Nonlinear Analysis 67 (2007) 3013–3036
www.elsevier.com/locate/na
Existence and uniqueness results for the 2-D dissipative
quasi-geostrophic equation
Jiahong Wu ∗
Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK 74078, USA
Received 3 July 2006; accepted 25 September 2006
Abstract
This paper concerns itself with Besov space solutions of the 2-D quasi-geostrophic (QG) equation with dissipation induced by
a fractional Laplacian (−1)α . The goal is threefold: first, to extend a previous result on solutions in the inhomogeneous Besov
r [J. Wu, Global solutions of the 2D dissipative quasi-geostrophic equation in Besov spaces, SIAM J. Math. Anal.
space B2,q
36 (2004–2005) 1014–1030] to cover the case when r = 2 − 2α; second, to establish the global existence of solutions in the
homogeneous Besov space B̊ rp,q with general indices p and q; and third, to determine the uniqueness of solutions in any one of
s+ 2α
q
s , B̊ r , L q ((0, T ); B
the four spaces: B2,q
p,q
2,q
c 2006 Elsevier Ltd. All rights reserved.
2α
r+ q
) and L q ((0, T ); B̊ p,q ), where s ≥ 2 − 2α and r = 1 − 2α + 2p .
MSC: 35Q35; 76B03
Keywords: 2-D quasi-geostrophic equation; Besov space; Existence and uniqueness
1. Introduction
The 2-D dissipative quasi-geostrophic (QG) equation concerned here assumes the form
∂t θ + u · ∇θ + κ(−1)α θ = 0,
(1.1)
where κ > 0 and α ≥ 0 are parameters, θ = θ (x, t) is a scalar function of x ∈ R2 and t ≥ 0, and u is a 2-D velocity
field determined by θ through the relations
u = (u 1 , u 2 ) = (−∂x2 ψ, ∂x1 ψ)
and
1
(−1) 2 ψ = θ.
The fractional Laplacian operator (−1)β for a real number β is defined through the Fourier transform, namely
\
β f (ξ ) = (2π|ξ |)2β b
(−1)
f (ξ )
∗ Tel.: +1 405 744 5788; fax: +1 405 744 8275.
E-mail address: [email protected].
c 2006 Elsevier Ltd. All rights reserved.
0362-546X/$ - see front matter doi:10.1016/j.na.2006.09.050
(1.2)
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J. Wu / Nonlinear Analysis 67 (2007) 3013–3036
where the Fourier transform b
f is given by
Z
b
f (x) e−2πix·ξ dx.
f (ξ ) =
R2
1
For notational convenience, we write Λ for (−1) 2 and combine the relations in (1.2) into
u = ∇ ⊥ Λ−1 θ,
where ∇ ⊥ = (−∂x2 , ∂x1 ). Physically, (1.1) models the temperature evolution on the 2-D boundary of a 3-D quasigeostrophic flow and is sometimes referred to as the surface QG equation [8,13].
Fundamental mathematical issues concerning the 2-D dissipative QG equation (1.1) include the global existence
of classical solutions and the uniqueness of solutions in weaker senses. In the subcritical case α > 21 , these issues
have been more or less resolved [9,14]. When α ≤ 21 , the issue on the global existence of classical solutions becomes
extremely difficult. For the critical case α = 12 , this issue was first dealt with by Constantin et al. [7] and later studied
in [3,6,10,11,17] and other works. A recent work of Kiselev et al. [12] appears to have resolved this issue (in the
periodic case) by removing the L ∞ -smallness condition of [7]. Another recent progress on the critical dissipative QG
equation was given in the work by Caffarelli and Vasseur [1]. We also mention other interesting investigations on
related issues (see cf. [2,15,16]). The supercritical case α < 12 remains a big challenge. This paper is mainly devoted
to understanding the behavior of solutions of (1.1) with α < 21 . Although our attention is mainly focused on the case
when α < 12 , the results presented here also hold for α ≥ 12 . We attempt to accomplish three major goals that we now
describe.
r with
In [17], we established the global existence of solutions of (1.1) in the inhomogeneous Besov space B2,q
1 ≤ q ≤ ∞ and r > 2 − 2α when the corresponding initial data θ0 satisfies
r
kθ0 k B2,q
≤ Cκ
for some suitable constant C. Our first goal is to extend this result to cover the case when r = 2−2α. For this purpose,
2−2α
. When combined with a procedure detailed in [17], this
we derive a new a priori bound on solutions of (1.1) in B2,q
2−2α
new bound yields the global existence of solutions in B2,q
. As a special consequence of this result, the 2-D critical
QG equation ((1.1) with α = 21 ) possesses a global H 1 -solution if the initial datum is comparable to κ.
Our second goal is to explore solutions of (1.1) in the homogeneous Besov space B̊ rp,q with general indices
2 ≤ p < ∞ and 1 ≤ q ≤ ∞. This study was partially motivated by the lower bound
Z
p
| f | p−2 f · (−1)α f dx ≥ C22α j k f k L p
(1.3)
Rd
valid for any function f that decays sufficiently fast at infinity and satisfies
supp b
f ⊂ {ξ ∈ Rd : K 1 2 j ≤ |ξ | ≤ K 2 2 j },
where 0 < K 1 ≤ K 2 are constants and j is an integer. This inequality, recently established in [5,18], provides a lower
bound for the integral generated by the dissipative term when we estimate solutions of (1.1) in B̊ rp,q . Combining this
lower bound with suitable upper bounds for the nonlinear term, we are able to derive a priori estimates for solutions
of (1.1) in B̊ rp,q . Applying the method of successive approximation, we then establish the existence and uniqueness of
solutions emanating from initial data θ0 satisfying
kθ0 k B̊ r
p,q
≤ Cκ,
where 2 ≤ p < ∞, 1 ≤ q ≤ ∞ and r = 1 − 2α + 2p . Setting q = p, we obtain as a special consequence the global
solutions in the homogeneous Sobolev space W̊ p,r , where 2 ≤ p < ∞ and r = 1 − 2α + 2p .
The third goal is to determine the uniqueness of solutions of (1.1) in the spaces
r + 2α
s+ 2α
q
q
s
r
q
q
and L (0, T ); B̊ p,q
B2,q ,
B̊ p,q ,
L (0, T ); B2,q
(1.4)
J. Wu / Nonlinear Analysis 67 (2007) 3013–3036
3015
where s ≥ 2 − 2α and r = 1 − 2α + 2p . For two solutions in any one of these spaces, we establish suitable bounds
for their difference which yield as a special consequence the uniqueness. We conclude that two solutions θ and e
θ
emanating from the same initial datum must coincide if they satisfy
r
kθk B2,q
≤ Cκ
and
r
ke
θk B2,q
≤ Cκ
(1.5)
for some constant C. A parallel result holds for solutions in B̊ rp,q . In addition, we prove that any two solutions in
r + 2α
the space L q ((0, T ); B2,q q ) must be identical if they satisfy the same initial condition. The same conclusion can be
r + 2α
drawn for solutions in L q ((0, T ); B̊ p,q q ). A special consequence is the uniqueness of H 1 (or H̊ 1 ) solutions of the
critical QG equation if their norms are comparable to κ. A more significant corollary is the uniqueness of solutions
3
3
of the critical QG equation in L 2 ((0, T ); H 2 ) (or L 2 ((0, T ); H̊ 2 )). Since H 1 (or H̊ 1 ) solutions of the critical QG
3
3
equation are in general also in L 2 ((0, T ); H 2 ) (or L 2 ((0, T ); H̊ 2 )), this corollary indicates the uniqueness of H 1 (or
H̊ 1 ) solutions that do not necessarily satisfy (1.5).
The rest of this paper is divided into four sections and an Appendix A. Section 2 is further divided into three
subsections. Section 2.1 recalls the definitions of Besov spaces, Section 2.2 presents the definitions of two spaces
involving time and the relations between them, and Section 2.3 provides the lower bound (1.3). Section 3 focuses on
r while Section 4 is devoted to solutions in B̊ r . Section 5 deals with the uniqueness of solutions
solutions in B2,q
p,q
in the spaces in (1.4). The Appendix A proves a Bernstein inequality involving fractional Laplacians and derives a
commutator estimate that is used in Sections 3 through 5.
We finally remark that after the completion of this manuscript, we learned that a result related to Theorem 4.2 in
Section 4 was obtained by Chen et al. [5].
2. Function spaces and a lower bound
This section makes necessary preparations for the subsequent sections. It is divided into three subsections.
Section 2.1 provides the definition of Besov spaces. Section 2.2 presents two spaces involving time and their relations.
The last subsection recalls a lower bound for an integral involving fractional Laplacians.
2.1. Besov spaces
We start with several notations. S denotes the usual Schwarz class and S 0 its dual, the space of tempered
distributions. S0 denotes a subspace of S defined by
Z
S0 = φ ∈ S :
φ(x)x γ dx = 0, |γ | = 0, 1, 2, . . .
Rd
and S00 denotes its dual. S00 can be identified as
S00 = S 0 /S0⊥ = S 0 /P
where P denotes the space of multinomials.
To introduce the Littlewood–Paley decomposition, we write for each j ∈ Z
A j = {ξ ∈ Rd : 2 j−1 ≤ |ξ | < 2 j+1 }.
