculation shows that the Fourier transform of Du as a function of ω b is given by c b b Du(ξ) = −[(∂i (∆−1 ∇ ∧ ω)j )b (ξ)]i,j = G(ξ) + H(ξ)
(2.11)
where ξ1 ξ2 ω b 3 − ξ1 ξ3 ω b2 1 b ξ ξ ω b G(ξ) := 1 3 1 2|ξ|2 −ξ1 ξ2 ω b1
−ξ2 ξ3 ω b2 ξ2 ξ3 ω b 1 − ξ1 ξ2 ω b3 ξ1 ξ2 ω b2
ξ2 ξ3 ω b3 (2.12) −ξ1 ξ3 ω b3 ξ1 ξ3 ω b2 − ξ2 ξ3 ω b1
and 0 1 b −ξ12 ω b3 H(ξ) := 2|ξ|2 ξ12 ω b2
ξ22 ω b3 0 −ξ22 ω b1
−ξ32 ω b2 ξ32 ω b1 , 0
(2.13)
using the notation ω bj ≡ ω bj (ξ) for brevity. Clearly, every component of G is given by a sum of Fourier multiplication opξ i ξj erators with symbols of the form |ξ| 2 , i 6= j, applied to a component of ω. For instance, Z dy G21 (x) = const. P.V. yb1 yb3 ω1 (x + y) 3 (2.14) |y| corresponds to the component G21 . It is easy to see that every component Gij is a sum of Calderon-Zygmund operators applied to components of ω, with kernel
´ T. CHEN AND N. PAVLOVIC
6
satisfying the asserted properties (2.8) ∼ (2.10). The same is true for the symmetric part, G+ = 12 (G + GT ). b The symmetric part of H(ξ) is given by ω3 0 (ξ22 − ξ12 )b 1 2 + 2 b (ξ − ξ )b ω 0 H (ξ) = 3 2 1 2|ξ|2 (ξ12 − ξ32 )b ω2 (ξ32 − ξ22 )b ω1
(ξ12 − ξ32 )b ω2 (ξ32 − ξ22 )b ω1 0
(2.15)
so that each component defines a Fourier multiplication operator with symbol of ξi2 −ξj2 |ξ|2 , i 6= j, acting on a x2i −x2j |x|n+2 ). That is, for instance,
the form form
component of ω (with associated kernel of the
+ H12 (x) = const P.V.
Z
(b y22 − yb12 ) ω3 (x + y)
dy . |y|3
(2.16)
The properties (2.8) ∼ (2.10) follow immediately. ` The Fourier transforms of the integral kernels Kij can be read off from the + + ` b b components Gij + Hij . In position space, one finds that σij (b y ) is obtained from a sum of terms proportional to terms of the form ybi1 ybj1 and (b yi22 − ybj22 ).
For the antisymmetric part Du− , one generally has Du− v = 21 (∇ ∧ u) ∧ v for any v ∈ R3 , and from u = −∆−1 ∇ ∧ ω, we get Du− v = 12 ω ∧ v, using that ∇ · u = 0. As a side remark, we note that while H − does not by itself exhibit the properties (2.8) ∼ (2.10), it combines with G− in a suitable manner to yield the stated properties of Du− , thanks to the condition ∇ · ω = 0. Next, Lemma 2.2 below provides an upper bound in terms of the quantity `δ (t) on singular integral operators applied to ω of the type appearing in (2.7). We note that similar bounds were used in [2] and [4] for the antisymmetric part Du− . Here, we observe that they also hold for the symmetric part Du+ . As shown in [4] for Du− , the proof of such a bound follows standard steps based on decomposing the singular integral into an inner and outer contribution. The inner contribution can be bounded based on a certain mean zero property, while the outer part is controlled via integration by parts. Lemma 2.2. For L > 0 fixed, and δ > 0, let `δ (t) be defined as above. Moreover, let ω` , ` = 1, 2, 3, denote the components of the vorticity vector ω(t). Then, any singular integral operator Z 1 dy T ω` (x) = P.V. σT (b y ) ω` (x + y) 3 (2.17) 4π |y| with Z σT (b y )dµS 2 (y) = 0
,
kσT kC 1 (S 2 ) < C ,
(2.18)
S2
satisfies 5
kT ω` kL∞ ≤ C(δ) ku0 kL2 `δ (t)− 2 for ` ∈ {1, 2, 3}, for a constant C(δ) = O(δ
−1
) independent of u and t.
