Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 185, pp. 1–37.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu
GLOBAL MILD SOLUTIONS TO MODIFIED NAVIER-STOKES EQUATIONS WITH SMALL INITIAL DATA IN CRITICAL
BESOV-Q SPACES
PENGTAO LI, JIE XIAO, QIXIANG YANG
Abstract. This article is devoted to establishing the global existence and uniqueness of a mild solution of the modified Navier-Stokes equations with a small initial data in the critical Besov-Q space.
1. Statement of the main results
For β >1/2, the Cauchy problem of the modified Navier-Stokes equations on the half-spaceR1+n+ = (0,∞)×Rn, n≥2, consists of studying the existence of a solutionuto
∂u
∂t + (−∆)βu+u· ∇u− ∇p= 0, in R1+n+ ;
∇ ·u= 0, inR1+n+ ; u|t=0=a, in Rn,
(1.1)
where (−∆)β represents theβ-order Laplace operator defined by the Fourier trans- form in the space variable:
(−∆)\βu(·, ξ) =|ξ|2βˆu(·, ξ).
Here, we point out that (1.1) is a generalization of the classical Navier-Stokes system and two-dimensional quasi-geostrophic equation which have continued to attract attention extensively, and that the dissipation (−∆)βu still retains the physical meaning of the nonlinearity u· ∇u+∇p and the divergence-free condi- tion∇ ·u= 0.
Upon lettingRj,j= 1,2, . . . n, be the Riesz transforms, writing P={δl,l0 +RlRl0}, l, l0= 1, . . . , n;
P∇(u⊗u) =X
l
∂
∂xl
(ulu)−X
l
X
l0
RlRl0∇(ulul0);
e−t(−∆)\βf(ξ) =e−t|ξ|2βfˆ(ξ),
(1.2)
2000Mathematics Subject Classification. 35Q30, 76D03, 42B35, 46E30.
Key words and phrases. Modified Navier-Stokes equations; Besov-Q spaces; mild solutions;
existence; uniqueness.
c
2014 Texas State University - San Marcos.
Submitted April 29, 2014. Published September 2, 2014.
1
and using ∇ ·u= 0, we can see that a solution of the above Cauchy problem is then obtained via the integral equation
u(t, x) =e−t(−∆)βa(x)−B(u, u)(t, x);
B(u, u)(t, x)≡ Z t
0
e−(t−s)(−∆)βP∇(u⊗u)ds,
(1.3) which can be solved by a fixed-point method whenever the convergence is suitably defined in a function space. Solutions of (1.3) are called mild solutions of (1.1). The notion of such a mild solution was pioneered by Kato-Fujita [14] in 1960s. During the latest decades, many important results about the mild solutions to (1.1) have been established; see for example, Cannone [3, 4], Germin-Pavlovic-Staffilani [11], Giga-Miyakawa [12], Kato [13], Koch-Tataru [16], Wu [30, 31, 32, 33], and their references including Kato-Ponce [15] and Taylor [28].
The main purpose of this paper is to establish the following global existence and uniqueness of a mild solution to (1.1) with a small initial data in the critical Besov-Q space.
Theorem 1.1. Assume that β > 1/2; 1 < p, q < ∞; γ1 = γ2−2β + 1; m >
max{p,2βn};0< m0 <min{1,2βp }. If the index (β, p, γ2)satisifes 1< p≤2 and 2β−2
p < γ2≤ n p or
2< p <∞andβ−1< γ2≤n p,
then (1.1) has a unique global mild solution in (Bγp,q,m,m1,γ2 0)n for any initial data a with kak( ˙Bγ1,γ2
p,q )n being small. Here the symbols B˙p,qγ1,γ2, and Bγp,q,m,m1,γ2 0 stand for the so-called Besov-Q spaces and their induced tent spaces, and will be determined properly in Sections 3 and 4.