(2.1)
The Littlewood–Paley decomposition asserts the existence of a sequence of functions {Φ j } j∈Z ∈ S such that
bj ⊂ A j ,
suppΦ
b j (ξ ) = Φ
b0 (2− j ξ )
Φ
and
∞
X
j=−∞
1,
b
Φ j (ξ ) =
0,
if ξ ∈ Rd \ {0},
if ξ = 0.
or
Φ j (x) = 2 jd Φ0 (2 j x),
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J. Wu / Nonlinear Analysis 67 (2007) 3013–3036
Therefore, for a general function ψ ∈ S, we have
∞
X
b j (ξ )ψ(ξ
b ) = ψ(ξ
b )
Φ
for ξ ∈ Rd \ {0}.
j=−∞
In addition, if ψ ∈ S0 , then
∞
X
b j (ξ )ψ(ξ
b ) = ψ(ξ
b )
Φ
for any ξ ∈ Rd .
j=−∞
That is, for ψ ∈ S0 ,
∞
X
Φj ∗ ψ = ψ
j=−∞
and hence
∞
X
Φ j ∗ f = f,
f ∈ S00
j=−∞
in the sense of weak-∗ topology of S00 . For notational convenience, we define
1 j f = Φ j ∗ f,
(2.2)
j ∈ Z.
For s ∈ R and 1 ≤ p, q ≤ ∞, the homogeneous Besov space B̊ sp,q consists of f ∈ S00 satisfying
k f k B̊ s
p,q
≡ k2 js k1 j f k L p kl q < ∞.
We now choose Ψ ∈ S such that
b (ξ ) = 1 −
Ψ
∞
X
b j (ξ ),
Φ
ξ ∈ Rd .
j=0
Then, for any ψ ∈ S,
Ψ ∗ψ +
∞
X
Φj ∗ ψ = ψ
j=0
and hence
Ψ∗ f +
∞
X
Φj ∗ f = f
(2.3)
j=0
in S 0 for any f ∈ S 0 .
To define the inhomogeneous Besov space, we set

0, if j ≤ −2,
10j f = Ψ ∗ f, if j = −1,

Φ j ∗ f, if j = 0, 1, 2, . . . .
(2.4)
The inhomogeneous Besov space B sp,q with 1 ≤ p, q ≤ ∞ and s ∈ R consists of functions f ∈ S 0 satisfying
k f k B sp,q ≡ k2 js k10j f k L p kl q < ∞.
For notational convenience, we will write 1 j for 10j . There will be no confusion if we keep in mind that 1 j ’s
associated with the homogeneous Besov spaces is defined in (2.2) while those associated with the inhomogeneous
Besov spaces are defined in (2.4).
J. Wu / Nonlinear Analysis 67 (2007) 3013–3036
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We will need the following characteristic properties of the 1 j ’s defined in (2.2) or in (2.4)
1 j1 1 j2 = 0
Sk ≡
k−1
X
if | j1 − j2 | ≥ 2,
1j → I
as k → ∞,
j>−∞
1 j (Sk f 1k f ) = 0
if | j − k| ≥ 3.
(2.5)
In addition, the following embedding relations of Besov spaces will be useful.
Proposition 2.1. Let s ∈ R, 1 ≤ p ≤ ∞ and 1 ≤ q ≤ ∞.
(1) If s > 0, then B sp,q ⊂ B̊ sp,q and
k f k B sp,q = k f k L p + k f k B̊ s .
p,q
(2) If
(3) If
(4) If
2
B sp,q
1
s1 ≤ s2 , then
⊂ B sp,q
. This inclusion relation is false for
1 ≤ q1 ≤ q2 ≤ ∞, then B sp,q1 ⊂ B sp,q2 and B̊ sp,q1 ⊂ B̊ sp,q2 .
1 ≤ p1 ≤ p2 ≤ ∞ and s1 = s2 + d( p11 − p12 ), then
B sp11 ,q (Rd ) ⊂ B sp21 ,q (Rd ),
the homogeneous Besov spaces.
B̊ sp11 ,q (Rd ) ⊂ B̊ sp21 ,q (Rd ).
Finally, we remark that many frequently used function spaces are special cases of Besov spaces. The Sobolev
spaces H̊ s and H s defined by
s
f (ξ )| ∈ L 2 }
H s = { f ∈ S 0 : (1 + |ξ |2 ) 2 | b
H̊ s = { f ∈ S 0 : |ξ |s | b
f (ξ )| ∈ L 2 },
can be identified as
s
H̊ s = B̊2,2
,
s
H s = B2,2
.
s
s
For 0 < s < 1, B∞,∞
and B̊∞,∞
are the same as the usual Hölder spaces C s and C̊ s , respectively, where C̊ s is a
1
subspace of continuous functions with a seminorm. B∞,∞
is bigger than the space of Lipschitz functions and can be
identified with the Zygmund class Z yg characterized by the inequality
| f (x − h) − 2 f (x) + f (x + h)| ≤ c|h| for some constant c and all x.
2.2. Two types of spaces involving time and their relations
In this subsection, we analyze the relationship between two types of function spaces that map time to Besov spaces.
For 1 ≤ ρ ≤ ∞, −∞ ≤ a < b ≤ ∞ and a real Banach space X , the space L ρ (a, b; X ) consists of measurable
functions f : (a, b) → X with
Z
k f k L ρ ((a,b);X ) = kk f k X k L ρ (a,b) ≡
a
b
ρ
k f (·, t)k X
ρ1
dt
< ∞.
We are mainly interested in the cases when X = B sp,q or B̊ sp,q .
fρ ((a, b); B sp,q ), which consists of functions f for which the norm
In [4], Chemin introduced the space L
k f k Lfρ ((a,b);B s
p,q )
≡ k2 js k1 j f k L ρ ((a,b);L xp ) kl q
fρ (B s )
fρ ((a, b); B̊ sp,q ) is similarly defined. For notational convenience, we sometimes write L
in finite. The space L
t
p,q
fρ ((0, t); B sp,q ), L ρt (B sp,q ) for L ρ ((0, t); B sp,q ), etc.
for L
fρ ((a, b); B̊ sp,q ) and how L ρ ((a, b); B sp,q ) is related to
We investigate how L ρ ((a, b); B̊ sp,q ) is related to L
s
ρ
f
L ((a, b); B p,q ). We start with some elementary facts.
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J. Wu / Nonlinear Analysis 67 (2007) 3013–3036
Lemma 2.2. Let { f j } be a sequence of measurable functions on an interval (a, b). Assume f j ≥ 0 on (a, b) for each
j. Then
b
XZ
f j (τ ) dτ =
b
Z
sup
f j (τ ) dτ ≤
f j (τ )dτ,
(2.6)
sup f j (τ ) dτ.
(2.7)
j
b
Z
a
a
j
X
a
a
j
b
Z
j
Proof. (2.6) can be obtained by applying the Monotone Convergence theorem to the sequence {gk }, where
X
f j.
gk =
j≤k
(2.7) also follows from the Monotone Convergence theorem. In fact,
Z
sup
j
b
Z
f j (τ ) dτ = lim max
b
Z
f j dτ ≤ lim
k→∞ j≤k
a
b
k→∞ a
a
b
Z
max f j dτ =
j≤k
sup f j (τ ) dτ.
a
j
The following proposition is a consequence of (2.6).
Proposition 2.3. Let s ∈ R and ρ, p, q ∈ [1, ∞]. If ρ = q, then
fρ ((a, b); B̊ sp,q ),
L ρ ((a, b); B̊ sp,q ) = L
fρ ((a, b); B sp,q ).
L ρ ((a, b); B sp,q ) = L
Proof. In the case when 1 ≤ ρ = q < ∞,
Z
k f k L ρ ((a,b); B̊ s
p,q )
= k f k Lfρ ((a,b); B̊ s
p,q )
b
=
a
!1
q
X
2
jsq
q
k1 j f k L p dτ
j
according to (2.6). In the case when ρ = q = ∞,
= sup sup 2 js k1 j f k L p . k f k L ∞ ((a,b); B̊ sp,∞ ) = k f k Lg
∞ ((a,b); B̊ s
p,∞ )
j t∈(a,b)
The inclusion relation in the following proposition follows from (2.7).
Proposition 2.4. For any s ∈ R and p ∈ [1, ∞],
L 1 ((a, b); B̊ sp,∞ ) ⊂ f
L 1 ((a, b); B̊ sp,∞ ),
L 1 ((a, b); B sp,∞ ) ⊂ f
L 1 ((a, b); B sp,∞ ).
Proof. By (2.7),
k f kf
L 1 ((a,b); B̊ s
p,∞ )
= sup 2
js
Z
a
b
k1 j f k L p dτ
a
j
≤
Z
b
sup 2 js k1 j f k L p dτ = k f k L 1 ((a,b); B̊ sp,∞ ) .
j
The proof for L 1 ((a, b); B sp,∞ ) ⊂ f
L 1 ((a, b); B sp,∞ ) is the same.
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J. Wu / Nonlinear Analysis 67 (2007) 3013–3036
2.3. A lower bound for an integral of fractional Laplacians
When we estimate solutions to partial differential equations with fractional Laplacian dissipation in L p -related
spaces, we often encounter an integral of the form
Z
| f | p−2 f · (−1)α f dx.