(2.19)
BLOWUP RATE FOR EULER EQUATION
7
Proof. Let χ1 (x) be a smooth cutoff function which is identical to 1 on [0, 1], and identically 0 for x > 2. Moreover, let χR (x) = χ1 (x/R), and χcR = 1 − χR . We consider Z σT (b y ) ω` (x + y) |y|>
for > 0 arbitrary, where Z (I) :=
dy = (I) + (II) |y|3 dy |y|3
(2.21)
dy . |y|3
(2.22)
σT (b y ) ω` (x + y) χ`δ (t) (|y|)
|y|>
(2.20)
and Z (II) :=
σT (b y ) ω` (x + y) χc`δ (t) (|y|)
From the zero average property (2.18), we find Z dy σT (b y ) (ω` (x + y) − ω` (x)) χ`δ (t) (|y|) k(I)kL∞ = |y|3 |y|> Z dy ≤ kω` kC δ 3−δ |y| 52 , we recall the definitions of the homogenous and inhomogenous Besov norms for 1 ≤ p, q ≤ ∞, X q1 kukB˙ s = 2jqs kuj kqLp , (2.34) p,q
j∈Z
BLOWUP RATE FOR EULER EQUATION
9
respectively, s kukBp,q =
kukqLp + kukqB˙ s
q1
,
(2.35)
p,q
where uj = Pj u is the Paley-Littlewood projection of u of scale j. In analogy to s (1.6), we obtain the bound on the B2,2 Besov norm of u(t) given by 1 ∂t ku(t)k2B2,2 ≤ kDu(t)kL∞ ku(t)k2B2,2 , (2.36) s s 2 from a straightforward application of estimates obtained in [8]; details are given in the Appendix. Accordingly, since the left hand side yields s ∂t ku(t)kB s ∂t ku(t)k2B2,2 = 2ku(t)kB2,2 , s 2,2
(2.37)
s s ∂t ku(t)kB2,2 ≤ kDu(t)kL∞ ku(t)kB2,2 .
(2.38)
we get
However, Corollary 2.3 implies that kDu(t)kL∞
≤
kDu+ (t)kL∞ + kDu− (t)kL∞
≤
Cδ ku0 kL2 (`δ (t))− 2 .
5
(2.39)
Therefore, by combining (2.38) and (2.39) we obtain s ∂t ku(t)kB2,2
≤
5
s Cδ ku0 kL2 (`δ (t))− 2 ku(t)kB2,2 ,
which implies that ku(t)kH s
∼ ≤ ∼
s ku(t)kB2,2
t i 5 s ku0 kB2,2 exp Cδ ku0 kL2 `δ (s)− 2 ds 0 Z t h i 5 ku0 kH s exp Cδ ku0 kL2 `δ (s)− 2 ds ,
Z
h
0
for s ≥ 0, where we recall from (2.23) that Cδ = O(δ
−1
).
This completes the proof of Theorem 1.2.