Needless to say, our current work grows from the already-known results. Lions [21] proved the global existence of the classical solutions of (1.1) whenβ ≥ 54 and n= 3. This existence result was extended toβ≥ 12+n4 by Wu [30], and moreover, for the important caseβ < 12+n4. Wu [31, 32] established the global existence for (1.1) in the Besov spaces ˙B1+
n p−2β,q
p (Rn) for 1≤q≤ ∞and for either 12 < β and p= 2 or 12 < β ≤1 and 2< p <∞and in ˙B2r,∞(Rn) withr >max{1,1 +np−2β};
see also [33] concerning the corresponding regularity. Importantly, Koch-Tataru [16] studied the global existence and uniqueness of (1.1) withβ= 1 via introducing BM O−1(Rn). Extending Koch-Tataru’s work [16], Xiao [35, 36] introduced the Q-spaces Q−10<α<1(Rn) to investigate the global existence and uniqueness of the classical Navier-Stokes system. The ideas of [35] were developed by Li-Zhai [18]
to study the global existence and uniqueness of (1.1) with small data in a class of Q-type spaces Qβ,−1α (Rn) under β ∈ (12,1). Recently, Lin-Yang [20] got the global existence and uniqueness of (1.1) with initial data being small in a diagonal Besov-Q space forβ ∈(12,1).
In fact, the above historical citations lead us to make a decisive two-fold obser- vation. On the one hand, thanks to that (1.1) is invariant under the scaling
uλ(t, x) =λ2β−1u(λ2βt, λx);
pλ(t, x) =λ4β−2p(λ2βt, λx),
the initial data space ˙Bp,qγ1,γ2 is critical for (1.1) in the sense that the space is invariant under the scaling
fλ(x) =λ2β−1f(λx). (1.4)
A simple computation, along with lettingβ = 1 in (1.4), indicates that the function spaces
L˙2n
2−1(Rn) = ˙B−1+
n 2,2
2 (Rn); Ln(Rn);
B˙−1+
n p,q
p (Rn); BM O−1(Rn),
are critical for (1.1) withβ = 1. Moreover, (1.4) underβ >1/2 is valid for functions in the homogeneous Besov spaces ˙B21+n2−2β,1(Rn) and ˙B1+2 n2−2β,∞(Rn) attached to (1.1). On the other hand, it is suitable to mention the following relations:
B˙γ1,
n
p,qp = ˙Bpγ1,q(Rn) for 1≤p, q <∞,−∞< γ1<∞;
B˙1+
n p−2β,np p0,q0 ⊇B˙1+
n p−2β,q
p (Rn) for 1< p≤p0, 1< q≤q0<∞, β >0;
B˙α−β+1,α+β−1
2,2 =Qβα(Rn) forα∈(0,1), β∈(1/2,1), α+β−1≥0.
To briefly describe the argument for Theorem 1.1, we should point out that the function spaces used in [16, 35, 18] have a common trait in the structure; i.e., these spaces can be seen as the Q-spaces withL2norm, and the advantage of such spaces is that Fourier transform plays an important role in estimating the bilinear term on the corresponding solution spaces. Nevertheless, for the global existence and uniqueness of a mild solution to (1.1) with a small initial data in ˙Bp,qγ1,γ2, we have to seek a new approach. Generally speaking, a mild solution of (1.1) is obtained by using the following method. Assume that the initial data belongs to ˙Bγp,q1,γ2. Via the iteration process:
u(0)(t, x) =e−t(−∆)βa(x);
u(j+1)(t, x) =u(0)(t, x)−B(u(j), u(j))(t, x) for j= 0,1,2, . . . ,
we construct a contraction mapping on a space in R1+n+ , denoted by X(R1+n+ ).
With the initial data being small, the fixed point theorem implies that there exists a unique mild solution of (1.1) in X(R1+n+ ). In this paper, we chooseX(R1+n+ ) = Bγp,q,m,m1,γ2 0associated with ˙Bp,qγ1,γ2. Owing to Theorem 4.5, we know that iff ∈B˙γp,q1,γ2 then e−t(−∆)βf(x) ∈ X(R1+n+ ). Hence the construction of contraction mapping comes down to prove the following assertion: The bilinear operator
B(u, v) = Z t
0
e−(t−s)(−∆)βP∇(u⊗v)ds is bounded from (X(R1+n+ ))n×(X(R1+n+ ))n to (X(R1+n+ ))n.