Dp( f ) ≡
Rd
For α = 1, lower bounds for this integral can be derived through integration by parts. For a general fraction α > 0,
(−1)α is a nonlocal operator and lower bounds no longer follow from integration by parts. The following lower bound
was recently established in [5,18].
Proposition 2.5. Assume either α ≥ 0 and p = 2 or 0 ≤ α ≤ 1 and 2 < p < ∞. Assume f decays sufficiently fast
f satisfies
at infinity and b
supp b
f ⊂ {ξ ∈ Rd : K 1 2 j ≤ |ξ | ≤ K 2 2 j }
for some integer j and real numbers 0 < K 1 ≤ K 2 . Then we have the lower bound
p
D p ( f ) ≥ C22α j k f k L p
for some constant C depending on d, α, K 1 and K 2 .
r for r ≥ 2 − 2α
3. Solutions in B2,q
This section is concerned with solutions of the initial-value problem (IVP) for the 2-D dissipative QG equation

α
2
∂t θ + u · ∇θ + κ(−1) θ = 0, x ∈ R , t > 0,
⊥ −1
2
(3.1)
u = ∇ Λ θ, x ∈ R , t > 0,

θ (x, 0) = θ0 (x), x ∈ R2
r . The major results are presented in Theorem 3.2, which asserts the global
in the inhomogeneous Besov space B2,q
r . The proof of this theorem relies on several a priori estimates, which are
existence and uniqueness of solutions in B2,q
stated in Theorem 3.1.
Theorem 3.1. Let α > 0. Let θ solve the IVP (3.1). Then θ satisfies the following a priori estimates.
(1) In the case when r = 2 − 2α and q = ∞, we have
kθ (t)k B 2−2α + Cκkθkf
L 1 (B 2
2,∞
t
2,∞ )
≤ kθ0 k B 2−2α + Ckθk L ∞ (B 2−2α ) kθkf
L 1 (B 2
t
2,∞
2,∞
t
2,∞ )
.
(3.2)
(2) In the case when r > 2 − 2α and q = ∞, we have for any s ∈ R,
s
s
s
kθ (t)k B2,∞
+ Cκkθkf
≤ kθ0 k B2,∞
+ Ckθk L ∞
kθkf
.
t (B2,∞ )
L 1 (B s+2α )
L 1 (B r +2α )
t
t
2,∞
(3.3)
2,∞
In particular, we have by setting s = r ,
r
r
r
kθ (t)k B2,∞
+ Cκkθkf
≤ kθ0 k B2,∞
+ Ckθk L ∞
kθkf
.
t (B2,∞ )
L 1 (B r +2α )
L 1 (B r +2α )
t
t
2,∞
2,∞
(3) In the case when r ≥ 2 − 2α and 1 ≤ q < ∞, we have
q
q
kθ (t)k B r + Cqκkθk
2,q
2α
fq (B r + q
L
t
2,q
q
)
q
r
≤ kθ0 k B r + Cqkθk L ∞
kθk
t (B2,q )
2,q
q
fq (B r +
L
t
2,q
2α
q
.
(3.4)
)
r + 2α
q
r + 2α
Remark. Because of Proposition 2.3, the norm f
L t (B2,q q ) in (3.4) can be replaced by L t (B2,q q ).
Proof of Theorem 3.1. For j ∈ Z, we apply 1 j to the first equation in (3.1) to obtain
∂t 1 j θ + u · ∇1 j θ + κ(−1)α 1 j θ = [u · ∇, 1 j ]θ,
(3.5)
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J. Wu / Nonlinear Analysis 67 (2007) 3013–3036
where the brackets [ ] represent the commutator operator, namely
[u · ∇, 1 j ]θ ≡ u · ∇(1 j θ ) − 1 j (u · ∇θ ).
Multiplying both sides by 21 j θ and integrating over R2 , we obtain
Z
Z
d
α
2
1 j θ (−1) 1 j θ dx = 2
1 j θ [u · ∇, 1 j ]θ dx.
k1 j θk L 2 + 2κ
dt
R2
R2
(3.6)
By Bernstein’s inequality (Theorem A.1),
Z
Z
|Λα 1 j θ|2 dx ≥ 22α j k1 j θ k2L 2 .
1 j θ (−1)α 1 j θ dx =
(3.7)
By Hölder’s inequality,
Z
2
1 j θ[u · ∇, 1 j ]θ dx ≤ C k1 j θ k L 2 k[u · ∇, 1 j ]θk L 2 .
(3.8)
R2
R2
R2
Inserting (3.7) and (3.8) in (3.6), we find
d
k1 j θk L 2 + Cκ22α j k1 j θ k L 2 ≤ Ck[u · ∇, 1 j ]θ k L 2 .
dt
Applying Proposition A.2 to the right-hand side yields
!
X
X
d
2α j
2j
2
2k
2j
2
k1 j θk L 2 + Cκ2 k1 j θ k L 2 ≤ C 2 k1 j θk L 2 + k1 j θ k L 2
2 k1k θk L 2 + 2
k1k θk L 2 .
dt
k≤ j−1
k≥ j−1
(3.9)
We now prove (1). Multiplying (3.9) by 2(2−2α) j , integrating over [0, t] and taking supremum over
≤ kθ0 k B 2−2α + I1 + I2 + I3 ,
kθ (t)k B 2−2α + Cκkθkf
L 1 (B 2
2,∞
2,∞ )
t
j ≥ −1, we obtain
(3.10)
2,∞
where I1 and I2 are given by
Z t
I1 = sup
2(2−2α) j 22 j k1 j θ k2L 2 dτ,
0
j
t
Z
I2 = sup
j
0
I3 = sup
j
X
22k k1k θk L 2 dτ,
k≤ j−1
t
Z
2(2−2α) j k1 j θ k L 2
2(2−2α) j 22 j
0
X
k1k θk2L 2 dτ.
k≥ j−1
Clearly,
I1 ≤ sup sup {2(2−2α) j k1 j θk L 2 }
≤ sup sup 2
0≤τ ≤t
j
22 j k1 j θk L 2 dτ
0
j 0≤τ ≤t
(2−2α) j
t
Z
k1 j θk L 2 sup 2
2j
j
t
Z
0
k1 j θ k L 2 dτ
= kθ k L ∞ (B 2−2α ) kθkf
2 ).
L 1t (B2,∞
t
2,∞
Z t
X
I2 ≤ sup
22 j k1 j θ k L 2
22α(k− j) 2(2−2α)k k1k θk L 2 dτ
j
0
≤ sup sup
k≤ j−1
X
j 0≤τ ≤t k≤ j−1
2
2α(k− j) (2−2α)k
2
k1k θk L 2 sup
j
t
Z
0
22 j k1 j θ k L 2 dτ
(3.11)
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J. Wu / Nonlinear Analysis 67 (2007) 3013–3036
≤ C sup sup 2(2−2α) j k1 j θk L 2 kθkf
L 1 (B 2
0≤τ ≤t
2,∞ )
t
j
(3.12)
= Ckθk L ∞ (B 2−2α ) kθkf
2 )
L 1t (B2,∞
t
2,∞
Z t X
I3 ≤ sup
2(2−2α)k k1k θk L 2 22k k1k θk L 2 2(4−2α)( j−k) dτ
0 k≥ j−1
j
(2−2α)k
≤ sup sup 2
k1k θk L 2 sup
t
X Z
j k≥ j−1 0
0≤τ ≤t k
22k k1k θk L 2 dτ 2(4−2α)( j−k) .
By Young’s inequality for the convolution of sequences,
Z t
X
22 j k1 j θk L 2 dτ
2(4−2α) j
I3 ≤ Ckθk L ∞ (B 2−2α ) sup
t
2,∞
0
j
j≤0
= Ckθk L ∞ (B 2−2α ) kθkf
L 1 (B 2
t
2,∞
2,∞ )
t
.
(3.13)
Combining (3.10)–(3.13) yields the estimate (3.2).
To prove (2), we multiply both sides of (3.9) by 2s j , integrate with respect to t and take the supremum over all
j ≥ −1 to obtain
s
s
kθ (t)k B2,∞
+ Cκkθkf
≤ kθ0 k B2,∞
+ I4 + I5 + I6 ,
L 1 (B s+2α )
t
(3.14)
2,∞
where
t
Z
I4 = sup
j
0
t
Z
I5 = sup
j
0
I6 = sup
j
X
2s j k1 j θk L 2
22k k1k θk L 2 dτ,
k≤ j−1
t
Z
2s j 22 j k1 j θ k2L 2 dτ,
2s j 22 j
0
X
k1k θk2L 2 dτ.
k≥ j−1
We bound I4 and I5 as follows
I4 ≤ sup sup {2s j k1 j θ k L 2 }
t
Z
22 j k1 j θ k L 2 dτ
0
j 0≤τ ≤t
(r +2α) j
≤ sup sup 2 k1 j θ k L 2 sup 2
sj
0≤τ ≤t
j
t
Z
0
j
k1 j θk L 2 dτ
s
= kθk L ∞
kθkf
for any r ≥ 2 − 2α.
r +2α
t (B2,∞ )
L 1t (B2,∞
)
Z t
X
I5 ≤ sup
2s j k1 j θ k L 2
22k k1k θk L 2 dτ
j
0
k≤ j−1
≤ sup sup 2 k1 j θ k L 2 sup
sj
j 0≤τ ≤t
j−1
X
2k
2
max
j −1≤k≤ j−1
j
s
= Ckθk L ∞
kθkf
t (B2,∞ )
L 1 (B r +2α )
t
t
Z
j k=−1
≤ sup sup 2s j k1 j θ k L 2 sup
0≤τ ≤t
(3.15)
0
k1k θk L 2 dτ
2(r +2α)k
t
Z
0
k1k θk L 2 dτ
j−1
X
2(2−2α−r )k
k=−1
for any r > 2 − 2α.
