Global well-posedness for the incompressible viscoelastic fluids in the critical Lp framework Ting Zhang∗, Daoyuan Fang†
arXiv:1101.5864v2 [math.AP] 2 Feb 2011
Department of Mathematics, Zhejiang University, Hangzhou 310027, China
Abstract We investigate global strong solutions for the incompressible viscoelastic system of Oldroyd–B type with the initial data close to a stable equilibrium. We obtain the existence and uniqueness of the global solution in a functional setting invariant by the scaling of the associated equations, where the initial velocity has the same critical regularity index as for the incompressible Navier–Stokes equations, and one more derivative is needed for the deformation tensor. Like the classical incompressible NavierStokes, one may construct the unique global solution for a class of large highly oscillating initial velocity. Our result also implies that the deformation tensor F has the same regularity as the density of the compressible Navier–Stokes equations.
1
Introduction
In this paper, we consider the following system describing incompressible viscoelastic fluids. N ∇ · v = 0, x ∈ R , N ≥ 2, h i vt + v · ∇v + ∇p = µ∆v + ∇ · ∂W∂F(F ) F ⊤ , Ft + v · ∇F = ∇vF, F (0, x) = I + E0 (x), v(0, x) = v0 (x).
(1.1)
Here, v, p, µ > 0, F and W (F ) denote, respectively, the velocity field of materials, pressure, viscosity, deformation tensor and elastic energy functional. The third equation is simply the consequence of the chain law. It can also be regarded as the consistence condition of the flow trajectories obtained from the velocity field v and also of those obtained from the deformation tensor F [11, 17, 22, 27, 29]. Moreover, on the right-hand side of the momentum equation, ∂W∂F(F ) is the Piola–Kirchhoff stress tensor and ∂W∂F(F ) F ⊤ is the Cauchy–Green tensor. The latter is the change variable (from Lagrangian to Eulerian coordinates) form of the former one [22]. The above system is equivalent to the usual Oldroyd–B model for viscoelastic fluids in the case of infinite Weissenberg number [21]. On the other hand, without the viscosity term, it represents exactly the incompressible elasticity in Eulerian coordinates. We refer to [3, 11, 21, 26, 28, 29] and their references for the detailed derivation and physical background of the above system. Throughout this paper, we will use the notations of (∇v)ij =
∂vi , (∇vF )ij = (∇v)ik Fkj , (∇ · F )i = ∂j Fij , ∂xj
and summation over repeated indices will always be understood. For incompressible viscoelastic fluids, Lin et al. [27] proved the global existence of classical small solutions for the two–dimensional case with the initial data v0 , E0 = F0 − I ∈ H k (R2 ), k ≥ 2, by introducing an auxiliary vector field to replace the transport variable F . Using the method in [19] for the damped wave equation, they [27] also obtained the global existence of classical small solutions for the three–dimensional case with the initial data v0 , E0 ∈ H k (R3 ), k ≥ 3. Lei and Zhou [24] obtained the same ∗ E-mail: † E-mail:
[email protected] [email protected]
1
results for the two–dimensional case with the initial data v0 , E0 ∈ H s (R2 ), s ≥ 4. via the incompressible limit working directly on the deformation tensor F . Then, Lei et al. [23] proved the global existence for two–dimensional small-strain viscoelasticity with H 2 (R2 ) initial data and without assumptions on the smallness of the rotational part of the initial deformation tensor. It is worth noticing that the global existence and uniqueness for the large solution of the two–dimensional problem is still open. Recently, by introducing an auxiliary function w = ∆v + µ1 ∇ · E, Lei et al. [22] obtained a weak dissipation on the deformation F and the global existence of classical small solutions to N –dimensional system with the initial data v0 , E0 ∈ H 2 (RN ) and N = 2, 3. All these results need that the initial data v0 and E0 have the same regularity, and the regularity index of the initial velocity v0 is bigger than the critical regularity index for the classical incompressible Navier–Stokes equations. There are some classical results for the following incompressible Navier–Stokes equations. vt + v · ∇v + ∇p = µ∆v, ∇ · v = 0, (1.2) v(0, x) = v0 (x).