For this purpose, using multi-resolution analysis, we decompose Bl(u, v) into several parts based on the relation between t and 2−2jβ, and expanse every part in terms of {Φεj,k}. More importantly, Lemmas 6.1 & 6.2 enable us to obtain an estimate from (Bγp,q,m,m1,γ2 0)n×(Bγp,q,m,m1,γ2 0)n to (Bγp,q,m,m1,γ2 0)n.
Remark 1.2. (i) Our initial spaces in Theorem 1.1 include both ˙B1+
n p−2β,q p (Rn) in Wu [30, 31, 32, 33],Qβ,−1α (Rn) in Xiao [35] and Li-Zhai [18]. Moreover, in [18, 20], the scope ofβ is (12,1). Our method is valid for β >1/2.
(ii) We point out that ˙Bγp,q1,γ2 provide a lot of new critical initial spaces where the well-posedness of equations (1.1) holds. By Lemma 3.7, forβ = 1, Theorem 1.1 holds for the initial spaces ˙Bp,qγ1,γ2 satisfying
B˙2,q−1+w,w0 ⊂Q−1α (Rn), q0 <2, w >0 or
Q−1α (Rn)⊂B˙2,q−1+w,w00 ⊂BM O−1(Rn), 2< q00<∞, w >0.
In some sense, ˙Bp,qγ1,γ2 fill the gap between the spaces Q−1α (Rn) and BM O−1(Rn).
See also Lemma 3.7, Corollary 3.8 and Remark 3.9.
(iii) For a initial data a ∈ B˙p,qγ1,γ2, the index γ1 represents the regularity of a.
In Theorem 1.1, taking ˙Bp,qγ1,γ2 = ˙B−1+w,wp,q with w > 0, p > 2 and q > 2 yields B˙p,q−1+w,w⊂BM O−1(Rn). Compared with the ones inBM O−1(Rn), the elements of ˙Bp,q−1+w,w have higher regularity. Furthermore, Theorem 1.1 implies that the regularity of our solutions becomes higher along with the growth ofγ1.
(iv) Interestingly, Federbush [8] employed the divergence-free wavelets to study the classical Navier-Stokes equations, while the wavelets used in this paper are classical Meyer wavelets. In addition, when constructing a contraction mapping, Federbush’s method was based on the estimates of “long wavelength residues”.
Nevertheless, our wavelet approach based on Lemmas 5.1-5.2 and the Cauchy- Schwarz inequality is to convert the bilinear estimate ofB(u, v) into various efficient computations involved in the wavelet coefficients ofuandv.
The remaining of this article is organized as follows. In Section 2, we list some preliminary knowledge on wavelets and give the wavelet characterization of the Besov-Q spaces. In Sections 3-4 we define the initial data spaces and the corre- sponding solution spaces. Section 5 carries out a necessary analysis of some non- linear terms and a prior estimates. In Section 6, we verify Theorem 1.1 via Lemmas 5.1-5.2 which will be demonstrated in Sections 7-8 respectively.
Notation. U≈Vindicates that there is a constantc >0 such thatc−1V≤U≤cV whose right inequality is also written as U . V. Similarly, one writes V& U for V≥cU.
2. Preliminaries
First of all, we would like to say that we will always utilize tensorial orthogonal wavelets which may be regular Daubechies wavelets (only used for characterizing Besov and Besov-Q spaces) and classical Meyer wavelets, but also to recall that the regular Daubechies wavelets are such Daubechines wavelets that are smooth enough and have more sufficient vanishing moments than the relative spaces do;
see Lemma 2.3 and the part before Lemma 3.2.
Next, we present some preliminaries on Meyer wavelets Φ(x) in detail and refer the reader to [22], [29] and [38] for further information. Let Ψ0be an even function inC0∞([−4π3,4π3 ]) with
0≤Ψ0(ξ)≤1;
Ψ0(ξ) = 1 for|ξ| ≤ 2π 3 . If
Ω(ξ) = r
(Ψ0(ξ
2))2−(Ψ0(ξ))2, then Ω is an even function inC0∞([−8π3 ,8π3]). Clearly,
Ω(ξ) = 0 for|ξ| ≤ 2π 3 ;
Ω2(ξ) + Ω2(2ξ) = 1 = Ω2(ξ) + Ω2(2π−ξ) forξ∈[2π 3 ,4π
3 ].