3. Lower bounds on the blowup rate In this section, we prove Theorem 1.3. Recalling the energy bound (2.38), s s ∂t ku(t)kB2,2 ≤ kDu(t)kL∞ ku(t)kB2,2 ,
(3.1)
we invoke the Sobolev embedding kDukL∞
≤
c L1 kDuk Z 21 ≤ dξ hξi−3−2δ kDuk
3
H 2 +δ
≤ Cδ kuk
5
∼ Cδ kuk
5 +δ
H 2 +δ 2 B2,2
,
(3.2)
´ T. CHEN AND N. PAVLOVIC
10 1
with Cδ = O(δ − 2 ), to get, for s =
5 2
+ δ,
2 s s ) . ∂t ku(t)kB2,2 ≤ Cδ (ku(t)kB2,2
(3.3)
Straightforward integration implies 1 1 − ≤ Cδ (t − t0 ) . − s s ku(t)kB2,2 ku(t0 )kB2,2
(3.4)
Hence, ku(t)kH s
∼ ≤
s ku(t)kB2,2 s ku(t0 )kB2,2 s 1 − (t − t0 )Cδ ku(t0 )kB2,2
ku(t0 )kH s , (3.5) 1 − (t − t0 )Cδ ku(t0 )kH s where a possible trivial modification of Cδ is implicit in passing to the last line. This implies that the solution u(t) is locally well-posed in H s , with s = 25 + δ, for ∼
1 . (3.6) Cδ ku(t0 )kH s In particular, this infers that if T ∗ is the first time beyond which the solution u cannot be continued, one necessarily has that 1 T ∗ > t0 + . (3.7) Cδ ku(t0 )kH s This in turn implies an a priori lower bound on the blowup rate given by 1 (3.8) ku(t)kH s > Cδ (T ∗ − t) for all 0 ≤ t < T ∗ . The lower bound on the blowup rate stated in Theorem 1.3 is stronger than this estimate, and we shall prove it in the sequel. t0 ≤ t < t0 +
To begin with, we note that kω(t)kC δ
≤ Cδ kω(t)k ≤ Cδ ku(t)k
3
H 2 +δ 5
H 2 +δ
≤
Cδ ku(t0 )k
5
H 2 +δ
1 − (t − t0 )Cδ ku(t0 )k
.
(3.9)
5
H 2 +δ
5
That is, local well-posedness of u in H 2 +δ implies δ-Holder continuity of the vorticity. The parameter L in the definition (1.11) of `δ (t) is arbitrary. Thus, in view of (3.9), we may now let L → ∞ for convenience. Then, 2 kω(t)k δ 2δ+5 · 52 5 C `δ (t)− 2 = ku0 kL2 Cδ ku(t)k 5 +δ 1−δ˜ H2 ≤ ku0 kL2 C 1−δ˜ 1−δ˜ ku(t0 )kH s δ , (3.10) ≤ ku0 kL2 1 − (t − t0 )Cδ ku(t0 )kH s
BLOWUP RATE FOR EULER EQUATION
11
where 2δ 5 and s = + δ . 5 + 2δ 2 We note that while the right hand side of (3.10) diverges as t approaches δ˜ :=
1 , Cδ ku(t0 )kH s
t1 := t0 + the integral Z t1 5 `δ (t)− 2 dt
Cδ 1−δ˜ ku0 kL2 =: B0 (δ)
≤
t0
Z
t1
t0
(3.11)
(3.12)
1−δ˜ ku(t0 )kH s dt 1 − (t − t0 )Cδ ku(t0 )kH s (3.13)
converges for δ > 0 (⇔ δe > 0). This implies that the solution u(t) for t ∈ [t0 , t1 ) can be extended to t > t1 . In particular, we obtain that ku(t1 )k
H
5 +δ 2
≤
ku(t0 )k
5 +δ 2
≤
ku(t0 )k
5
H
H 2 +δ
Z t1 5 exp Cδ ku0 kL2 (`δ (t))− 2 dt t0 exp Cδ ku0 kL2 B0 (δ)
(3.14)
from Theorem 1.2. 5
We may now repeat the above estimates with initial data u(t1 ) in H 2 +δ , thus obtaining a local well-posedness interval [t1 , t2 ]. Accordingly, we may set t2 to be given by 1 t2 := t1 + . (3.15) Cδ ku(t1 )kH s More generally, we define the discrete times tj by 1 . Cδ ku(tj )kH s
tj+1 := tj +
(3.16)
We then have ku(tj+1 )kH s ≤ exp Cδ ku0 kL2 Bj (δ) ku(tj )kH s ,
(3.17)
where Bj (δ) is defined by Cδ ku0 kL2 Bj (δ) Cδ 1−δ˜ ku0 kL2 ku k 2 δ˜ ˜
:= Cδ ku0 kL2
Z
tj+1
tj
1−δ˜ ku(tj )kH s dt 1 − (t − tj )Cδ ku(tj )kH s
1 1−δ 0 L Cδ ˜ ku(t δ j )kH s ku k 2 δ˜ 0 L =: bδ . ku(tj )kH s =
(3.18)
Letting ku k 2 δ˜ 0 L ρj := exp bδ , ku(tj )kH s
(3.19)
´ T. CHEN AND N. PAVLOVIC
12
we have ku(tj )kH s ≤ ρj−1 ku(tj−1 )kH s ,
(3.20)
and we remark that (ρj )j satisfy the recursive estimates ρj
δ˜ ku0 kL2 ρj−1 ku(tj−1 )kH s
≥
exp bδ
=
( ρj−1 )ρj−1 ˜ δ exp ρ− j−1 ln ρj−1 .
