1. Asymptotic Analysis 38 (2004) 339–358
IOS Press
339
Local existence and blow-up criterion
for the Euler equations in the Besov spaces
Dongho Chae
Department of Mathematics, Seoul National University, Seoul 151-742, Korea
E-mail: dhchae@math.snu.ac.kr
Abstract. We prove the local in time existence and a blow-up criterion of solutions in the Besov spaces for the Euler equations
of inviscid incompressible fluid flows in Rn , n 2. As a corollary we obtain the persistence of Besov space regularity for the
solutions of the 2-D Euler equations with initial velocity belonging to the Besov spaces. For the proof of the results we establish
a logarithmic inequality of the Beale–Kato–Majda type and a Moser type of inequality in the Besov spaces.
1. Introduction and main results
We are concerned by the Euler equations for homogeneous incompressible fluid flows.
∂v
+ (v · ∇)v = −∇p, (x, t) ∈ Rn × (0, ∞),
∂t
div v = 0, (x, t) ∈ Rn × (0, ∞),
v(x, 0) = v0 (x),
x ∈ Rn ,
(1.1)
(1.2)
(1.3)
where v = (v1 , . . . , vn ), vj = vj (x, t), j = 1, . . . , n, is the velocity of the fluid, p = p(x, t) is the
scalar pressure, and v0 is the given initial velocity satisfying div v0 = 0. Given v0 ∈ H m (Rn ), for
integer m > n + 1, Kato proved the local in time existence and uniqueness of solution in the class
2
C([0, T ]; H m (Rn )), where T = T ( v0 H m ) [13]. Temam extended this result to bounded domains, in
H m (Ω) and in W m,p (Ω) (see [22,23]). Later, Kato and Ponce obtained a local existence result in the
more general Sobolev space, W s,p (Rn ) = (1 − ∆)−s/2 Lp (Rn ) for real number s > n/p + 1 [14]. One of
the most outstanding open problems in the mathematical fluid mechanics is to prove the global in time
continuation of the local solution, or to find an initial data v0 ∈ H m (Rn ) such that the associated local
solution blows up in finite time for n = 3. One of the most significant achievements in this direction is
the Beale–Kato–Majda criterion [3] for the blow-up of solutions, which states that
lim sup v(t)
t
T∗
=∞
(1.4)
ds = ∞,
(1.5)
Hm
if and only if
T∗
0
ω(s)
L∞
0921-7134/04/$17.00  2004 – IOS Press and the authors. All rights reserved

2. 340
D. Chae / Euler equations in the Besov spaces
where ω = curl v is the vorticity of the flow. Recently this criterion has been refined by Kozono and
Taniuch [16], replacing the L∞ norm of the vorticity by the BMO norm. (We note L∞ → BMO. See,
e.g., [21] for a detailed description of the space BMO.) In the Hölder space, Lichtenstein [17], Chemin
[10] proved the local existence of solution. Bahouri and Dehman [2] also obtained the blow-up criterion
for the local solution in the Hölder space. We also mention that there is a geometric type of blow-up
criterion, using the deep structure of the nonlinear term of the Euler equation [11]. For n = 2 it is
well-known that the local solution can be continued for all time, and one of the main questions in this
case is persistence problem of the regularity of initial data. Kato and Ponce proved the persistence of
the Sobolev space regularity for the super critical Sobolev space initial data, i.e., v0 ∈ W s,p (R2 ) with
s > 2/p+1 [15]. All the above results are concerned only in the super-critical cases, namely s > n/p+1.
In the current paper we study the initial value problem for the initial data belonging to the both supers
critical and critical Besov spaces, namely v0 ∈ Bp,q , s > n/p + 1 with p ∈ (1, ∞), q ∈ [1, ∞],
or s = n/p + 1, q = 1, and obtain the blow-up criterion in the Besov space as well as the local
in time existence of solutions. Our criterion is sharper than the Beale–Kato–Majda’s and the Kozono–
˙0
Taniuch’s result, in the sense that the BMO norm of vorticity is replaced by the B∞,∞ norm, which is
0
s
˙
weaker than the BMO norm (remember BMO → B∞,∞ ). We also recall the relation, B2,2 = H s (Rn )
between the Besov space and the fractional order Sobolev space. As a corollary of our criterion we
prove the global in time existence and persistence of Besov space regularity in the 2-dimensional Euler
s
equations for v0 ∈ Bp,q , s > 2/p + 1. We mention that Vishik obtained the global well-posedness of the
2-dimensional Euler equations in the critical Besov space [25,26]. We also mention that in the case of the
Navier–Stokes equations in the Besov spaces there are many studies by Cannone and his collaborators
(see [6,7] and references therein). In order to prove our blow-up criterion we establish the following
logarithmic Besov space inequality with features similar to the logarithmic Sobolev inequality in [3].
Proposition 1.1. Let s > n/p with p ∈ (1, ∞), q ∈ [1, ∞]. There exists a constant C such that for all
s
f ∈ Bp,q the following inequality holds:
f
L∞
C 1+ f
˙0
B∞,∞
log+ f
s
Bp,q
+1 .
(1.6)
The following is our main theorem.
Theorem 1.1 (Main theorem).
(i) Local in time existence: Let s > n/p + 1 with p ∈ (1, ∞), q ∈ [1, ∞], or s = n/p + 1
s
with p ∈ (1, ∞), q = 1. Suppose v0 ∈ Bp,q , satisfying div v0 = 0, is given. Then, their exists
s
s
T = T ( v0 Bp,q ) such that a unique solution v ∈ C([0, T ]; Bp,q ) of the system (1.1)–(1.3) exists.
(ii) Blow-up criterion:
(A) Super-critical case: Let s > n/p + 1, p ∈ (1, ∞), q ∈ [1, ∞]. Then, the local in time solution
s
s
v ∈ C([0, T ]; Bp,q ) blows up at T∗ > T in Bp,q , namely
lim sup v(t)
t
T∗
=∞
(1.7)
dt = ∞.
(1.8)
s
Bp,q
if and only if
T∗
0
ω(t)
˙0
B∞,∞