Let Ψ1(ξ) = Ω(ξ)e−iξ2. For any= (1, . . . , n)∈ {0,1}n, define Φ(x) via the Fourier transform ˆΦ(ξ) = Qn
i=1Ψi(ξi). For j ∈ Z and k ∈ Zn, set Φj,k(x) = 2nj2 Φ(2jx−k). Furthermore, we put
En={0,1}n\{0};
Fn ={(, k) :∈En, k∈Zn};
Λn ={(, j, k), ∈En, j∈Z, k∈Zn},
and for any∈ {0,1}n, k ∈Zn and a functionf onRn, we writefj,k =hf,Φj,ki.
The following result is well-known.
Lemma 2.1. The Meyer wavelets{Φj,k}(,j,k)∈Λn form an orthogonal basis in the spaceL2(Rn). Consequently, for anyf ∈L2(Rn), the following wavelet decomposi- tion holds in theL2 convergence sense:
f = X
(,j,k)∈Λn
fj,k Φj,k. Moreover, forj∈Z, let
Pjf = X
k∈Zn
fj,k0 Φ0j,k, Qjf = X
(,k)∈Fn
fj,k Φj,k.
For the above Meyer wavelets, by Lemma 2.1, the product of any two functionsu andv can be decomposed as
uv=X
j∈Z
Pj−3uQjv+X
j∈Z
QjuQjv+ X
0<j−j0≤3
QjuQj0v
+ X
0<j0−j≤3
QjuQj0v+X
j∈Z
QjuPj−3v.
(2.1)
Suppose thatϕis a function onRn satisfying
supp ˆϕ⊂ {ξ∈Rn:|ξ| ≤1}, ˆ
ϕ(ξ) = 1 for{ξ∈Rn :|ξ| ≤ 1 2}, and that
ϕv(x) = 2n(v+1)ϕ(2v+1x)−2nvϕ(2vx) ∀v∈Z, are the Littlewood-Paley functions; see [25].
Definition 2.2. Given constants −∞ < α < ∞, 0 < p, q < ∞. A function f ∈S0(Rn)/P(Rn) belongs to ˙Bpα,q(Rn) if
kfkB˙α,qp =h X
v∈Z
2qvαkϕv∗fkqpi1/q
<∞.
The following lemma is essentially known.
Lemma 2.3 ([22]). Let {Φ,1j,k}(,j,k)∈Λn and {Φ,2j,k}(,j,k)∈Λn be different wavelet bases which are sufficiently regular. If
a,j,k,j0 0,k0 =hΦ,1j,k,Φj00,2,k0i,
then for any natural number N there exists a positive constant CN such that for j, j0∈Z andk, k0∈Zn,
|a,j,k,j0 0,k0| ≤CN2−|j−j0|(n2+N) 2−j+ 2−j0
2−j+ 2−j0+|2−jk−2−j0k0| n+N
. (2.2)
According to Lemma 2.3 and Peetre’s paper [25], we see that this definition of B˙pα,q(Rn) is independent of the choice of {ϕv}v∈Z, whence reaching the following description of ˙Bpα,q(Rn).
Theorem 2.4. Givens∈R and0< p, q <∞. A function f belongs to B˙ps,q(Rn) if and only if
h X
j∈Z
2qj(s+n2−np) X
,k
|fj,k |pq/pi1/q
<∞.
3. Besov-Q spaces via wavelets
3.1. Definition and wavelet formulation. The forthcoming Besov-Q spaces cover many important function spaces, for example, Besov spaces, Morrey spaces and Q-spaces and so on. Such spaces were first introduced by wavelets in Yang [38] and were studied by several authors. For a related overview, we refer to Yuan- Sickel-Yang [40].
Let ϕ∈C0∞(B(0, n)) andϕ(x) = 1 for x∈ B(0,√
n). Let Q(x0, r) be a cube parallel to the coordinate axis, centered atx0and with side lengthr. For simplicity, sometimes, we denote byQ=Q(r) the cubeQ(x0, r) and letϕQ(x) =ϕ(x−xrQ). For 1< p, q <∞andγ1, γ2∈R, letm0=mγp,q1,γ2 be a positive constant large enough.