2,∞
I6 can be similarly bounded. Putting these bounds together gives the estimate (3.3).
(3.16)
3022
J. Wu / Nonlinear Analysis 67 (2007) 3013–3036
q−1
To prove (3), we multiply both sides of (3.9) by q2rq j k1 j θk L 2 , integrate over [0, t] and sum over j to obtain
q
q
kθ (t)k B r + Cqκkθk
2,q
q
fq (B r +
L
t
2,q
2α
q
≤ kθ0 k B r + I7 + I8 + I9 ,
(3.17)
2,q
)
where
I7 = q
0
j
I8 = q
t
XZ
0
j
I9 = q
t
XZ
q
X
2rq j 22 j
0
j
X
2rq j k1 j θk L 2
22k k1k θk L 2 dτ,
k≤ j−1
t
XZ
q+1
2rq j+2 j k1 j θk L 2 dτ,
k1k θk2L 2 dτ.
k≥ j−1
We bound I7 and I8 as follows
XZ t
(r + 2−r
q )q j k1 θkq dτ
I7 = q
2r j k1 j θk L 2 2
j
L2
0
j
≤q
X
sup 2 k1 j θk L 2
rj
q
k1 j θ k L 2 dτ
0
(r + 2−r
q )q j
X
j 0≤τ ≤t
t
Z
q
k1 j θk L 2 dτ
2
0
j
≤ Cq sup sup 2r j k1 j θk L 2
0≤τ ≤t
(r + 2−r
q )q j
2
j 0≤τ ≤t
≤ q sup sup 2r j k1 j θk L 2
t
Z
X
j
(r + 2α
q )q j
t
Z
2
0
j
q
k1 j θ k L 2 dτ
!1/q
X
≤ Cq sup
0≤τ ≤t
q
2r jq k1 j θk L 2
j
q
r
= Cqkθk L ∞
kθk
t (B2,q )
I8 = q
t
XZ
j
0
fq (B r +
L
t
2,q
2α
q
)
X
22α(k− j) 2(2−2α)k k1k θk L 2 dτ
k≤ j−1
X
j 0≤τ ≤t k≤ j−1
22α(k− j) 2(2−2α)k k1k θk L 2
≤ Cq sup sup 2
j
(r + 2α
q )q j
2
0
q
k1 j θ k L 2 dτ
q
k1 j θk L 2 kθk
fq (B r +
L
t
2,q
2α
q
)
q
≤ Cq sup sup 2 k1 j θk L 2 kθk
rj
j
r
L∞
t (B2,q )
t
XZ
j
(2−2α) j
= Cqkθk
2α
q
.
q
≤ q sup sup
0≤τ ≤t
fq (B r +
L
t
2,q
)
22α j+rq j k1 j θ k L 2
0≤τ ≤t
q
kθk
q
kθk
fq (B r +
L
t
2,q
2α
q
fq (B r +
L
t
2,q
2α
q
)
.
)
I9 is similarly bounded. After inserting these estimates in (3.17), we obtain (3.4).
As in the proof of Theorem 3.1 of [17], we can establish the following theorem concerning solutions of the IVP
r . The proof combines the a priori estimates of Theorem 3.1 and the
(3.1) in the inhomogeneous Besov space B2,q
method of successive approximation. We omit the details of the proof.
3023
J. Wu / Nonlinear Analysis 67 (2007) 3013–3036
r (R2 ).
Theorem 3.2. Let κ > 0 and α > 0. Let 1 ≤ q ≤ ∞ and r ≥ 2 − 2α. Consider the IVP (3.1) with θ0 ∈ B2,q
There exists a constant C0 depending on α, r and q only such that if
r
≤ C0 κ,
kθ0 k B2,q
r (R2 ) satisfying
then the IVP (3.1) has a unique solution θ ∈ B2,q
r
kθ (·, t)k B2,q
≤ C1 κ
r +2α
for any t > 0 and some constant C1 depending on α, r and q. In addition, θ ∈ f
L 1 ((0, ∞); B2,∞
) in the case when
r + 2α
q = ∞ and θ ∈ L q ((0, ∞); B2,q q ) in the case when q < ∞.
Setting q = 2 and α =
QG equation.
1
2
in Theorem 3.2, we obtain the following corollary on H 1 -solutions of the 2-D critical
Corollary 3.3. Consider the IVP (3.1) with α =
and θ0 ∈ H 1 (R2 ). There exists a constant C2 such that if
1
2
kθ0 k H 1 ≤ C2 κ,
then the IVP (3.1) has a unique solution θ satisfying
3
θ ∈ C([0, ∞); H 1 ) ∩ L 2 ((0, ∞); H 2 )
and, for some constant C3 ,
kθ (·, t)k H 1 ≤ C3 κ
for any t > 0.
4. Solutions in B̊ rp,q
1−2α+ 2p
In this section we study solutions of the IVP (3.1) in the homogeneous Besov space B̊ p,q
q ∈ [1, ∞]. We start with the following a priori estimates.
for p ∈ [2, ∞) and
Theorem 4.1. Assume either α > 0 and p = 2 or 0 < α ≤ 1 and 2 < p < ∞. Let 1 ≤ q ≤ ∞, and −∞ < r < ∞.
Consider the IVP (3.1) with θ0 ∈ B̊ rp,q (R2 ). Then the corresponding solution θ of (3.1) obeys the following a priori
estimates.
(1) In the case when q < ∞, we have
q
B̊ rp,q
kθ (t)k
q
+ C1 κkθk
r + 2α
q
q
L t ( B̊ p,q
q
B̊ rp,q
≤ kθ0 k
)
+ C2 kθk
q
1−2α+ dp
L∞
t ( B̊ p,q
kθk
)
q
r + 2α
q
L t ( B̊ p,q
.
)
(2) In the case when q = ∞, we have
kθ (t)k B̊ r
p,∞
≤ kθ0 k B̊ r
+ C1 κkθkf
L 1 ( B̊ r +2α )
t
p,∞
p,∞
+ C2 kθk
1−2α+ dp
L∞
t ( B̊ p,∞
)
.
kθkf
L 1 ( B̊ r +2α )
t
p,∞
Proof. We first consider the case when 1 ≤ q < ∞. Multiplying (3.5) by p|1 j θ| p−2 1 j θ and integrating over R2 ,
we obtain
d
p
k1 j θk L p + κ p J1 = p J2 ,
dt
where J1 and J2 are given by
Z
J1 =
|1 j θ | p−2 1 j θ (−1)α 1 j θ dx,
2
ZR
J2 =
|1 j θ | p−2 1 j θ[u · ∇, 1 j ]θ dx.
R2
(4.1)
3024
J. Wu / Nonlinear Analysis 67 (2007) 3013–3036
Applying the lower bound in Proposition 2.5, we have
p
J1 ≥ C 22α j k1 j θk L p .
(4.2)
By Hölder’s inequality,
p−1
J2 ≤ C k1 j θk L p k[u · ∇, 1 j ]θ k L p .
(4.3)
Inserting (4.2) and (4.3) in (4.1), we get
d
k1 j θk L p + Cκ22α j k1 j θ k L p ≤ Ck[u · ∇, 1 j ]θ k L p .
dt
Applying Proposition A.2 to the right-hand side yields
!
X
d
(1+ 2p ) j
(1+ 2p )k
2α
j
2
k1 j θk L p + Cκ2 k1 j θ k L p ≤ C 2
k1 j θ k L p + k1 j θ k L p
2
k1k θk L p .
dt
−∞<k≤ j−1
(4.4)
For the sake of brevity, we have intentionally omitted the interaction term of high–high frequencies. As we have
q−1
seen in the previous section, this term can be similarly dealt with. Multiplying this inequality by q2rq j k1 j θk L p ,
integrating over [0, t] and summing over j ∈ Z, we obtain
q
B̊ rp,q
kθ (t)k
q
+ Cκqkθk
r + 2α
q
q
L t ( B̊ p,q
q
B̊ rp,q
≤ kθ0 k
)
+ J3 + J4 ,
where J3 and J4 are given by
XZ t
q+1
(1+ 2 ) j
J3 = C q
2rq j 2 p k1 j θk L p dτ,
0
j
J4 = C q
XZ
j
t
0
q
(1+ 2p )k
X
2rq j k1 j θk L p
2
k1k θk L p dτ.
−∞<k≤ j−1
To bound J3 , we first rewrite it as
X Z t (1−2α+ 2 ) j
p k1 θk p 2rq j+2α j k1 θ kq dτ.