In 1934, J. Leray proved the existence of global weak solutions for (1.2) with divergence–free u0 ∈ L2 N (see [25]). Then, H. Fujita and T. Kato (see [16]) obtain the uniqueness with u0 ∈ H˙ 2 −1 . The index s s = N/2 − 1 is critical for (1.2) with initial data in H˙ : this is the lowest index for which uniqueness has been proved (in the framework of Sobolev spaces). This fact is closely linked to the concept of scaling invariant space. Let us precise what we mean. For all l > 0, system (1.2) is obviously invariant by the transformation u(t, x) → ul (t, x) := lu(l2 t, lx), u0 (x) → u0l (x) := lu0 (lx), and a straightforward computation shows that ku0 k ˙ N2 −1 = ku0l k ˙ N2 −1 . This idea of using a functional H H setting invariant by the scaling of (1.2) is now classical and originated many works. In [5], M. Cannone, Y. Meyer and F. Planchon proved that: if the initial data satisfy kv0 k
−1+ N p
B˙ p,∞
≤ cν,
for p > N and some constant c small enough, then the classical Navier-Stokes system (NS ) is globally wellposed. Refer to [4, 9, 10, 20] for a recent panorama. In this paper, using the method in [6, 7] studying the compressible Navier–Stokes system, we are concerned with the existence and uniqueness of a solution for the initial data in a functional space with −1+ N minimal regularity order v0 ∈ B˙ p,1 p . Although the equation (1.1)3 for the deformation tensor F is identical to the equation for the vorticity ω = ∇ × v of the Euler equations ∂t ω + v · ∇ω = ∇vω, our result implies that the deformation tensor F has the same regularity as the density of the compressible Navier–Stokes equations. At this stage, we will use scaling considerations for (1.1) to guess which spaces may be critical. We observe that (1.1) is invariant by the transformation (v0 (x), F0 (x)) → (lv0 (lx), F0 (lx)), (v(t, x), F (t, x), P (t, x)) → (lv(l2 t, lx), F (l2 t, lx), l2 P (l2 t, lx)), up to a change of the elastic energy functional W into l2 W . Definition 1.1. A functional space E ⊂ (S ′ (RN ))N × (S ′ (RN )N ×N is called a critical space if the associated norm is invariant under the transformation (v(x), F (x)) → (lv(lx), F (lx)) (up to a constant independent of l). N
N
p −1 N p ) × (B˙ p,1 )N ×N , p ∈ [2, ∞), is a critical space. In this paper, (B˙ p,1 Assume that E0 := F0 − I and v0 satisfy the following constraints:
∇ · v0 = 0, det(I + E0 ) = 1, ∇ · E0⊤ = 0, 2
(1.3)
and ∂m E0ij − ∂j E0im = E0lj ∂l E0im − E0lm ∂l E0ij .
(1.4)
The first three of these expressions are just the consequences of the incompressibility condition and the last one can be understood as the consistency condition for changing variables between the Lagrangian and Eulerian coordinates [22]. For simplicity, we only consider the case of Hookean elastic materials: W (F ) = |F 2 |. Define the usual strain tensor by the form E = F − I. (1.5) Then, the system (1.1) is ∇ · v = 0, x ∈ RN , N ≥ 2, vit + v · ∇vi + ∂i p = µ∆vi + Ejk ∂j Eik + ∂j Eij , Et + v · ∇E = ∇vE + ∇v, (v, E)(0, x) = (v0 , E0 )(x).