For arbitrary functionf, let Sp,q,fγ1,γ2 be the class of the polynomial functions PQ,f
such that
Z
xαϕQ(x)(f(x)−PQ,f(x))dx= 0 ∀|α| ≤m0.
Definition 3.1. Given 1< p, q <∞andγ1, γ2∈R. We say thatf belongs to the Besov-Q space ˙Bp,qγ1,γ2:= ˙Bγp,q1,γ2(Rn) provided
sup
Q
|Q|γn2−1p inf
PQ,f∈Sγp,q,f1,γ2kϕQ(f−PQ,f)kB˙γp1,q <∞, (3.1) where the superum is taken over all cubesQwith centerxQ and lengthr.
As a generalization of the Morrey spaces, the forthcoming Besov-Q spaces cover many important function spaces, for example, Besov spaces, Morrey spaces and Q-spaces and so on. Such spaces were first introduced by wavelets in Yang [38].
On the other hand, our Lemma 3.2 as below and Yang-Yuan’s [37, Theorem 3.1]
show that our Besov-Q spaces and their Besov type spaces coincide; see also Liang- Sawano-Ullrich-Yang-Yuan [19] and Yuan-Sickel-Yang [40] for more information on the so-called Yang-Yuan’s spaces.
Given 1< p, q <∞andγ1, γ2∈R. Letm0=mγp,q1,γ2be a sufficiently big integer.
For the regular Daubechies wavelets Φ(x), there exist two integers m ≥ m0 = mγp,q1,γ2 andM such that for∈En, Φ(x)∈C0m([−2M,2M]n) andR
xαΦ(x)dx= 0 ∀ |α| ≤m. By applying the regular Daubechies wavelets, we have the following wavelet characterization for ˙Bγp,q1,γ2.
Lemma 3.2. (i)f =P
,j,kaj,kΦj,k∈B˙γp,q1,γ2 if and only if sup
Q
|Q|γn2−1ph X
nj≥−log2|Q|
2jq(γ1+n2−np) X
(,k):Qj,k⊂Q
|aj,k|pq/pi1/q
<+∞, (3.2) where the supremum is taken over all dyadic cubes inRn.
(ii)The wavelet characterization in (i) is also true for the Meyer wavelets.
A direct application of Lemma 3.2 gives the following assertion.
Corollary 3.3. Given 1< p, q <∞,γ1, γ2∈R. (i) Each B˙p,qγ1,γ2 is a Banach space.
(ii) The definition ofB˙p,qγ1,γ2 is independent of the choice ofφ.
Now we recall some preliminaries on the Calder´on-Zygmund operators (cf. [22, 23]). Forx6=y, letK(x, y) be a smooth function such that there exists a sufficiently largeN0≤msatisfying
|∂xα∂yβK(x, y)|.|x−y|−(n+|α|+|β|) ∀|α|+|β| ≤N0. (3.3) A linear operator
T f(x) = Z
K(x, y)f(y)dy
is said to be a Calder´on-Zygmund one if it is continuous fromC1(Rn) to (C1(Rn))0, where the kernelK(·,·) satisfies (3.3) and
T xα=T∗xα= 0 ∀α∈Nn with|α| ≤N0. For such an operator, we writeT ∈CZO(N0).
The kernel K(·,·) may have a high singularity on the diagonal x = y, so ac- cording to the Schwartz kernel theorem, it is only a distribution in S0(R2n). For (, j, k),(0, j0, k0)∈Λn, let
a,j,k,j0 0,k0=hK(x, y),Φj,k(x)Φj00,k0(y)i.
IfT is a Calder´on-Zygmund operator, then its kernelK(·,·) and the related coeffi- cients satisfy the following relations (cf. [22, 23, 38]).
Lemma 3.4. (i)If T ∈CZO(N0), then the coefficientsa,j,k,j0 0,k0 satisfy
|a,j,k,j0 0,k0|.