J3 = Cq
2
j
L
j
Lp
j
0
According to Lemma 2.2,
Z tX
J3 ≤ Cq
kθ (τ )k
0
j
≤ Cq sup kθ (τ )k
τ ∈[0,t]
= Cqkθk
1−2α+ 2p
B̊ p,q
(r + 2α
q )q j
2
Z tX
1−2α+ 2p
B̊ p,q
0
1−2α+ 2p
kθk
)
q
(r + 2α
q )q j
2
q
k1 j θk L p dτ
j
q
L∞
t ( B̊ p,q
q
k1 j θk L p dτ
r + 2α
q
L t ( B̊ p,q
.
)
To bound J4 , we start by writing it as
X Z t (r + 2α )q j
X
q
(1−2α+ 2p )k
q
J4 = Cq
k1 j θk L p
22α(k− j) 2
k1k θk L p dτ.
2
j
0
k≤ j−1
Since
X
k≤ j−1
2
2
2α(k− j) (1−2α+ p )k
2
k1k θk L p ≤ C
X
(1−2α+ 2p )k
[2
k≤ j−1
≤ C kθk
1−2α+ 2p
B̊ p,q
,
!1
q
k1k θk L p ]
q
3025
J. Wu / Nonlinear Analysis 67 (2007) 3013–3036
we have
J4 ≤ C qkθk
q
1−2α+ 2p
L∞
t ( B̊ p,q
kθk
)
r + 2α
q
q
L t ( B̊ p,q
.
)
This completes the proof for (1).
In the case when q = ∞, we multiply (4.4) by 2r j , integrate over [0, t] and take the supremum over j ∈ Z. This
results in the inequality
kθ (t)k B̊ r
+ Cκkθkf
≤ kθ0 k B̊ r
L 1 ( B̊ r +2α )
p,∞
p,∞
p,∞
t
+ C J5 + C J6 ,
where J5 and J6 are given by
Z t
1+ 2p j
2r j 2
J5 = sup
k1 j θ k2L p dτ,
0
j
t
Z
J6 = sup
0
j
(1+ 2p )k
X
2r j k1 j θk L p
2
k1k θk L p dτ.
k≤ j−1
J5 is bounded by
(1−2α+ 2p ) j
J5 ≤ sup sup 2
τ ∈[0,t]
j
= kθk
k1 j θ (τ )k L p kθkf
L 1 ( B̊ r +2α )
t
p,∞
1−2α+ 2p
L∞
t ( B̊ p,∞
)
.
kθkf
L 1 ( B̊ r +2α )
t
p,∞
J6 can be bounded as follows
Z t
X
(1−2α+ 2p )k
J6 = sup
2(r +2α) j k1 j θk L p
22α(k− j) 2
k1k θk L p dτ
0
j
k≤ j−1
t
Z
≤ sup
j
(1−2α+ 2p )k
2(r +2α) j k1 j θk L p sup 2
0
= C kθk
k1k θk L p
k
1−2α+ 2p
L∞
t ( B̊ p,∞
)
X
22α(k− j) dτ
k≤ j−1
.
kθkf
L 1 ( B̊ r +2α )
t
p,∞
This completes the proof for the case q = ∞ and thus the proof of Theorem 4.1.
Combining these a priori estimates with the method of successive approximation allows us to prove the following
existence and uniqueness result.
Theorem 4.2. Assume either α > 0 and p = 2 or 0 < α ≤ 1 and 2 < p < ∞. Let 1 ≤ q ≤ ∞ and r = 1 − 2α + 2p .
Consider the IVP (3.1) with θ0 ∈ B̊ rp,q . Then there exists a constant C0 such that if
kθ0 k B̊ r
p,q
≤ C0 κ,
then the IVP (3.1) has a unique solution θ satisfying
q
L ∞ ([0,∞); B̊ rp,q )
kθk
q
+ C1 κkθk
L q ((0,∞), B̊
r + 2α
q
p,q
≤ C2 κ q
)
in the case when 1 ≤ q < ∞, and
kθk L ∞ ([0,∞); B̊ r
p,∞ )
+ C1 κkθkf
≤ C2 κ
L 1 ((0,∞), B̊ r +2α )
p,∞
in the case when q = ∞, where C1 and C2 are constants depending on α, p and q only.
Remark. Although Theorem 4.2 does not cover the case when 1 ≤ p < 2, the global existence of solutions for
θ0 ∈ B̊ rp,q with 1 ≤ p < 2 can be established by combining this theorem with the Besov embedding stated in
Proposition 2.1. In fact, for any 1 ≤ p1 < 2 and r1 = 1 − 2α + p21 , we can choose p2 ≥ 2 and r2 = 1 − 2α + p22 such
that r1 −
2
p1
= r2 −
2
p2 .
By the Besov embedding,
θ0 ∈ B̊ rp11 ,q (R2 ) ⊂ B̊ rp22 ,q (R2 ).
3026
J. Wu / Nonlinear Analysis 67 (2007) 3013–3036
Theorem 4.2 then concludes that θ0 leads to a global solution.
Proof of Theorem 4.2. We sketch the proof of this theorem very briefly. It consists of two major steps. The first step
is to consider a successive approximation sequence {θ (n) }∞
n=1 satisfying
 (1)
θ = S2 θ0 ,


u (n) = ∇ ⊥ Λ−1 θ (n) ,
∂ θ (n+1) + u (n) · ∇θ (n+1) + κ(−1)α θ (n+1) = 0,


 t(n+1)
(n+1)
θ
(x, 0) = θ0
(x) = Sn+2 θ0 (x)
∞
r
and show that {θ (n) }∞
n=1 is bounded uniformly in L ([0, ∞); B̊ p,q ). More precisely, we show that if kθ0 k B̊ rp,q ≤ C 0 κ,
then
kθ (n) k
q
L ∞ ([0,∞); B̊ rp,q )
+ C1 κkθ (n) k
q
L q ((0,∞), B̊
r + 2α
q
p,q
≤ C2 κ q
)
in the case when 1 ≤ q < ∞, and
kθ (n) k L ∞ ([0,∞); B̊ r
p,∞ )
≤ C2 κ
+ C1 κkθ (n) kf
L 1 ((0,∞), B̊ r +2α )
p,∞
in the case when q = ∞, where C1 and C2 are constants independent of κ and n.
−1 ). That is, we show the sequence
The second step proves that {θ (n) } is a Cauchy sequence in L ∞ ([0, ∞); B̊ rp,q
{η(n) } with η(n) = θ (n) − θ (n−1) satisfies
kη(n) k L ∞ ([0,∞); B̊ r −1 ) ≤ C kθ0 k B̊ r 2−n .
p,q
p,q
For this purpose, we consider the equations that {η(n) } satisfies
 (1)
η = S2 θ0 − θ0 ,


w (n) = ∇ ⊥ Λ−1 η(n) ,
∂ η(n+1) + u (n) · ∇η(n+1) + κ(−1)α η(n+1) = w (n) · ∇θ (n) ,


 t(n+1)
(n+1)
η
(x, 0) = η0
(x) = 1n+1 θ0
and prove that
kη(n) k
q
−1
L ∞ ([0,∞); B̊ rp,q
)
+ C1 κkη(n) k
q
L q ((0,∞), B̊
r −1+ 2α
q
p,q
≤ C2 κ q 2−qn
)
in the case when 1 ≤ q < ∞, and
≤ C2 κ2−n
kη(n) k L ∞ ([0,∞); B̊ r −1 ) + C1 κkη(n) kf
L 1 ((0,∞), B̊ r −1+2α )
p,∞
p,∞
in the case when q = ∞. Therefore, there exists
r + 2α
θ ∈ L ∞ ([0, ∞); B̊ rp,q ) ∩ L q (0, ∞), B̊ p,q q
for 1 ≤ q < ∞ and
+2α
θ ∈ L ∞ ([0, ∞); B̊ rp,∞ ) ∩ f
L 1 (0, ∞), B̊ rp,∞
for q = ∞ such that
θ
(n)
→θ
θ (n) → θ
in L ([0, ∞);
∞
−1
B̊ rp,q
)∩
L
q
(0, ∞),
r −1+ 2α
B̊ p,q q
−1
−1+2α
in L ∞ ([0, ∞); B̊ rp,∞
)∩f
L 1 (0, ∞), B̊ rp,∞
One can then easily verify that θ satisfies the 2-D QG equation
∂t θ + u · ∇θ + κ(−1)α θ = 0
−1 , where u = ∇ ⊥ Λ−1 θ. We omit further details.
in B̊ rp,q
for 1 ≤ q < ∞,
for q = ∞.
3027
J. Wu / Nonlinear Analysis 67 (2007) 3013–3036
5. Uniqueness
This section addresses the issue of the uniqueness of solutions to the IVP (3.1) in Besov spaces. The major results
are presented in four theorems.