(1.6)
N
2 −1 N The global well-poesdness result had been obtained with the small initial data v0 ∈ (B˙ 2,1 ) and N N 2 2 −1 N ×N ˙ ˙ independently in [30, 35]. E0 ∈ (B2,1 ∩ B2,1 )
Theorem 1.1 ([30, 35]). Suppose that the initial data satisfies the incompressible constraints (1.3), N N 2 −1 N 2 ) and E0 ∈ (B˙ 2,1 )N ×N . Then there exist T > 0 and a unique local solution for system (1.6) v0 ∈ (B˙ 2,1 that satisfies N (v, E) ∈ UT2 , k(v, E)k
≤ C(kE0 k
),
(1.7)
∇ · v = 0, det(I + E) = 1, ∇ · E ⊤ = 0,
(1.8)
N
UT2
N
2 B˙ 2,1
+ kv0 k
N
−1 2 B˙ 2,1
and where
N N ×N s+1 s−1 s UTs = L1 ([0, T ]; B˙ 2,1 ) ∩ C([0, T ]; B˙ 2,1 ) × C([0, T ]; B˙ 2,1 ) ,
Furthermore, suppose that the initial data satisfies the incompressible constraints (1.3)–(1.4), v0 ∈ N N −1 2 2 −1 N ×N ∩ B˙ 2,1 ) and (B˙ 2,1 )N , E0 ∈ (B˙ 2,1 N 2
kE0 k
N
N
2 ∩B ˙ 2 −1 B˙ 2,1 2,1
+ kv0 k
N
−1 2 B˙ 2,1
≤ λ,
(1.9)
where λ is a small positive constant. Then there exists a unique global solution for system (1.6) that satisfies N (v, E) ∈ V 2 k(v, E)k
V
N 2
≤ C(kE0 k
B
N 2
N −1
2 ∩B˙ 2,1
+ kv0 k
N
−1 2 B˙ 2,1
),
where N N ×N s+1 s−1 s−1 s s V s = L1 (R+ ; B˙ 2,1 ) ∩ C(R+ ; B˙ 2,1 ) × L2 (R+ ; B˙ 2,1 ) ∩ C(R+ ; B˙ 2,1 ∩ B˙ 2,1 ) .
In this paper, we will obtain the following theorem, where X X ∆k f, ∆k f, f h = fl = 2k >R0
2k ≤R0
and R0 is as in Proposition 4.2.
3
(1.10)
Theorem 1.2. Suppose that the initial data satisfies the incompressible constraints (1.3)–(1.4), E0 ∈ N N −1 s+1 s s B˙ pp11,1 , E0l ∈ B˙ 2,r , E0h ∈ B˙ 2,r , v0 ∈ B˙ pp11,1 ∩ B˙ 2,r , for some p1 ∈ [2, ∞), p2 ∈ [p1 , ∞), r ∈ [1, ∞] and s ∈ R such that (−1, N2 − 1) if r > 1, (1.11) s∈ (−1, N2 − 1] if r = 1. There exists a constant λ depending only on N , p2 and s such that if kE0l kB˙ s + kE0h k 2,r
N p s+1 ∩B˙ p 2,1 B˙ 2,r 2
+ kv0 kB˙ s + kv0h k 2,r
N −1
p B˙ p22,1
≤ λ,
(1.12)
then there exists a global solution for system (1.6) that satisfies (v, E) ∈ Vps1 ,r where s Vp,r
=
(
N N N p −1 p +1 1 + ˙ s+2 + ˙s e ˙ e ˙ v ∈ C(R ; B2,r ∩ Bp,1 ) ∩ L (R ; B2,r ∩ Bp,1 )
E∈
N ×N N N e + ; B˙ s+1 ∩ B˙ p ) ∩ L2 (R+ ; B˙ p ) C(R , 2,r p,1 p,1
N ×N N ×N h e 2 (R+ ; B˙ s+1 ) ∩ L e 1 (R+ ; B˙ s+2 ) e 1 (R+ ; B˙ s+1 ) El ∈ L , E ∈ L . 2,r 2,r 2,r
Furthermore, the solution is unique in Vps1 ,r if p1 ≤ 2N .
Remark 1.1. This result allows us to construct the unique global solution for the high oscillating initial velocity v0 . For example, N x1 v0 (x) = ε p −1 sin( )φ(x), φ(x) ∈ S(RN ), p ∈ [N, ∞), ε which satisfies kv0 kB˙ s + kv0h k 2,r
N −1
p B˙ p 2,1
≪ 1,
2
if s < −1 +
N p,
p2 > p, ε is small enough, (Proposition 2.9, [7]).