=
˜ −δ
(3.21)
We note that from its definition, ρj > 1 for all j. We shall now assume that T ∗ > 0 is the first time beyond which the solution u(t) cannot be continued. Thus, by choosing t0 close enough to T ∗ , (3.8) implies that ku(t0 )kH s can be made sufficiently large that the following hold: (1) The quantity bδ
ku k 2 δ˜ 0 L 1 ku(t0 )kH s
(3.22)
is small. e independent of j such that (2) There is a positive, finite constant C e ku(t0 )kH s ku(tj )kH s ≥ C
(3.23)
holds for all j ∈ N. Without any loss of generality (by a redefinition of the e = 1. constant bδ if necessary), we can assume that C e = 1 implies that ρj ≤ ρ0 for all j. Then, for any Accordingly, (3.23) with C N ∈ N, T ∗ − t0
≥
N X (tj+1 − tj ) j=0
= = ≥ ≥
1 1 1 + ··· + Cδ ku(t0 )kH s ku(tN )kH s 1 ku(t0 )kH s ku(t0 )kH s 1+ + ··· + Cδ ku(t0 )kH s ku(t1 )kH s ku(tN )kH s 1 1 1 1+ + ··· + Cδ ku(t0 )kH s ρ0 ρ0 · · · ρN 1 1 1 1+ + ··· + N Cδ ku(t0 )kH s ρ0 ρ0
(3.24)
from ρ1j ≥ ρ10 for all j, and the fact that ρ0 > 1 since the argument in the exponent (3.19) is positive.
BLOWUP RATE FOR EULER EQUATION
13
Then, letting N → ∞, we obtain 1 ∗ T − t0
1 ≤ Cδ ku(t0 )kH s 1 − ρ0 ku k 2 δ˜ 0 L = Cδ ku(t0 )kH s 1 − exp − bδ . ku(t0 )kH s
(3.25)
Next, we deduce a lower bound on the blowup rate. Invoking (3.22), we obtain
T∗
1 − t0
ku k 2 δ˜ 0 L ≤ Cδ ku(t0 )kH s 1 − exp − bδ ku(t0 )kH s ku k 2 δ˜ 0 L ≈ Cδ ku(t0 )kH s bδ ku(t0 )kH s =
˜
˜
δ Cδ bδ ku0 kδL2 ku(t0 )k1− Hs .
(3.26)
This implies a lower bound on the blowup rate of the form ku(t0 )k
H
5 +δ 2
≥ =
1 1−1 δ˜ T ∗ − t0 1 2δ+5 5 C(δ, ku0 kL2 ) , T ∗ − t0 C(δ, ku0 kL2 )
(3.27)
under the condition that (3.22) and (3.23) hold. This concludes our proof of Theorem 1.3.