Theorem 5.1. Let α > 0. Let r ≥ 2 − 2α and 1 ≤ q ≤ ∞. Let T > 0 and let
r
θ ∈ L ∞ ([0, T ); B2,q
(R2 ))
r
and e
θ ∈ L ∞ ([0, T ); B2,q
(R2 ))
be the solutions of the IVP (3.1) corresponding to the initial data
r
θ0 ∈ B2,q
(R2 )
r
and e
θ0 ∈ B2,q
(R2 ),
respectively. Let s < 1 −
2α
q .
s , then the difference
If η0 = e
θ0 − θ0 is in B2,q
η=e
θ −θ
satisfies for 1 ≤ q < ∞
q
kηk L ∞ (B s )
t
2,q
q
+ C κkηk
q
s+ 2α
q
L t (B2,q
≤
)
q
kη0 k B s
2,q
q
e
+ C kθk L ∞ (B 2−2α ) + kθk L ∞ (B 2−2α ) kηk
t
2,q
t
2,q
q
s+ 2α
q
L t (B2,q
(5.1)
)
and for q = ∞
e
s
+
C
kθk
kηk L ∞
+ C κkηkf
∞ (B 2−2α ) + kθk L ∞ (B 2−2α ) kηk f
s+2α ≤ kη0 k B s
1
L
t (B2,∞ )
2,∞
L 1 (B s+2α )
)
L (B
t
t
2,∞
2,∞
t
2,∞
t
(5.2)
2,∞
for any t ≤ T . In particular, if θ0 = e
θ0 and
kθk L ∞ (B 2−2α ) ≤ C κ,
T
2,q
ke
θk L ∞ (B 2−2α ) ≤ C κ
T
2,q
for some suitable constant C, then θ = e
θ.
In the special case when α =
of the 2-D critical QG equation.
Corollary 5.2. Let α =
1
2
1
2
and r = 1, this theorem reduces to a corollary on the uniqueness of H 1 solutions
and T > 0. Assume that
θ ∈ L ∞ ([0, T ); H 1 (R2 ))
and e
θ ∈ L ∞ ([0, T ); H 1 (R2 ))
θ0 ∈ H 1 (R2 ), respectively. Then
are solutions of the IVP (3.1) corresponding to the initial data θ0 ∈ H 1 (R2 ) and e
there exists a constant C such that if
kθk L ∞ (H 1 ) ≤ C κ
T
and
ke
θk L ∞ (H 1 ) ≤ C κ,
T
then θ0 = e
θ0 implies θ = e
θ.
Proof of Theorem 5.1. Let u and e
u be the corresponding velocities, namely
u = ∇ ⊥ Λ−1 θ
and e
u = ∇ ⊥ Λ−1e
θ.
The difference η = e
θ − θ satisfies the equation
∂t η + e
u · ∇η + w · ∇θ + κ(−1)α η = 0,
where w = e
u − u. Applying 1 j to this equation, we have
∂t 1 j η + e
u · ∇1 j η + κ(−1)α 1 j η = [e
u · ∇, 1 j ]η − 1 j (w · ∇θ ).
Multiplying by 1 j η, integrating over R2 , bounding the dissipative term from below and applying the Hölder inequality
to the terms on the right, we find
d
k1 j ηk L 2 + Cκ22α j k1 j ηk L 2 ≤ Ck[e
u · ∇, 1 j ]ηk L 2 + Ck1 j (w · ∇θ )k L 2 .
dt
(5.3)
3028
J. Wu / Nonlinear Analysis 67 (2007) 3013–3036
q−1
For 1 ≤ q < ∞, we multiply this inequality by 2s jq k1 j ηk L 2 , integrate over [0, t] and sum over j to obtain
q
q
kη(t)k B s + C κkηk
2,q
q
s+ 2α
q
L t (B2,q
≤ K1 + K2,
)
where K 1 and K 2 are given by
XZ t
q−1
u · ∇, 1 j ]ηk L 2 dτ,
2s jq k1 j ηk L 2 k[e
K1 = C
0
j
K2 = C
t
XZ
0
j
q−1
2s jq k1 j ηk L 2 k1 j (w · ∇θ )k L 2 dτ.
To estimate K 1 , we first change the order of summation and time integration and then apply Proposition A.2 to obtain
Z t
(K 11 + K 12 + K 13 ) dt
K1 ≤
0
with K 11 , K 12 and K 13 being given by
X
X
q
K 11 = C
2s jq k1 j ηk L 2
22m k1m e
ukL 2 ,
j
m≤ j−1
q−1
2s jq k1 j ηk L 2 k1 j e
ukL 2
X
K 12 = C
X
j
m≤ j−1
X
K 13 = C
22m k1m ηk L 2 ,
s jq
2
q−1
k1 j ηk L 2 k1 j e
ukL 2
22 j k1 j ηk L 2 .
j
For K 11 , we have
X (s+ 2α ) jq
q
k1 j ηk L 2
K 11 = C
2 q
j
≤C
X
2
(s+ 2α
q ) jq
!1− 1
q
q
k1 j ηk L 2
q
s+ 2α
q
22α(m− j)q/(q−1)
2,q
(5.4)
2,q
K 12 can be bounded as follows
X
(s+ 2α
q ) j (q−1)
2
k1 j ηk L 2 2(2−2α) j k1 j e
ukL 2
q−1
j
≤ C kηk
s+ 2α
q
B2,q
q
≤ C kηk
s+ 2α
q
ke
u k B 2−2α
2,q
s+ 2α
q jq
2
X
(2−s− 2α
q )(m− j)q/(q−1)
2
!1− 1
q
X
2
2,q
k1 j ηk L 2 2(2−2α) j k1 j e
u k L 2 ≤ C kηk
q
To bound K 2 , we obtain after applying Proposition A.2
Z t
K2 ≤ C
(K 21 + K 22 + K 23 ) dτ,
q
s+ 2α
q
B2,q
ke
u k B 2−2α .
2,q
(s+ 2α
q )mq
2
k≤ j−1
k≤ j−1
j
0
2α
(2−s− 2α
q )(m− j) (s+ q )m
2
ke
u k B 2−2α .
B2,q
X
X
k1m ηk L 2
m≤ j−1
q−1
K 13 =
ke
u k B 2−2α
ke
u k B 2−2α .
B2,q
K 12 = C
X
k≤ j−1
= C kηk
2α
q ,
22α(m− j) 2(2−2α)m k1m e
ukL 2
m≤ j−1
j
For s < 2 −
X
!1
q
q
k1m ηk L 2
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J. Wu / Nonlinear Analysis 67 (2007) 3013–3036
where K 21 , K 22 and K 23 are given by
X
q−1
2s jq k1 j ηk L 2 kS j−1 wk L ∞ k∇1 j θ k L 2 ,
K 21 =
j
K 22 =
X
K 23 =
X
q−1
2s jq k1 j ηk L 2 k1 j wk L 2 k∇ S j−1 θk L ∞ ,
j
q−1
2s jq k1 j ηk L 2 k1 j wk L 2 k∇1 j θk L ∞ .
j
For s < 1 −
K 21
2α
q ,
K 21 is bounded by
X
q−1
2s jq k1 j ηk L 2 2 j k1 j θk L 2
≤
X
j
=
X
2m k1m wk L 2
m≤ j−1
(s+ 2α
q ) j (q−1)
2
k1 j ηk L 2 2(2−2α) j k1 j θk L 2
q−1
j
2α
(1−s− 2α
q )(m− j) (s+ q )m
X
2
2
k1m wk L 2
m≤ j−1
q−1
≤ C kηk
s+ 2α
q
kθk B 2−2α kwk
2,q
B2,q
.
s+ 2α
q
B2,q
For K 22 and K 23 , we have
X (s+ 2α ) j (q−1)
X
q−1 (s+ 2α ) j
K 22 ≤
2 q
k1 j ηk L 2 2 q k1 j wk L 2
22α(m− j) 2(2−2α)m k1m θ k L 2
j
m≤ j−1
q−1
≤ C kηk
s+ 2α
q
kwk
K 23 ≤
X
kθk B 2−2α ,
s+ 2α
q
2,q
B2,q
B2,q
(s+ 2α
q ) j (q−1)
2
(s+ 2α
q )j
q−1
k1 j ηk L 2 2
k1 j wk L 2 2(2−2α) j k1 j θk L 2
j
q−1
≤ C kηk
s+ 2α
q
kwk
kθk B 2−2α .
s+ 2α
q
2,q
B2,q
B2,q
Combining the estimates for K 1 and K 2 , we obtain
q
q
kη(t)k B s + C κkηk
2,q
s+ 2α
q
q
L t (B2,q
)
t
Z
q
≤ kη0 k B s + C
2,q
t
Z
+C
q−1
kηk
0
q
kηk
0
s+ 2α
q
ke
u k B 2−2α dτ
s+ 2α
q
2,q
B2,q
kwk
kθk B 2−2α dτ.
s+ 2α
q
(5.5)
2,q
B2,q
B2,q
Since e
u = ∇ ⊥ Λ−1e
θ and w = ∇ ⊥ Λ−1 η are Riesz transforms of e
θ and η, respectively,
ke
u k B 2−2α ≤ ke
θk B 2−2α
2,q
and
2,q
kwk
s+ 2α
q
≤ kηk
B2,q
s+ 2α
q
B2,q
according to the boundedness of Riesz transforms on L 2 . Inserting these estimates in (5.5), we establish (5.1).
In the case when q = ∞, we integrate (5.3) with respect to t, multiply by 2s j and take the supremum over j to
obtain
s
kη(t)k B2,∞
+ C κkηkf
≤ K3 + K4,
L 1 (B s+2α )
t
2,∞
where K 3 and K 4 are given by
Z t
K 3 = sup 2s j
k[e
u · ∇, 1 j ]ηk L 2 dτ,
j
0
K 4 = C sup 2s j
j
Z
0
t
k1 j (w · ∇θ )k L 2 dτ.