3. D. Chae / Euler equations in the Besov spaces
341
n/p+1
(B) Critical case: Let p ∈ (1, ∞). Then, the local in time solution v ∈ C([0, T ]; Bp,1
up at T∗ > T in
n/p+1
Bp,1 ,
lim sup v(t)
t
T∗
) blows
namely
n/p+1
Bp,1
=∞
(1.9)
if and only if
T∗
ω(t)
0
˙0
B∞,1
dt = ∞.
(1.10)
s
˙0
Remark 1.2. Since B2,2 = H s (Rn ), and L∞ → BMO → B∞,∞ , the above theorem improves the
original Beale–Kato–Majda criterion [3], and its refined version by Kozono and Taniuchi [16] for the
s
case of H s (Rn ). On the other hand, since B∞,∞ = C 0,s (Rn ), we find that the result of blow-up criterion
in the Hölder space by Bahouri and Dehman [2] corresponds to the extreme case of Theorem 1.1(ii)(A).
If n = 2 the conservation of vorticity, ω(t) L∞ = ω0 L∞ for all t > 0, combined with the embeds−
˙0
ding Bp,q 1 → L∞ (R2 ) → B∞,∞ for s > 1 (see Remark 2.1 below), implies immediately the following
corollary.
Corollary 1.1 (Persistence of the Besov space regularity). Let s > 2/p + 1 with p ∈ (1, ∞), q ∈ [1, ∞].
s
Suppose v0 ∈ Bp,q , satisfying div v0 = 0, is given. Then there exists a unique solution v ∈ C([0, ∞) :
s ) to the system (1.1)–(1.3) with n = 2.
Bp,q
s
Remark 1.3. In the case p = q = 2, since B2,2 = H s (R2 ), the Lebesgue space, we recover the result
by Kato–Ponce [15].
The results in this paper were announced in [8].
2. Proof of the main results
We first set our notations, and recall definitions on the Besov spaces. We follow [20,24]. Let S be the
ˆ
Schwartz class of rapidly decreasing functions. Given f ∈ S its Fourier transform F (f ) = f is defined
by
ˆ
f (ξ) =
1
(2π)n/2
Rn
e−ix·ξ f (x) dx.
We consider ϕ ∈ S satisfying Supp ϕ ⊂ {ξ ∈ Rn | 1 |ξ| 2}, and ϕ(ξ) > 0 if 2 < |ξ| < 3 . Setting
ˆ
ˆ
2
3
2
−j ξ) (in other words, ϕ (x) = 2jn ϕ(2j x)), we can adjust the normalization constant in front
ϕj = ϕ(2
ˆ
ˆ
j
of ϕ so that (see, e.g., [4, Lemma 6.1.7])
ˆ
ϕj (ξ) = 1
ˆ
j∈Z
∀ξ ∈ Rn {0}.

4. 342
D. Chae / Euler equations in the Besov spaces
Given k ∈ Z, we define the function Sk ∈ S by its Fourier transform
Sk (ξ) = 1 −
ϕj (ξ).
ˆ
j k+1
ˆ
In particular we set S−1 (ξ) = Φ(ξ).
We observe
Supp ϕj ∩ Supp ϕj = ∅
ˆ
ˆ
if |j − j |
2.
(2.11)
Let s ∈ R, p, q ∈ [0, ∞]. Given f ∈ S , we denote ∆j f = ϕj ∗ f , and then the homogeneous Besov
norm f B˙ is defined by
s
p,q





f
˙s
Bp,q
=
∞
1/q
2jqs ϕj ∗ f
−∞
q
Lp
if q ∈ [1, ∞),

 sup 2js ϕ ∗ f p


j
L
if q = ∞.
j
˙s
The homogeneous Besov space Bp,q is a semi-normed space with the semi-norm given by ·
s
s > 0, p, q ∈ [0, ∞] we define the inhomogeneous Besov space norm f Bp,q of f ∈ S as
f
s
Bp,q
= f
Lp
+ f
·
s
Bp,q
(see, e.g., [4,20,
ˆ
Lemma 2.1 (Bernstein’s lemma). Assume that f ∈ Lp , 1 p ∞, and Supp f ⊂ {2j−2
then there exists a constant Ck such that the following inequality holds
Dkf
Lp
For
˙s
Bp,q .
The inhomogeneous Besov space is a Banach space equipped with the norm,
24]). We now recall the following lemma [10].
−
Ck 1 2jk f
˙s
Bp,q .
Ck 2jk f
Lp
|ξ| < 2j },
(2.12)
Lp .
As an immediate corollary of the above lemma we have the equivalence of norms,
Dkf
˙s
Bp,q
∼ f
(2.13)
˙ s+k
Bp,q .
We are now ready to prove Proposition 1.1.
Proof of Proposition 1.1. We decompose
f (x) = S−N ∗ f (x) +
∆j f (x) +
|j|<N
∆j f (x) = {1} + {2} + {3},
(2.14)
j N
and estimate the first term as
{1}
S−N
Lp/(p−1)
f
Lp
C2−N n/p f
Lp ,
(2.15)

5. D. Chae / Euler equations in the Besov spaces
343
using simply the Hölder inequality. The estimate of the second term is immediate.
∆j f
{2}
CN f
L∞
|j|<N
(2.16)
˙0
B∞,∞ .
For the third term we have
∆j f
{3}
2nj/p ∆j f
C
L∞
j N
Lp
j N




2sjq ∆j f q p

L


 j N



1/q
−(s−n/p)j
2−(s−n/p)jq/(q−1)
(q−1)/q
if q ∈ (1, ∞),
j N
sup 2−(s−n/p)j
C f
Lp 2
j N
2sj ∆j f
Lp
j N

j N




 sup 2sj ∆ f p

2−(s−n/p)j

j
L

j N
2sj ∆j f
=
if q = 1,
if q = ∞
j N
˙s
Bp,q 2
−(s−n/p)N
(2.17)
.
Combining (2.15), (2.16), we obtain
f
C 2−aN f
L∞
s
Bp,q
+N f
˙0
B∞,∞
,
(2.18)
where we set
a = min s −
n n 1
.
, ,
p p log 2
Choosing
N=
log+ f
s
Bp,q
a log 2
+1 ,
we finally have (1.6).
Remark 2.1. If we decompose
f (x) = S−1 ∗ f (x) +
∆j f (x) = {I} + {II},
j 0
and using the similar estimates to (2.15), (2.17) for {I} and {II} respectively, we have immediately
f
L∞
C f
s
Bp,q
(2.19)
for s > n/p with p, q ∈ [1, ∞], or s = n/p with p ∈ [1, ∞], q = 1. This corresponds to the Sobolev
embedding in the Besov space version.