Appendix A. Proof of inequality (2.38) for s >
5 2
In this Appendix, we prove (2.38) which follows from (2.36), 1 ∂t ku(t)k2B2,2 . kDu(t)kL∞ ku(t)k2B2,2 , s s 2
(A.1)
for s > 25 . We invoke Eq. (26) in the work [8] of F. Planchon, which is valid for s > 1 + n2 in n dimensions (thus, s > 25 in our case of n = 3), for parameter values p = q = 2 in the notation of that paper. It yields 1 ∂t 22js kuj k2L2 2
. 22js
X
kSj+1 DukL∞ kuk kL2 kuj kL2
k∼j
+ 22js
X
kuk kL2 kuk0 kL2 kDuj kL∞
(A.2)
j.k∼k0
where uk = Pk u is the Paley-Littlewood projection of u at scale k, and Sj = P 0 P 0 j ≤j j is the Paley-Littlewood projection to scales ≤ j.
´ T. CHEN AND N. PAVLOVIC
14
Summing over j, 1 X 2js ∂t 2 kuj k2L2 2 j
. sup kSj+1 DukL∞
X
j
22js
j
+
kuk kL2 kuj kL2
k∼j 0
X X
22s(j−k) 2ks kuk kL2 2k s kuk0 kL2
k∼k0 &j
j
. kDukL∞
X
X
22js kuj k2L2
j
+
XX k
. kDukL∞
X
22s(j−k) 2ks kuk k2L2
j.k
22js kuj k2L2 .
(A.3)
j
To pass to the second inequality, we used that kSj+1 DukL∞ = kmj+1 ∗ DukL∞ . kDukL∞ kmj+1 kL1 ,
(A.4)
where m cj is the symbol of the Fourier multiplication operator Sj , and the fact that kmj kL1 ∼ 1 uniformly in j. Accordingly, we get 1 ∂t ku(t)k2B˙ s . kDu(t)kL∞ ku(t)k2B˙ s . 2,2 2,2 2
(A.5)
ku(t)k2B2,2 = ku(t)k2L2 + ku(t)k2B˙ s , s
(A.6)
From 2,2
and energy conservation, ∂t ku(t)k2L2 = 0, we obtain 1 ∂t ku(t)k2B2,2 s 2
1 ∂t ku(t)k2B˙ s 2,2 2 . kDu(t)kL∞ ku(t)k2B˙ s
=
2,2
. kDu(t)k
L∞
This proves (A.1).
ku(t)k2B2,2 s
.
(A.7)
Acknowledgments. The work of T.C. was supported by NSF grant DMS 1009448. The work of N.P. was supported NSF grant number DMS 0758247 and an Alfred P. Sloan Research Fellowship.
References [1] J.T. Beale, T. Kato, A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys. 94 (1), 61–66, 1984. [2] P. Constantin, Geometric statistics in turbulence, SIAM Rev. 36 (1), 73 – 98, 1994. [3] P. Constantin, On the Euler equations of incompressible fluids, Bull. Amer. Math. Soc. (N.S.) 44 (4), 603 – 621, 2007. [4] P. Constantin, C. Fefferman, A. Majda, Geometric constraints on potentially singular solutions for the 3-D Euler equations, Comm. Part. Diff. Eq., 21 (3-4), 559 – 571 [5] J. Deng, T.Y. Hou, X. Yu, Improved Geometric Conditions for Non-Blowup of the 3D Incompressible Euler Equation, Commun. Part. Diff. Eq. 31, 293 – 306, 2006.
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[6] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, Spectral theory and differential equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad J¨ orgens), 25–70. Lecture Notes in Mathematics 448. Springer, Berlin, 1975. [7] H. Kozono,Y. Taniuchi, Limiting case of the Sobolev inequality in BMO, with application to the Euler equations. Commun. Math. Phys. 214 (1), 191 – 200, 2000. [8] F. Planchon, An Extension of the Beale-Kato-Majda Criterion for the Euler Equations, Commun. Math. Phys. 232, 319 – 326, 2003.
T. Chen, Department of Mathematics, University of Texas at Austin. E-mail address: [email protected] ´, Department of Mathematics, University of Texas at Austin. N. Pavlovic E-mail address: [email protected]