3030
J. Wu / Nonlinear Analysis 67 (2007) 3013–3036
As in the estimates for K 1 and K 2 , we split each of K 3 and K 4 into three terms. For the sake of brevity, we shall only
provide the details for K 31 , a term in K 3 .
Z t
X
sj
k1 j ηk L 2
22m k1m e
u k L 2 dτ
K 31 = C sup 2
0
j
m≤ j−1
= C sup 2(s+2α) j
t
Z
0
j
X
k1 j ηk L 2
22α(m− j) 2(2−2α)m k1m e
u k L 2 dτ
m≤ j−1
≤ C sup sup 2(2−2α)m k1m e
u k L 2 sup 2(s+2α) j
0≤τ ≤t m
t
Z
0
j
k1 j ηk L 2 dτ
u k L ∞ (B 2−2α ) kηkf
= C ke
.
L 1 (B s+2α )
t
2,∞
t
2,∞
(5.2) is then established after combining the estimates for K 3 and K 4 .
Theorem 5.3. Let α > 0. Let r ≥ 2 − 2α and 1 ≤ q < ∞. Let T > 0 and let
r + 2α
r + 2α
and e
θ ∈ L q (0, T ); B2,q q (R2 )
θ ∈ L q (0, T ); B2,q q (R2 )
be the solutions of the IVP (3.1) corresponding to the initial data θ0 and e
θ0 , respectively. Let s < 2. If η0 = e
θ0 − θ0
s , then the difference
is in B2,q
η=e
θ −θ
satisfies, for any t ≤ T ,
q
Z
q
kη(t)k B s ≤ kη0 k B s + C
2,q
2,q
0
t


q
q
kθ (τ )kq
+ ke
θ (τ )k 2−2α+ 2α  kη(τ )k B s dτ.
2−2α+ 2α
B2,q
q
B2,q
(5.6)
2,q
q
In particular, if θ0 = e
θ0 , then
θ =e
θ.
(5.7)
For the sake of brevity, we did not include the case when q = ∞ in this theorem. We now state as a corollary a
special consequence of this theorem.
Corollary 5.4. Let α =
1
2
and T > 0. Let θ and e
θ satisfying
3
θ ∈ L 2 ((0, T ); H 2 (R2 ))
3
and e
θ ∈ L 2 ((0, T ); H 2 )
be two solutions of the IVP for the 2-D critical QG equation (3.1) corresponding to the initial data θ0 and e
θ0 ,
respectively. If θ0 = e
θ0 , then θ = e
θ.
s . We start with the inequality
Proof of Theorem 5.3. We estimate the difference η = e
θ − θ in B2,q
d
k1 j ηk L 2 + Cκ22α j k1 j ηk L 2 ≤ Ck[e
u · ∇, 1 j ]ηk L 2 + Ck1 j (w · ∇θ )k L 2 .
dt
Integrating with respect to t, we obtain
Z t
Z t
k1 j ηk L 2 ≤ E j (t)k1 j η0 k L 2 + C
E j (t − τ )k[e
u · ∇, 1 j ]ηk L 2 dτ + C
E j (t − τ )k1 j (w · ∇θ )k L 2 dτ,
0
where E j (t) = exp(−C
we obtain
q
κ22α j t).
q
Multiplying both sides by
kη(t)k B s ≤ kη0 k B s + L 1 + L 2 ,
2,q
2,q
0
2s j ,
raising them to the qth power and summing over j,
(5.8)
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J. Wu / Nonlinear Analysis 67 (2007) 3013–3036
where
L1 = C
X
2s jq
E j (t − τ )k[e
u · ∇, 1 j ]ηk L 2 dτ
0
j
L2 = C
t
Z
X
s jq
t
Z
E j (t − τ )k1 j (w · ∇θ )k L 2 dτ
2
0
j
q
q
,
.
Applying Proposition A.2, we have
L 1 ≤ L 11 + L 12 + L 13 ,
where
L 11 = C
X
2
0
j
L 12 = C
X
2
0
X
s jq
2
0
j
E j (t − τ )k1 j ηk L 2
2m
2
,
k1m e
u k L 2 dτ
!q
X
E j (t − τ )k1 j e
ukL 2
2m
2
k1m ηk L 2 dτ
,
m≤ j−1
t
Z
!q
X
m≤ j−1
t
Z
s jq
j
L 13 = C
t
Z
s jq
E j (t − τ )k1 j e
u k L 2 2 k1 j ηk L 2 dτ
2j
q
.
We now provide the estimate for L 11 . By Hölder’s inequality,
!q
q−1 Z t
Z t
q
X
X
q−1
s jq
2m
L 11 ≤ C
E j (t − τ )ds
2
2 k1m e
u k L 2 dτ
k1 j ηk L 2
0
j
=C
X
2s jq−2α j (q−1)
0
Z tX
0
2
≤C
q
k1 j ηk L 2
dτ
2α
(2α− 2α
q )(m− j) (2−2α+ q )m
X
2
2
!q
dτ
k1m e
ukL 2
m≤ j−1
q
q
kη(τ )k B s ke
u (τ )k
2,q
0
22m k1m e
ukL 2
m≤ j−1
j
t
Z
s jq
m≤ j−1
!q
X
k1 j ηk L 2
0
j
=C
t
Z
(2−2α+ 2α
q )
dτ.
B2,q
For L 12 and L 13 , we have
L 12 ≤ C
X
2s jq 2−2α j (q−1)
0
j
=C
Z tX
0
2
≤C
X
(2−2α+ 2α
q )
B2,q
2s jq 2−2α j (q−1)
0
0
kη(τ )k B s dτ.
2,q
Z t
q
k1 j e
u k L 2 22 j k1 j ηk L 2 dτ
(2−2α+ 2α
q ) jq
2
q
q
k1 j e
u k L 2 2s jq k1 j ηk L 2 dτ
j
t
≤C
2(2−s)(m− j) 2sm k1m ηk L 2
0
Z tX
Z
X
q
q
ke
u (τ )k
q
ke
u (τ )k
(2−2α+ 2α
q )
B2,q
dτ
m≤ j−1
q
k1 j e
ukL 2
j
=C
22m k1m ηk L 2
m≤ j−1
0
L 13 ≤ C
X
k1 j e
ukL 2
j
t
Z
(2−2α+ 2α
q ) jq
!q
t
Z
q
kη(τ )k B s dτ.
2,q
!q
dτ
3032
J. Wu / Nonlinear Analysis 67 (2007) 3013–3036
L 2 can be similarly estimated and is bounded by
Z t
q
q
kθ (τ )k (2−2α+ 2α ) kw(τ )k B s dτ.
L2 ≤ C
0
2,q
q
B2,q
Combining (5.8) with the estimates for L 1 and L 2 and using the fact that
q
≤ ke
θ (τ )k (2−2α+ 2α )
q
ke
u (τ )k
(2−2α+ 2α
q )
B2,q
and
B2,q
q
q
kw(τ )k B s ≤ kη(τ )k B s ,
2,q
q
2,q
we establish (5.6). (5.7) is obtained by applying Gronwall’s inequality to (5.6).
The following two theorems assert the uniqueness of solutions of the 2-D QG equation in homogeneous Besov
spaces. We omit their proofs since they are similar to those of Theorems 5.1 and 5.3.
Theorem 5.5. Assume either α > 0 and p = 2 or 0 < α ≤ 1 and 2 < p < ∞. Let 1 ≤ q ≤ ∞ and r = 1 − 2α + 2p .
Let T > 0. Let
θ ∈ L ∞ ([0, T ); B̊ rp,q (R2 ))
and e
θ ∈ L ∞ ([0, T ); B̊ rp,q (R2 ))
be the solutions of the IVP (3.1) corresponding to the initial data
θ0 ∈ B̊ rp,q (R2 )
and e
θ0 ∈ B̊ rp,q (R2 ),
respectively. Let s < 1 −
2α
q .
If η0 = e
θ0 − θ0 is in B̊ sp,q (R2 ), then the difference
η=e
θ −θ
satisfies for 1 ≤ q < ∞
q
s )
L∞
t ( B̊
kηk
q
+ C κkηk
B̊,q
s+ 2α
q
q
L t ( B̊ p,q
q
B̊ sp,q
≤ kη0 k
)
+ C kθk L ∞ ( B̊ r
p,q )
t
+ ke
θk L ∞ ( B̊ r
p,q )
t
q
kηk
s+ 2α
q
q
L t ( B̊ p,q
(5.9)
)
and for q = ∞
≤
kη
k
+
C
kθk L ∞ ( B̊ r
kηk L ∞ ( B̊ sp,∞ ) + C κkηkf
s
0
s+2α
B̊ p,∞
)
L 1 ( B̊
t
t
t
p,∞
p,∞
e
) + kθk L ∞ ( B̊ r
t
p,∞ )
kηkf
L 1 ( B̊ s+2α )
t
(5.10)
p,∞
for any t ≤ T . In particular, if θ0 = e
θ0 and
kθk L ∞ ( B̊ r
T
2,q )
≤ C κ,
ke
θk L ∞ ( B̊ r
T
2,q )
≤Cκ
for some suitable constant C, then
θ =e
θ.