2−j+2−j0 2−j+2−j0+|k2−j−k02−j0|
n+N0
2|j−j0|(n2+N0) ∀(, j, k),(0, j0, k0)∈Λn. (3.4)
(ii) If a,j,k,j0 0,k0 satisfy (3.4), then K(·,·), the kernel of the operator T, can be written as
K(x, y) = X
(,j,k),(0,j0,k0)∈Λn
a,j,k,j0 0,k0Φj,k(x)Φj00,k0(y)
in the distribution sense. Moreover, T belongs to CZO(N0−δ) for any small positive numberδ.
By the above lemma, we can prove the following result.
Corollary 3.5. For any1/2≤λ≤2 we have kf(λ·)kB˙p,qγ1,γ2 ≈ kfkB˙p,qγ1,γ2. 3.2. Critical spaces and their inclusions.
Definition 3.6. An initial data space is called critical for (1.1), if it is invariant under the scalingfλ(x) =λ2β−1f(λx).
Note that, ifu(t, x) is a solution of (1.1) and we replace u(t, x), p(t, x), a(x) by uλ(t, x) =λ2β−1u(λ2βt, λx), pλ(t, x) =λ4β−2u(λ2βt, λx), aλ(x) =λ2β−1a(λx) respectively, uλ(t, x) is also a solution of (1.1). So, the critical spaces occupy a significant place for (1.1). Forβ = 1,
L˙2n
2−1(Rn) = ˙B−1+2 n2,2(Rn); Ln(Rn);
B˙−1+
n p,∞
p (Rn), p <∞; BM O−1(Rn); B˙α−1,α2,2 (Rn), are critical spaces. For the generalβ,
B˙1+
n p−2β,∞
p (Rn), p <∞; B˙α−β+1,α+β2,2 (Rn), are critical spaces.
By Corollary 3.5, it is easy to see that each ˙Bp,qγ1,γ2 enjoys following dilation- invariance. Forβ > 12 andγ1−γ2= 1−2β, each ˙Bγp,q1,γ2 is a critical space; i.e.,
kλγ2−γ1f(λ·)kB˙γp,q1,γ2 ≈ kfkB˙γp,q1,γ2 ∀λ >0.
To better understand why the Besov-Q spaces are larger than many spaces cited in the introduction, we should observe the basic fact below.
Lemma 3.7. Given 1< p, q <∞ andγ1, γ2∈R. (i) If q1≤q2, then B˙p,qγ1,γ2
1 ⊂B˙p,qγ1,γ2
2 . (ii) ˙Bp,qγ1,γ2 ⊂B˙∞γ1−γ2,∞(Rn).
(iii) Given p1 ≥ 1. For w = 0, q1 = 1 or w > 0,1 ≤ q1 ≤ ∞, one has B˙p,qγ1,γ2+w⊂B˙γp1−w,γ2
p1,qq
1
.
For 0≤α−β+ 1 andα+β−1≤n/2, we say thatf belongs to the Q-type spaceQβα(Rn) provided
sup
Q
r2(α+β−1)−n Z
Q
Z
Q
|f(x)−f(y)|2
|x−y|n+2(α−β+1)dxdy <∞,
where the supremum is taken over all cubes with sidelengthr. This definition was used in [18] to extend the results in [35] which initiated a PDE-analysis of the original Q-spaces introduced in [7] (cf. [5, 6, 26, 34, 38] for more information). The following is a direct consequence of Lemmas 3.2 and 3.7.
Corollary 3.8. (i) If 0 ≤ α−β+ 1 < 1, α+β −1 ≤ n2, then Qβα(Rn) = B˙α−β+1,α+β−1
2,2 .
(ii) If p= γn
2, thenB˙p,qγ1,γ2 = ˙Bpγ1,q(Rn).
(iii) Given w= 0, v= 1or w >0,1≤v≤ ∞. Ifp=n/(γ2+w), then B˙pγ1,q(Rn)⊂B˙γ1−w,γn 2
u(w+γ2 ),qv.
Remark 3.9. Wu [31] obtained the well-posedness of (1.1) with an initial data in the critical Besov space ˙B1+
n p−2β,q
p (Rn). Given 1< p0, q0<∞. By Lemma 3.7, we can see that if 1< p≤p0, 1< q ≤q0 andβ >0,
B˙1+
n p−2β,q
p (Rn)⊂B˙1+
n p−2β,np p0,q0 .