6. 344
D. Chae / Euler equations in the Besov spaces
Next, we will prove the following Moser type of inequality for the Besov spaces.
Lemma 2.2. Let s > 0, q ∈ [1, ∞], then there exists a constant C such that the following inequalities
hold
fg
C f
˙s
Bp,q
Lp1
g
˙s
Bp2 ,q
+ g
Lr1
f
Lr1
f
(2.20)
˙s
Br2 ,q
for the homogeneous Besov space, and
fg
C f
s
Bp,q
Lp1
g
s
Bp2 ,q
+ g
(2.21)
s
Br2 ,q
for the inhomogeneous Besov space respectively, where p1 , r1 ∈ [1, ∞] such that 1/p = 1/p1 + 1/p2 =
1/r1 + 1/r2 .
Proof. We use Bony’s formula for paraproduct of two functions
f g = Tf g + Tg f + R(f , g),
(2.22)
where
Sj−2 f ∆j g,
Tf g =
Sj−2 g∆j f ,
Tg f =
j
(2.23)
j
and
∆i f ∆j g.
R(f , g) =
(2.24)
|i−j| 1
From (2.10) we infer that
Supp F (Sj −2 f ∆j g) ⊂ 2j −2
and ∆j (Sj −2 f ∆j g) = 0 if |j − j |
∆j (f g) =
|ξ|
2 j +1 ,
4. Thus we have
∆j (Sj −2 f ∆j g) +
|j−j | 3
∆j (Sj
−2 g∆j
∆j (∆i f ∆j g)
f) +
|j−j | 3
|i −j | 1, max{i ,j } j−3
= I + II + III
(2.25)
Now we observe, thanks to Hölders’ and Young’s inequalities
I
sup Sj −2 f
Lp
Lp1
∆j g
|j−j | 3 j
C f
Lp2
Φ
L1
f
∆j g
Lp1
Lp2
|j−j | 3
∆j g
Lp1
|j−j | 3
Lp2 ,
(2.26)

7. D. Chae / Euler equations in the Besov spaces
345
and similarly
II
Lp
C g
∆j f
Lr1
Lr2 .
(2.27)
∆j g
Lp2 .
(2.28)
|j−j | 3
On the other hand,
III
C
Lp
f
j
Lp1
j−5
Thus, by the Minkowski inequality
fg
˙s
Bp,q
2qsj ∆j (f g)
=
j∈Z
q
Lp
1/q
1/q
f
2qsj ∆j g
Lp1
q
Lp2
j∈Z |j−j | 3
1/q
+ g
q
Lr2
2qsj ∆j f
Lr1
1/q
+ f
2qsj ∆j g
Lp1
j∈Z |j−j | 3
j∈Z j
q
Lp2
j−5
= {1} + {2} + {3}.
(2.29)
Since, in general,
3
q
Lp2
2qsj ∆j h
2−qks
=
j∈Z |j−j | 3
2qs(j+k) ∆j+k h
q
Lp2
j∈Z
k=−3
2qsj ∆j h
C
q
Lp2 ,
(2.30)
j∈Z
we have
{1} + {2}
C f
Lp1
g
˙s
Bp2 ,q
+C g
Lr1
f
(2.31)
˙s
Br2 ,q .
We estimate the third term as
{3} = f
2qs(j−j ) 2j s ∆j g
Lp1
j∈Z j
= f
−kqs
k −5
= f
k −5
2
∆j+k g
Lp2
2jqs ∆j g
q
Lp2
(j+k )s
q
1/q
j∈Z
2−kqs
Lp1
1/q
q
j−5
2
Lp1
Lp2
1/q
1/q
j∈Z
C f
Lp1
g
˙s
Bp2 ,q .
(2.32)

8. 346
D. Chae / Euler equations in the Besov spaces
We have proved (2.20). The inhomogeneous inequality (2.21), is obtained by adding the obvious estimate
fg
1
2
Lp
f
Lp1
g
Lp2
+ g
f
Lr1
(2.33)
Lr2
to (2.20).
We are now ready to prove our main theorem.
Proof of Theorem 1.1. We first derive a priori estimate for solutions of the system (1.1)–(1.3). Taking
operation, ∆j on the both sides of (1.1), we have
∂t ∆j v + (Sj−2 v · ∇)∆j v = (Sj−2 v · ∇)∆j v − ∆j (v · ∇)v − ∆j ∇p.
Since div Sj−2 v = 0, we deduce
∆j v(t)
t
∆j v0
Lp
Lp
+
0
(Sj−2 v · ∇)∆j v − ∆j (v · ∇)v (s)
Lp
+ ∆j ∇p
Lp
ds. (2.34)
Thus, by the Minkowski inequality we obtain
t
v(t)
s
Bp,q
v0
s
Bp,q
+
t
∇p
s
Bp,q
ds
2jqs (Sj−2 v · ∇)∆j v − ∆j (v · ∇)v (s)
+
0
0
j∈Z
q
Lp
1/q
ds.
(2.35)
We will now prove estimate
2
jqs
(Sj−2 v · ∇)∆j v − ∆j (v · ∇)v
j∈Z
q
Lp
1/q
C ∇v
L∞
v
(2.36)
s
Bp,q .
For this we follow closely the argument in [25] (see also [1,9]). Using Bony’s paraproduct formula [5],
we write
n
(Sj−2 v · ∇)∆j vk − ∆j (v · ∇)vk = −
n
∆j T∂i vk vi +
i=1
n
[Tvi ∂i , ∆j ]vk −
i=1
Tvi −Sj−2 vi ∂i ∆j vk
i=1
n
−
∆j R(vi , ∂i vk ) − R(Sj−2 vi , ∆j ∂i vk )
i=1
= I + II + III + IV.
(2.37)
We estimate term by term of (2.27). From the definition of paraproduct
n
∆j Sj −2 (∂i vk )∆j vi .
I=
i=1 j
(2.38)