Theorem 5.6. Assume either α > 0 and p = 2 or 0 < α ≤ 1 and 2 < p < ∞. Let 1 ≤ q < ∞ and r = 1 − 2α + 2p .
Let T > 0. Let
r + 2α
r + 2α
θ ∈ L q (0, T ); B̊ p,q q (R2 )
and e
θ ∈ L q (0, T ); B̊ p,q q (R2 )
be the solutions of the IVP (3.1) corresponding to the initial data θ0 and e
θ0 , respectively. Let s < 2. If η0 = e
θ0 − θ0
s
2
is in B̊ p,q (R ), then the difference
η=e
θ −θ
satisfies, for any t ≤ T ,
q
kη(t)k s
B̊ p,q
≤
q
kη0 k s
B̊ p,q
Z
+C
0
t
!
q
kθ (τ )k
r + 2α
q
B̊ p,q
q
+ ke
θ (τ )k
r + 2α
q
B̊ p,q
q
B̊ sp,q
kη(τ )k
dτ.
J. Wu / Nonlinear Analysis 67 (2007) 3013–3036
3033
In particular, if θ0 = e
θ0 , then
θ =e
θ.
Acknowledgments
The author would like to thank Professor Changxing Miao for valuable communications and thank the referee for
pointing out the Refs. [1,2,12].
Appendix A
This appendix proves a Bernstein type inequality for fractional derivatives and a commutator estimate that has been
used in the previous sections.
Theorem A.1. Let α ≥ 0. Let 1 ≤ p ≤ q ≤ ∞.
(1) If f satisfies
supp b
f ⊂ {ξ ∈ Rd : |ξ | ≤ K 2 j },
for some integer j and a constant K > 0, then
2α j+ jd( 1p − q1 )
k(−1)α f k L q (Rd ) ≤ C1 2
k f k L p (Rd ) .
(2) If f satisfies
supp b
f ⊂ {ξ ∈ Rd : K 1 2 j ≤ |ξ | ≤ K 2 2 j }
(A.1)
for some integer j and constants 0 < K 1 ≤ K 2 , then
2α j+ jd( 1p − q1 )
C1 22α j k f k L q (Rd ) ≤ k(−1)α f k L q (Rd ) ≤ C2 2
k f k L p (Rd ) ,
where C1 and C2 are constants depending on α, p and q only.
Proof. We prove (2) and the proof of (1) is similar. To prove (2), it suffices to show
jd( 1p − q1 )
k f k L q ≤ C2
k f kL p
(A.2)
and
C22α j k f k L p ≤ k(−1)α f k L p ≤ C22α j k f k L p .
(A.3)
Because of (A.1), there exists a Φ j such that
bj b
b
f =Φ
f,
(A.4)
where Φ j is as defined in Section 2. That is, f = Φ j ∗ f . By Young’s inequality
k f k L q ≤ kΦ j k L p1 k f k L p ,
where
1
p1
=1+
1
q
− 1p . Noticing that Φ j (x) = 2 jd Φ0 (2 j x), we have
kΦ j k L p1 = 2
jd( 1p − q1 )
kΦ0 k L p1
and this proves (A.2).
To prove (A.3), we choose Φ j such that
α f (ξ ) = (2π |ξ |)2α b
b j (ξ )(2π|ξ |)2α b
\
(−1)
f (ξ ) = Φ
f (ξ ).
(A.5)
That is,
(−1)α f = K j ∗ f,
(A.6)
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J. Wu / Nonlinear Analysis 67 (2007) 3013–3036
where
K j (x) =
Z
Rd
b j (ξ )(2π|ξ |)2α dξ.
e2πix·ξ Φ
b j (ξ ) = Φ
b0 (2− j ξ ), we have
Since Φ
Z
j
2α j d j
2α
b0 (ξ )|ξ |2α dξ.
e2πi2 x·ξ Φ
K j (x) = 2 2 (2π )
Rd
By repeated integration by parts, we obtain
|K j (x)| ≤ C 22α j 2d j |2 j x|−s
for any s > 0. Therefore,
kK j k L 1 ≤ C 22α j .
(A.7)
Applying Young’s inequality to (A.6) and using (A.7), we prove the right half of (A.3). To prove the left half of (A.3),
we have from (A.5) that
α f (ξ ),
b j (ξ )(2π|ξ |)2α )−1 (−1)
b
\
f (ξ ) = (Φ
ξ ∈ Aj
where A j is defined in (2.1). This, in turn, implies
Z
b j (ξ )(2π|ξ |)2α )−1 dξ.
f = L j ∗ (−1)α f with L j (x) =
e2πix·ξ (Φ
Aj
The rest is then similar to the proof for the right half of (A.3). This completes the proof of Theorem A.1.
We now state and prove the commutator estimate.
Proposition A.2. Let j be an integer. Let 1 ≤ p < ∞ and 1 ≤ r ≤ ∞. Let u be a divergence free vector field. Then
k[u · ∇, 1 j ]θk L p ≤ C 2(1+ r ) j k1 j uk L p k1 j θ k L r + k1 j θk L p
d
X
2(1+ r )k k1k uk L r
d
k≤ j−1
+ k1 j uk L p
X
(1+ dr )k
2
!
(1+ dr ) j
X
k1k θk L r + 2
k1k uk L r k1k θk L p ,
(A.8)
k≥ j−1
k≤ j−1
where the brackets [ ] represent the commutator operator, namely
[u · ∇, 1 j ]θ ≡ u · ∇(1 j θ ) − 1 j (u · ∇θ ).
In particular, if d = 2, 2 ≤ p < ∞ and u = ∇ ⊥ Λ−1 θ, then
(1+ 2p ) j
k[u · ∇, 1 j ]θ k L p ≤ C 2
k1 j θk2L p + k1 j θk L p
X
(1+ 2p )k
2
k1k θk L p
k≤ j−1
(1+ 2p ) j
+ 2
!
X
k1k θk2L p
.
k≥ j−1
Proof. Splitting [u · ∇, 1 j ]θ into paraproducts, we have
[u · ∇, 1 j ]θ = I1 + I2 + I3 + I4 + I5 ,
where
I1 =
X
Sk−1 u · ∇1 j 1k θ − 1 j (Sk−1 u · ∇1k θ ),
k
I2 =
X
k
1k u · ∇1 j Sk−1 θ,
(A.9)
J. Wu / Nonlinear Analysis 67 (2007) 3013–3036
I3 =
X
3035
1 j (1k u · ∇ Sk−1 θ ),
k
I4 =
X X
1k u · ∇1 j 1l θ,
k |k−l|≤1
I5 =
X X
1 j (1k u · ∇1l θ ).
k |k−l|≤1
We bound the L p -norms of these terms. According to (2.5), the summation in I1 is only over k satisfying |k − j| ≤ 2.
Using the definition of 1 j , we can write
X Z
Φ j (x − y)(Sk−1 u(x) − Sk−1 u(y)) · ∇1k θ (y) dy.
I1 =
d
|k− j|≤2 R
We integrate by parts and use the fact that ∇ · u = 0 to obtain
X Z
I1 = −
∇Φ j (x − y) · (Sk−1 u(x) − Sk−1 u(y))1k θ (y) dy.
d
|k− j|≤2 R
By Young’s inequality,
Z
X
kI1 k L p ≤ C
k∇ Sk−1 uk L ∞ k1k θk L p
Rd
|k− j|≤2
X
=C
|x||∇Φ j (x)| dx
k1k θk L p k∇ Sk−1 uk L ∞ .
|k− j|≤2
Similarly, the summation in I2 , I3 and I4 are also only over k satisfying |k − j| ≤ 2. The estimates for these terms are
simple and their L p -norms are both bounded by
X
kI2 k L p , kI3 k L p ≤ C
k1k uk L p k∇ Sk−1 θk L ∞ ,
|k− j|≤2
X
kI4 k L p ≤ C
k1k uk L p k∇1l θk L ∞ .
|k− j|≤2,|k−l|≤1
The estimates for I5 are slightly different. The summation is over all k ≥ j − 1, namely
X
I5 =
1 j (1k u · ∇1l θ ).
k≥ j−1,|k−l|≤1
Since u is divergence free, we obtain after applying Bernstein’s inequality
X
d
kI5 k L p ≤ C2(1+ r ) j
k1 j (1k u1l θ )k L q ,
k≥ j−1,|k−l|≤1
where q satisfies
1
r
+
1
p
|I5 | ≤ C 2(1+ r ) j
d
= q1 . By Hölder’s inequality
X
k1k uk L r k1l θk L p .
k≥ j−1,|k−l|≤1
Since the summations in the bounds for I1 through I4 are only over a finite number of k’s, it suffices to consider the
typical term with k = j in our further estimates. It follows from Bernstein’s inequalities that
k∇1 j θk L ∞ ≤ C 2(1+ r ) j k1 j θk L r ,
X
X
d
k∇ S j−1 θk L ∞ ≤
k∇1k θk L ∞ ≤ C
2(1+ r )k k1k θk L r .
d
k≤ j−1
k≤ j−1
(A.9) is a consequence of (A.8) since k1 j uk L p ≤ C k1 j θk L p .
3036
J. Wu / Nonlinear Analysis 67 (2007) 3013–3036
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