4. Besov-Q spaces via semigroups
To establish a semigroup characterization of the Besov-Q spaces, recall the fol- lowing semigroup characterization ofQβα(Rn), see [18]: Given max{α,1/2}< β <1 andα+β−1≥0. f ∈Qβα(Rn) if and only if
sup
x∈Rn, r∈(0,∞)
r2α−n+2β−2 Z r2β
0
Z
|y−x|<r
|∇e−t(−∆)βf(y)|2t−αβdy dt <∞.
This characterization was used to derive the global existence and uniqueness of a mild solution to (1.1) with a small initial data in∇ ·(Qβα(Rn))n. Notice that
Qβα(Rn) = ˙Bα+β−1,α−β+1
2,2 .
So, to obtain the corresponding result of (1.1) with a small initial data in ˙Bγp,q1,γ2 under |p−2| +|q−2| 6= 0, we need a more meticulous relation among time, frequency and locality. For this purpose, by the Meyer wavelets and the fractional heat semigroups, we introduce some new tent spaces associated with ˙Bp,qγ1,γ2, and then establish some connections between these tent spaces and ˙Bp,qγ1,γ2.
4.1. Wavelets and semigroups. Forβ >0, let ˆKtβ(ξ) =e−t|ξ|2β. We have f(t, x) =e−t(−∆)βf(x) =Ktβ∗f(x).
For the Meyer wavelets {Φj,k}(,j,k)∈Λn, let aj,k(t) = hf(t,·),Φj,ki and aj,k = hf,Φj,ki. By Lemma 2.1 we obtain
f(x) =X
,j,k
aj,kΦj,k(x) and f(t, x) =X
,j,k
aj,k(t)Φj,k(x).
Iff(t, x) =Ktβ∗f(x), then aj,k(t) = X
0,|j−j0|≤1,k0
aj00,k0hKtβΦj00,k0,Φj,ki
= X
0,|j−j0|≤1,k0
aj00,k0
Z
e−t22jβ|ξ|2βΦˆ0(2j−j0ξ) ˆΦ(ξ)e−i(2j−j
0k0−k)ξdξ.
(4.1)
Lemma 4.1. Let{Φj,k}(,j,k)∈Λn be Meyer wavelets. For β >0 there exist a large constant Nβ>0and a small constant ˜c >0 such that ifN > Nβ then
|aj,k(t)|.e−˜ct22jβ X
0,|j−j0|≤1,k0
|aj00,k0|(1 +|2j−j0k0−k|)−N ∀t22βj≥1 (4.2) and
|aj,k(t)|. X
|j−j0|≤1
X
0,k0
|aj00,k0|(1 +|2j−j0k0−k|)−N ∀0< t22βj≤1. (4.3) Proof. Formally, we can write
aj,k(t) = X
0,|j−j0|≤1,k0
aj00,k0hKtβ∗Φj00,k0,Φj,ki
= X
0,|j−j0|≤1,k0
aj00,k0he−t(−∆)βΦj00,k0,Φj,ki
= X
0,|j−j0|≤1,k0
aj00,k0
Z
e−t22jβ|ξ|2βΦc0(2j−j0ξ)cΦ(ξ)e−i(2j−j
0k0−k)ξdξ.
We divide the rest of the argument into two cases.
Case 1: |2j−j0k0 −k| ≤ 2. Notice that Φc is supported on a ring. By a direct computation, we obtain
|aj,k(t)|. X
0,|j−j0|≤1,k0
|aj00,k0|e−t22jβ . X
0,|j−j0|≤1,k0
|aj00,k0|
(1 +|2j−j0k0−k|)Ne−t22jβ. Case 2: |2j−j0k0−k| ≥ 2. Denote by li0 the largest component of 2j−j0k0−k.
Then (1 +|li0|)N ∼(1 +|2j−j0k0−k|)N. We have aj,k(t)
= X
0,|j−j0|≤1,k0
aj00,k0
(li0)N Z
e−t22jβ|ξ|2βΦc0(2j−j0ξ)cΦ(ξ)[(−1
i ∂ξi0)Ne−i(2j−j
0k0−k)ξ]dξ.