9. D. Chae / Euler equations in the Besov spaces
347
As in the proof of Lemma 2.2 we observe
Supp F (Sj −2 ∂i vk ∆j vi ) ⊂ 2j −3
and ∆j (Sj −2 ∂i vk ∆j ) = 0 if |j − j |
|ξ|
2 j +1 ,
(2.39)
4. Thus, we can reduce the terms in the series
n
∆j Sj −2 (∂i vk )∆j vi .
I=
(2.40)
i=1 |j−j | 3
Thanks to Young’s inequality for convolutions we estimate
n
I
Lp
n
C
Sj −2 ∂i vk
L∞
∆j v
∆j vi
sup Sj −2 ∂i vk
∆j vi
Lp .
∇v
L∞
C
i=1 |j−j | 3 j
i=1 |j−j | 3
C
Lp
L∞
Lp
0
(2.41)
|j−j | 3
For the estimate of the second term of (2.37) we use the integral representation formula for the convolution. By the similar argument as the above we first note
[Sj −2 vi , ∆j ]∂i ∆j vk = 0
if |j − j |
4.
(2.42)
Thus, we have
n
II = −
[Sj
−2 vi , ∆j ]∂i ∆j
vk
i=1 |j−j | 3
n
=−
2jn
i=1 |j−j | 3
Rn
ϕ 2j (x − y) Sj −2 vi (x) − Sj −2 vi (y) ∂i ∆j vk (y) dy
n
=−
2j (n+1)
i=1 |j−j | 3
Rn
∂i ϕ 2j (x − y) Sj −2 vi (x) − Sj −2 vi (y) ∆j vk (y) dy
n
=−
2j (n+1)
i=1 |j−j | 3
Rn
∂i ϕ 2j (x − y)
1
×
n
=−
i=1 |j−j | 3
0
1
Rn
∂i ϕ(z)
0
(x − y) · ∇ (Sj −2 vi ) x + τ (y − x) dτ ∆j vk (y) dy
(z · ∇)(Sj −2 vi ) x − τ 2−j z ∆j vk x − 2−j z dz,
(2.43)

10. 348
D. Chae / Euler equations in the Besov spaces
where we used div Sj −2 v = 0 in the integration by part in the third line from the top. By the Minkowski
inequality we have
n
II
n
C
Lp
Sj −2 ∂m vi
∆j vk
L∞
Lp
∇v
C
i=1 m=1 |j−j | 3
L∞
∆j v
Lp .
(2.44)
|j−j | 3
For the estimate of the third term of (2.37) we first observe
n
j −1
n
Sj −2 (vi − Sj−2vi )∂i ∆j ∆j vk =
III =
i=1 |j−j | 1
∆m vi ∂i ∆j ∆j vk
Sj − 2
i=1 |j−j | 1
m=j−1
n
Sj −2 (∆j−1 vi + ∆j vi )∂i ∆j ∆j vk .
=
(2.45)
i=1 |j−j | 1
Using (2.12), we estimate
n
III
∆j−1 vi
Lp
L∞
+ ∆j vi
∂i ∆j ∆j vk
Lp
+ 2−j ∆j ∇vi
L∞
L∞
i=1 |j−j | 1
n
2−j+1 ∆j−1 ∇vi
C
L∞
· 2j ∆j ∆j vk
Lp
i=1 |j−j | 1
∇v
C
L∞
∆j v
(2.46)
Lp .
|j−j | 1
For the estimate of the fourth term of (2.28) we decompose
n
n
∆j ∂i R(vi − Sj−2 vi , vk ) +
IV =
i=1
∆j R(Sj−2 vi , ∂i vk ) − R(Sj−2 vi , ∆j ∂i vk )
i=1
= IV(1) + IV(2).
Since
n
i=1 ∂i ∆j
(2.47)
(vi − Sj−2 vi ) = 0, we have
n
∂i ∆j
IV(1) =
∆j (vi − Sj−2 vi )∆j vk
|j −j | 1
i=1
n
∆j
=
i=1
(∆j vi − Sj−2 ∆j vi )∆j ∂i vk .
|j −j | 1 j
j−3
(2.48)

11. D. Chae / Euler equations in the Besov spaces
349
Hence, we estimate
n
IV(1)
∆j vi
C
Lp
+ Sj ∆j vi
Lp
∆j vi
Lp
∆j ∂i vk
C
L∞
C
Lp
j−3 |j −j | 1
j
∆j ∂i vk
L∞
j−3 |j −j | 1
i=1 j
∇v
j
L∞
∆j v
(2.49)
Lp .
j−5
On the other hand, we note
n
∆j (∆j Sj−2 vi )∆j ∂i vk − (∆j Sj−2 vi )(∆j ∆j ∂i vk )
IV(2) =
i=1 |j −j | 1
n
[∆j , ∆j Sj−2 vi ]∆j ∂i vk .
=
i=1 j−1 j
j−3 |j −j | 1
Using the integral representation formula for the convolution as previously, we write
[∆j , ∆j Sj−2 vi ]∆j ∂i vk (x)
= 2jn
Rn
= 2j (n+1)
= 2j (n+1)
=−
∂i ϕ 2j (x − y) ∆j Sj−2 vi (y) − ∆j Sj−2 vi (x) ∆j ∂i vk (y) dy
Rn
Rn
∂i ϕ 2j (x − y) ∆j Sj−2 vi (y) − ∆j Sj−2 vi (x) ∆j vk (y) dy
1
Rn
1
∂i ϕ 2j (x − y)
∂i ϕ(z)
0
0
dτ (y − x) · ∇ (∆j Sj−2 vi ) x + τ (y − x) ∆j vk (y) dy
dτ (z · ∇)(∆j Sj−2 vi ) x − 2−j τ z ∆j vk x − 2−j z dz.
(2.50)
Hence,
n
[∆j , ∆j Sj−2 vi ]∆j ∂i vk
Lp
∆j Sj−2 ∂m vi
C
L∞
∆j vk
Lp
m=1
C ∇v
L∞
∆j v
Lp ,
(2.51)
and
IV(2)
Lp
C ∇v
∆j vk
L∞
(2.52)
Lp .
|j−j | 5
Summing up the estimates (2.41), (2.44), (2.46),(2.49) and (2.52), we obtain
(Sj−2 v · ∇)∆j v − ∆j (v · ∇)v
Lp
C ∇v
∆j v
L∞
|j−j | 5
Lp
∆j v
+
j >j−5
Lp
.
(2.53)

12. 350
D. Chae / Euler equations in the Besov spaces
Thus, by the Minkowski inequality
2
jqs
1/q
q
Lp
(Sj−2 v · ∇)∆j v − ∆j (v · ∇)v
j∈Z
1/q
C ∇v
2jqs ∆j v
L∞
q
Lp
1/q
+ C ∇v
2jqs ∆j v
L∞
j∈Z |j−j | 5
j∈Z j
q
Lp
j−5
= {1} + {2}.
(2.54)
By the same argument as in (2.29)–(2.32) we obtain
{1}
C ∇v
L∞
{2}
C ∇v
L∞
˙s
Bp,q ,
v
(2.55)
and
2(j−j )qs 2j s ∆j v
j∈Z j
Lp
q
1/q
C ∇v
L∞
v
˙s
Bp,q .
(2.56)
j−5
We have finished the proof of (2.36). Next, we estimate the pressure term in (2.35). We recall the wellknown representation of the pressure,
n
p=
(−∆)−1 ∂i vj ∂j vi .
(2.57)
i,j=1
Thus,
n
n
−1
∂i ∂j (−∆) ∂k vl ∂l vk =
∂i ∂j p =
Ri Rj (∂k vl ∂l vk ),
k ,l=1
(2.58)
k ,l=1
where Ri ’s are the n-dimensional Riesz transforms (see, e.g., [21]). Thus, using the norm equivalence
f B s ∼ Df Bp,q in (2.13), and the boundedness property of the singular integral operator from
˙
˙ s−1
p,q
s into itself [12], and (2.20), we have
˙
B
p ,q
n
∇p
˙s
Bp,q
n
C
∂i ∂j p
˙ s−1
Bp,q
i,j=1
C ∇v
n
Ri Rj (∂k vl ∂l vk )
C
i,j ,k ,l=1
L∞
∇v
˙ s−1
Bp,q
C ∇v
˙ s−1
Bp,q
C
∂k vl ∂l vk
˙ s−1
Bp,q
k ,l=1
L∞
v
˙s
Bp,q .
(2.59)
ds.
(2.60)
Substituting (2.59) and (2.36) into (2.35), we have
t
v(t)
˙s
Bp,q
v0
˙s
Bp,q
+C
0
∇v(s)
L∞
v(s)
˙s
Bp,q

13. D. Chae / Euler equations in the Besov spaces
351
This is our basic estimate in terms of the homogeneous Besov space semi-norm. In order to have similar
estimate in terms of the inhomogeneous Besov norm we derive the Lp estimate of solutions. From (1.1),
we obtain immediately
t
v(t)
v0
Lp
Lp
∇p(s)
+
0
Lp
ds.
(2.61)
On the other hand, by the Calderon–Zygmund inequality
n
∇p
∇∂j (−∆)−1 (v · ∇)vj
Lp
C (v · ∇)v
Lp
Lp
C ∇v
v
L∞
(2.62)
Lp
j=1
for 1 < p < ∞. Substituting (2.62) into (2.61), we find
t
v(t)
v0
Lp
Lp
+C
0
∇v(s)
L∞
v(s)
ds.
Lp
(2.63)
Adding (2.63) to (2.60), we obtain the inhomogeneous Besov space estimate;
t
v(t)
v0
s
Bp,q
s
Bp,q
+C
0
∇v(s)
L∞
v(s)
s
Bp,q
ds.
(2.64)
s−
This, combined with the embedding, Bp,q 1 (Rn ) → L∞ (Rn ), for s > n/p + 1 with p ∈ (1, ∞), q ∈
[1, ∞], or s = n/p + 1 with p ∈ (1, ∞), q = 1 (see Remark 2.1) yields the estimate
t
v(t)
v0
s
Bp,q
s
Bp,q
+C
0
∇v(s)
t
s−1
Bp,q
v(s)
s
Bp,q
ds
v0
s
Bp,q
+C
v(s)
0
2
s
Bp,q
ds. (2.65)
This completes the derivation of a priori estimate for the solutions of the system (1.1)–(1.3). We now set
s
s
XT := C [0, T ]; Bp,q .
Then, (2.65) implies
v
v0
s
XT
s
Bp,q
+ CT v
2
(2.66)
s
XT
In order to establish the local in time existence of solution we construct sequence {v(m) }, defined
recursively by solving the linear transport equations
∂t v(m) + v(m−1) · ∇ v(m) = −∇p(m−1) ,
div v(m) = 0,
v(m) (x, 0) = Sm v0 (x),
(2.67)
for m = 1, 2, . . . , where we set
n
p(m) =
(
(
(−∆)−1 ∂i vjm) ∂j vim) ,
i,j=1
m = 1, 2, . . . ,
(2.68)