By an integration-by-parts, we can obtain that if CNl is the binomial coefficient indexed byN andlthen
|aj,k(t)|=
X
0,|j−j0|≤1,k0
(−1)N aj00,k0
(li0)N Z N
X
l=0
CNl ∂ξli
0(e−t22jβ|ξ|2β)
×∂ξN−l
i0
(Φc0(2j−j0ξ)cΦ(ξ))e−i(2j−j
0k0−k)ξdξ
. X
0,|j−j0|≤1,k0
|aj00,k0| (1 +|2j−j0k0−k|)N
Z N X
l=0
CNl (−t22jβ|ξ|2β)l|ξ|2β−2ξi0
×e−t22jβ|ξ|2β∂ξN−l
i0 (Φc0(2j−j0ξ)cΦ(ξ))e−i(2j−j
0k0−k)ξdξ .
If t22jβ ≥1, there exists a constant c such that (t22jβ)le−t22jβ . e−ct22jβ. Since Φ0 is defined on a ring, we obtain
|aj,k(t)|. X
0,|j−j0|≤1,k0
|aj00,k0|
(1 +|2j−j0k0−k|)N(t22jβ)le−t22jβ
. X
0,|j−j0|≤1,k0
e−ct22jβ |aj00,k0| (1 +|2j−j0k0−k|)N. If 0< t22jβ ≤1, then we can deduce directly that
|aj,k(t)|. X
0,|j−j0|≤1,k0
|aj00,k0| (1 +|2j−j0k0−k|)N.
4.2. Tent spaces generated by Besov-Q spaces. Over R1+n+ we introduce a new tent type spaceBγp,q,m,m1,γ2 0 associated with ˙Bp,qγ1,γ2, and then establish a relation between ˙Bp,qγ1,γ2 andBγp,q,m,m1,γ2 0 via the fractional heat semigroupe−t(−∆)β.
Definition 4.2. Letγ1, γ2, m∈R,m0>0, 1< p, q <∞and a(t, x) = X
(,j,k)∈Λn
aj,k(t)Φj,k(x).
We say that: (i)f ∈Bγp,q,m1,γ2,I if supt≥0supx0,rIp,q,Qγ1,γ2
r,m(t)<∞, where Ip,q,Qγ1,γ2
r,m(t)
=:|Qr|qγn2−qp X
j≥max{−log2r,−log22β t}
2jq(γ1+n2−np)h X
(,k):Qj,k⊂Qr
|aj,k(t)|p(t22jβ)miq/p
;
(ii)f ∈Bγp,q1,γ2,II if supt≥0supx0,rIIp,q,Qγ1,γ2
r(t)<∞, where IIp,q,Qγ1,γ2
r(t) =:|Qr|qγn2−qp X
−log2r≤j<−log22β t
2jq(γ1+n2−np)h X
(,k):Qj,k⊂Qr
|aj,k(t)|piq/p
;
(iii)f ∈Bγp,q,m1,γ2,III if supx
0,rIIIp,q,Qγ1,γ2
r,m<∞, where IIIp,q,Qγ1,γ2
r,m=|Qr|qγn2−qp X
j≥−log2r
2jq(γ1+n2−np)
×Z r2β 2−2jβ
X
(,k):Qj,k⊂Qr
|aj,k(t)|p(t22jβ)mdt t
q/p
; (iv)f ∈Bγp,q,m1,γ2,IV0 if supx0,rIVp,q,Qγ1,γ2
r,m0 <∞, where IVp,q,Qγ1,γ2
r,m0 =:|Qr|qγn2−qp X
j≥−log2r
2jq(γ1+n2−np)
×Z 2−2jβ 0
X
(,k):Qj,k⊂Qr
|aj,k(t)|p(t22jβ)m0dt t
q/p .
Moreover, the associated tent type spaces are defined as
Bγp,q,m,m1,γ2 0 =Bγp,q,m1,γ2,I∩Bγp,q1,γ2,II∩Bγp,q,m1,γ2,III∩Bγp,q,m1,γ2,IV0 .
To continue our discussion, we need to introduce two more function spacesBγτ,∞
andBγ0,∞1 .