14. 352
D. Chae / Euler equations in the Besov spaces
and v(0) = 0, p(0) = 0. Writing
∂t ∆j v(m) + Sj−2 v(m−1) · ∇ ∆j v(m)
= Sj−2 v(m−1) · ∇ ∆j v(m) − ∆j v(m−1) · ∇ v(m) − ∆j ∇p(m) ,
by the same procedure of estimates leading to (2.66), we obtain
v(m)
Sm v0
s
XT
s
Bp,q
+ CT v(m−1)
2
C0 v0
s
XT
s
Bp,q
+ C0 T v(m−1)
2
s
XT ,
(2.69)
where for some constant C0 . Thus, by the standard induction argument we find
v(m)
2C0 v0
s
XT
(2.70)
s
Bp,q
0 if T ∈ [0, T0 ], where we set
for all m
T0 =
1
2
8C0 v0
(2.71)
.
s
Bp,q
Taking differences between the equations for the (m + 1)-step and the (m) step equations of (2.67), we
have
∂t v(m+1) − v(m) + v(m) · ∇ v(m+1) − v(m)
= v(m−1) − v(m) · ∇v(m) − ∇∆j p(m) − p(m−1) ,
div v(m) = 0,
(2.72)
v(m+1) − v(m) (x, 0) = ∆m+1 v0 .
(2.73)
We write
∂t ∆j v(m+1) − v(m) + Sj−2 v(m) · ∇ ∆j v(m+1) − v(m)
= Sj−2 v(m) · ∇ ∆j v(m+1) − v(m) − ∆j
+ ∆j
v(m) · ∇ v(m+1) − v(m)
v(m−1) − v(m) · ∇v(m) − ∆j ∇ p(m) − p(m−1) .
(2.74)
Since div Sj−2 v(m) = 0, we obtain as before
∆j v(m+1) − v(m) (t)
T
+
0
T
+
0
Lp
(
(
∆j v0m+1) − v0m)
Lp
Sj−2 v(m) · ∇ ∆j v(m+1) − v(m) − ∆j
∆j
v(m−1) − v(m) · ∇v(m)
v(m) · ∇ v(m+1) − v(m)
T
Lp
dt +
0
∆j ∇ p(m) − p(m−1)
Lp
Lp
ds
ds
(2.75)

15. D. Chae / Euler equations in the Besov spaces
353
for all t ∈ [0, T ]. Multiplying 2j (s−1) on both sides of (2.75), and taking lq norm, we obtain by use of
the Minkowski inequality
v(m+1) − v(m) (t)
∆m+1 v0
− ∆j
˙ s−1
Bp,q
T
˙ s−1
Bp,q
2qj (s−1) ∆j Sj−2 v(m) · ∇ ∆j v(m+1) − v(m)
+
0
j∈Z
q
v(m) · ∇ v(m+1) − v(m)
T
+
0
Lp
v(m−1) − v(m) · ∇v(m)
˙ s−1
Bp,q
1/q
ds
T
∇ p(m) − p(m−1)
dt +
0
˙ s−1
Bp,q
ds
(2.76)
= I1 + I2 + I3 + I4 .
By (2.13)
C2−m−1 ∆m+1 v0
I1
C2−m v0
˙s
Bp,q
˙s
Bp,q
C2−m v0
(2.77)
s
Bp,q .
Estimate of I2 is exactly same as before, and we have
T
∇v(m)
I2
0
v(m+1) − v(m)
L∞
T
˙ s−1
Bp,q
ds
C
0
v(m)
s
Bp,q
v(m+1) − v(m)
s−1
Bp,q
ds.
(2.78)
Using (2.20) of Lemma 2.2 we estimate
T
I3
C
0
T
C
v(m−1) − v(m)
v(m)
0
s
Bp,q
L∞
∇v(m)
v(m−1) − v(m)
˙ s−1
Bp,q
s−1
Bp,q
+ v(m−1) − v(m)
˙ s−1
Bp,q
∇v(m)
L∞
ds
ds.
(2.79)
Finally we estimate the pressure term.
Using the fact
n
n
(
∂j vjm−1) =
j=1
(
(
∂i vim) − vim−1) = 0,
i=1
we have
n
∇p(m) − ∇p(m−1) =
(
(
(
(
(
(
(−∆)−1 ∇∂i vjm) − vjm−1) ∂j vim) + vjm−1) ∂j vim) − vim−1)
i,j=1
n
=
(
(
(
(
(
(
(−∆)−1 ∇∂i vjm) − vjm−1) ∂j vim) + ∂j vjm−1) vim) − vim−1)
i,j=1
n
=
(
(
(
(−∆)−1 ∇ ∂i vjm) − vjm−1) ∂j vim) + ∂j
i,j=1
(
(
(
∂i vjm−1) vim) − vim−1)
(2.80)

16. 354
D. Chae / Euler equations in the Besov spaces
Since both of (−∆)−1 ∇∂i and (−∆)−1 ∇∂j are the Calderon–Zygmund SIO’s, we obtain
n
I4
T
(
(
(
vjm) − vjm−1) ∂j vim)
C
i,j=1 0
T
∇v(m)
C
0
T
L∞
∇v(m−1)
+C
0
+ v(m) − v(m−1)
T
C
v(m) − v(m−1)
v(m)
0
s
Bp,q
∇v(m−1)
L∞
+ v(m−1)
+ ∇v(m)
˙ s−1
Bp,q
v(m) − v(m−1)
L∞
˙ s−1
Bp,q
(
(
(
∂i vjm−1) vim) − vim−1)
+
˙ s−1
Bp,q
v(m) − v(m−1)
dτ
L∞
dτ
˙ s−1
Bp,q
dτ
v(m) − v(m−1)
s
Bp,q
˙ s−1
Bp,q
˙ s−1
Bp,q
s−1
Bp,q
dτ.
(2.81)
Summing up (2.76)–(2.81), we have
v(m+1) (t) − v(m) (t)
T
+C
v(m) + v(m−1)
0
T
C2−m v0
˙ s−1
Bp,q
s
Bp,q
+C
s
Bp,q
v(m)
s
Bp,q
0
v(m) − v(m−1)
s−1
Bp,q
v(m+1) − v(m)
s−1
Bp,q
ds
ds
(2.82)
for all t ∈ [0, T ]. In order to have inhomogeneous space estimate we derive Lp estimate for v(m+1) −v(m) .
From (2.72) we have
v(m+1) (t) − v(m) (t)
T
0
C2−m ∇v0
T
+C
0
T
+C
v(m)
C2−m v0
T
0
0
L∞
+ v(m−1)
T
s
Bp,q
+C
v(m) + v(m−1)
s
Bp,q
0
s
Bp,q
v(m) · ∇ v(m+1) − v(m)
∇ p(m) − p(m−1)
∇ v(m+1) − v(m)
L∞
L∞
v(m)
0
dt +
Lp
∇v(m)
Lp
+
0
Lp
v(m)
+C
v(m) − v(m−1)
0
+C
T
Lp
Lp
T
v(m−1) − v(m) · ∇v(m)
+
T
∆m+1 v0
Lp
L∞
Lp
Lp
ds
ds
ds
ds
v(m) − v(m−1)
Lp
v(m+1) − v(m)
v(m) − v(m−1)
s−1
Bp,q
ds,
ds
s−1
Bp,q
ds
(2.83)

17. D. Chae / Euler equations in the Besov spaces
355
where we estimated the pressure term, using (2.80) and the Calderon–Zygmund inequality,
n
∇ p(m) − p(m−1)
(
(
(
vjm) − vjm−1) ∂j vim)
C
Lp
Lp
+
(
(
(
∂i vjm−1) vim) − vim−1)
Lp
i,j=1
C ∇v(m)
C v
(m)
v(m) − v(m−1)
L∞
L∞
+ v
(m−1)
L∞
Lp
v
+ ∇v(m−1)
(m)
−v
(m−1)
L∞
v(m) − v(m−1)
Lp
Lp .
for 1 < p < ∞, where used the Calderon–Zygmund inequality. Adding (2.83) to (2.82) we find
v(m+1) (t) − v(m) (t)
T
s−1
Bp,q
C2−m v0
T
s
Bp,q
+C
v(m)
s
Bp,q
0
v(m) − v(m−1)
s−1
Bp,q
C1 2−m + C1 T sup v(m+1) (t) − v(m) (t)
v(m+1) − v(m)
s−1
Bp,q
ds
s−1
Bp,q
+C
v(m) + v(m−1)
0
s
Bp,q
t∈[0,T ]
+ C1 T sup v(m) (t) − v(m−1) (t)
t∈[0,T ]
s−1
Bp,q ,
for T ∈ (0, T0 ], and the constant C1 = C1 ( v0
if C1 T < 1 , then
2
v(m+1) − v(m)
ds
(2.84)
s
Bp,q ),
where we used the uniform estimate (2.70). Thus,
C1 2−m+1 + 2C1 T v(m) − v(m−1)
s−1
XT
s−1
XT ,
(2.85)
and, from which we deduce
v(m+1) − v(m)
s−1
XT
C2−m
(2.86)
s−
1
if C1 T
, and the sequence {v(m) } converges in XT1 1 for T1 min{T0 , 1/(4C1 )}. From Eqs (2.67),
4
s
s
(2.68) for {v(m) } we find the limit v ∈ XT1 is a solution of the Euler system for initial data v0 ∈ Bp,q .
s ). Moreover this local solutions
This completes the proof of local existence of solution in C([0, T1 ]; Bp,q
satisfy the estimate
sup
t∈[0,T1 ]
v(t)
s
Bp,q
2C0 v0
s
Bp,q
(2.87)
for the constant C0 in (2.70). We now prove uniqueness of solutions satisfying (2.62). Let us consider the
s
s
two solutions (u, p1 ), (v, p2 ) with u, v ∈ C([0, T1 ]; Bp,q ) associated with the initial datum u0 , v0 ∈ Bp,q
respectively, and take the difference between the equations satisfied by them, then we find
∂t [u − v] + (v · ∇)[u − v] = [v − u] · ∇ u − ∇(p1 − p2 ),
div(u − v) = 0,
(u − v)(x, 0) = u0 − v0 .
(2.88)
(2.89)

18. 356
D. Chae / Euler equations in the Besov spaces
By exactly same procedure leading to (2.84) we have
u−v
u0 − v0
s−1
XT
s−1
Bp,q
+ C2 T u − v
(2.90)
s−1
XT
s
s
for sufficiently small T ∈ (0, T2 ], and C2 = C2 ( v0 Bp,q , u0 Bp,q ). We note that the estimate (2.62)
s
in XT for solutions u, v is essential in the derivation of (2.90). This proves the uniqueness of the local
s−
solution in XT 1 for T ∈ (0, T3 ] for T3 = min{T1 , T2 , 1/(2C2 )}. Thus we have finished existence and
s
s
uniqueness of solutions in C([0, T3 ]; Bp,q ) to the system (1.1)–(1.3) for v0 ∈ Bp,q .
Finally, we prove the blow-up criterion. We recall the well known fact that the elliptic system, div v =
0 and curl v = ω implies the relation between the gradient of velocity and the vorticity as
∇v = P(ω) + C3 ω,
(2.91)
where P is a singular integral operator of the Calderon–Zygmund type, and C3 is a constant matrix
˙0
(see, e.g., [18,19]). By the boundedness of the singular integral operator from B∞,∞ into itself [24], and
Proposition 1.1 we estimate
∇v
L∞
C 1 + ∇v
C 1+ ω
˙0
B∞,∞
˙0
B∞,∞
log+ ∇v
log+ v
s−1
Bp,q
s
Bp,q
+1
+1
(2.92)
for s > n/p + 1. Then, substituting (2.92) into (2.64), we obtain
t
v(t)
s
Bp,q
v0
s
Bp,q
+C
ω(s)
˙0
B∞,∞
0
+ 1 log+ v(s)
s
Bp,q
+ 1 v(s)
s
Bp,q
ds.
We finally have by Gronwall’s lemma
t
v(t)
s
Bp,q
v0
s
Bp,q
exp C4 exp C5
ω(s)
0
˙0
B∞,∞
+ 1 ds
(2.93)
for some constants C4 and C5 . The inequality (2.93) implies the “only if” part of the blow-up criterion
in Theorem 1.1 in the super-critical case. The “if" part of the super-critical case follows from the easy
estimates
T
ω(s)
0
˙0
B∞,∞
ds
T sup
ω(s)
0 s T
˙0
B∞,∞
CT sup
0 s T
v(s)
s
Bp,q ,
(2.94)
s−
˙0
where we used the embeddings, Bp,q 1 → L∞ → B∞,∞ for s > n/p + 1. On the other hand, in the
critical case we substitute
∇v
L∞
C ∇v
˙0
B∞,1
C ω
˙0
B∞,1
(2.95)

19. D. Chae / Euler equations in the Besov spaces
357
˙0
into (2.64) to obtain the “only if” part, where we used the embedding B∞,1 → L∞ , and the fact that our
˙0
singular integral operator P(·) in (2.91) is bounded from B∞,1 into itself. The “if” part of Theorem 1.1
follows also from the easy estimates
T
ω(s)
0
˙0
B∞,1
ds
T sup
0 s T
ω(s)
˙0
B1,∞
CT sup
0 s T
v(s)
n/p+1
Bp,1
,
(2.96)
n/p
˙ n/p
˙0
where we used the embeddings, Bp,1 → Bp,1 → B∞,1 for p ∈ (1, ∞). This completes the proof of
Theorem 1.1.
Acknowledgements
The author would like to thank deeply to Professor R. Temam for his careful reading of the manuscript
and corrections. This research is supported partially by the grant no. 2000-2-10200-002-5 from the basic
research program of the KOSEF.
References
[1] H. Bahouri and J.-Y. Chemin, Equations de transport relatives à des champs de vecteurs non-lipshitziens et mé canique
des fluides, Arch. Rational Mech. Anal. 127 (1994), 159–199.
[2] H. Bahouri and B. Dehman, Remarques sur l’apparition de singularités dans les écoulements Eulériens incompressibles à
donnée initiale Höldérienne, J. Math. Pures Appl. 73 (1994), 335–346.
[3] J.T. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm.
Math. Phys. 94 (1984), 61–66.
[4] J. Bergh and J. Löfström, Interpolation Spaces, Springer-Verlag, 1970.
[5] J.M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann.
École Norm. Sup. 14 (1981), 209–246.
[6] M. Cannone, Ondelettes, Paraproduits et Navier–Stokes, Diderot, Paris, 1995.
[7] M. Cannone, Nombres de Reynolds, stabilité et Navier–Stokes, Banach Center Publication, 1999.
[8] D. Chae, On the well-posedness of the Euler equations in the Besov and Triebel–Lizorkin spaces, in: Tosio Kato’s Method
and Principle for Evolution Equations in Mathematical Physics, A Proceedings of the workshop held at Hokkaido University, Japan, Yurinsha, June 27–29, 2001, Tokyo 2002, pp. 42–57; See also: Local existence and blow-up criterion for
the Euler equations in the Besov space, RIM-GARC, Seoul National University, Korea, Preprint No. 8, June 2001.
[9] J.-Y. Chemin, Régularité des trajectoires des particules d’un fluide incompressible remplissant l’espace, J. Math. Pures
Appl. 71(5) (1992), 407–417.
[10] J.-Y. Chemin, Perfect Incompressible Fluids, Clarendon Press, Oxford, 1998.
[11] P. Constantin, C. Fefferman and A. Majda, Geometric constraints on potentially singular solutions for the 3-D Euler
equations, Comm. Partial Differential Equations 21(3–4) (1996), 559–571.
[12] B. Jawerth, Some observations on Besov and Lizorkin–Triebel spaces, Math. Scand. 40 (1977), 94–104.
[13] T. Kato, Nonstationary flows of viscous and ideal fluids in R3 , J. Funct. Anal. 9 (1972), 296–305.
[14] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier–Stokes equations, Comm. Pure Appl. Math. 41
(1988), 891–907.
[15] T. Kato and G. Ponce, Well posedness of the Euler and Navier–Stokes equations in Lebesgue spaces Ls (R2 ), Rev. Mat.
p
Iberoamericana 2 (1986), 73–88.
[16] H. Kozono and Y. Taniuchi, Limiting case of the Sobolev inequality in BMO, with applications to the Euler equations,
Comm. Math. Phys. 214 (2000), 191–200.
[17] L. Lichtenstein, Über einige Existenzprobleme der Hydrodynamik homogener unzusammendrückbarer, reibunglosser
Flüssikeiten und die Helmholtzschen Wirbelsalitze, Math. Z. 23 (1925), 89–154; 26 (1927), 193–323, 387–415; 32 (1930),
608–725.
[18] A. Majda, Vorticity and the mathematical theory of incompressible fluid flow, Comm. Pure Appl. Math. 39 (1986), 187–
220.

20. 358
D. Chae / Euler equations in the Besov spaces
[19] A. Majda, Mathematical Theory of Fluid Mechanics, Princeton University Graduate Course Lecture Note, 1986.
[20] J. Peetre, New Thoughts on Besov Spaces, Duke University Press, 1976.
[21] E.M. Stein, Harmonic Analysis; Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical
Series, Vol. 43, Princeton University Press, 1993.
[22] R. Temam, On the Euler equations of incompressible flows, J. Funct. Anal. 20 (1975), 32–43.
[23] R. Temam, Local existence of C ∞ solutions of the Euler equations of incompressible perfect fluids, in: Proc. Conf. on
Turblence and Navier–Stokes Equations, Lecture Notes in Math., Vol. 565, Springer-Verlag, New York, 1976.
[24] H. Triebel, Theory of Function Spaces II, Birkhäuser, 1992.
[25] M. Vishik, Hydrodynamics in Besov spaces, Arch. Rational Mech. Anal. 145 (1998), 197–214.
[26] M. Vishik, Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type, Ann. Sci. École Norm.
Sup. (4) 32 (1999), 769–812.