Remark 1.2. The L2 -decay in time for E is a key point in the proof of the global existence. We shall also N ×N N ×N e 1 (R+ ; B˙ s+2 ) e 1 (R+ ; B˙ s+1 ) get a L1 −decay in a space E l ∈ L , Eh ∈ L . 2,r
2,r
Remark 1.3. Theorem 1.2 implies that the deformation tensor F has similar property as the density of the compressible Navier–Stokes system [6, 7]. And we think that the incompressible viscoelastic system is similar to the compressible Navier–Stokes system.
Remark 1.4. Similar to the compressible Navier–Stokes system [6, 7], the initial data do not really belong s to a critical space in the sense of Definition 1.1. We indeed made the additional assumptions E0l ∈ B˙ 2,r , s+1 h s ˙ ˙ E0 ∈ B2,r , v0 ∈ B2,r . On the other hand, our scaling considerations do not take care of the Cauchy– Green tensor term. A careful study of the linearized system (see Proposition 4.2 below) besides indicates that such an assumption may be unavoidable. Remark 1.5. Considering the general viscoelastic model (1.1), if the strain energy function satisfies the strong Legendre–Hadamard ellipticity condition ∂ 2 W (I) = (α2 − 2β 2 )δil δjm + β 2 (δim δjl + δij δlm ), with α > β > 0, ∂Fil ∂Fjm
(1.13)
and the reference configuration stress–free condition ∂W (I) = 0, ∂F then we can obtain the same results as that in Theorem 1.2. 4
(1.14)
In this paper, we introduce the following function: c = Λ−1 ∇ · E, where Λs f = F −1 (|ξ|s fˆ). Then, the system (1.6) reads ∇ · v = ∇ · c = 0, x ∈ RN , N ≥ 2, vit + v · ∇vi + ∂i p = µ∆vi + Ejk ∂j Eik + Λci , ct + v · ∇c + [Λ−1 ∇·, v·]∇E = Λ−1 ∇ · (∇vE) − Λv, ∆Eij = Λ∂j ci + ∂k (∂k Eij − ∂j Eik ), (v, c)(0, x) = (v0 , Λ−1 ∇ · E0 )(x).
(1.15)
So, we need to study the following mixed parabolic–hyperbolic linear system with a convection term: ∇ · v = ∇ · c = 0, x ∈ RN , N ≥ 2, vit + u · ∇vi + ∂i p − µ∆vi − Λci = G, (1.16) ct + u · ∇c + Λv = L, (v, c)(0, x) = (v0 , c0 )(x), where divu = 0 and c0 = Λ−1 ∇ · E0 . This system is similar to the system ct + v · ∇c + Λd = F, dt + v · ∇d − µ ¯∆d − Λc = G,
(1.17)
for compressible Navier-Stokes system [6, 7]. Using the similar method studying the compressible Navier– Stokes system in [6, 7], we can obtain some important estimates for the system (1.16). As for the related studies on the existence of solutions to nonlinear elastic systems, there are works by Sideris [31] and Agemi [1] on the global existence of classical small solutions to three–dimensional compressible elasticity, under the assumption that the nonlinear terms satisfy the null conditions. The global existence for three–dimensional incompressible elasticity was then proved via the incompressible limit method in [32] and by a different method in [33]. It is worth noticing that the global existence and uniqueness for the corresponding two–dimensional problem is still open. As for the density-dependent incompressible viscoelastic fluids, Hu and Wang [18] obtained the existence and uniqueness of the global strong solution with small initial data in R3 . The rest of this paper is organized as follows. In Section 2, we state three lemmas describing the intrinsic properties of viscoelastic system. In Section 3, we present the functional tool box: Littlewood– Paley decomposition, product laws in Sobolev and hybrid Besov spaces. The next section is devoted to the study of some linear models associated to (1.6). In Section 5, we give some a priori estimates. At last, we will study the global well-posedness for (1.6).
2
Basic mechanics of viscoelasticity
Using the similar arguments as that in the proof of Lemmas 1–3 in [22], we can easily obtain the following three lemmas. Lemma 2.1. Assume that det(I + E0 ) = 1 is satisfied and (v, F ) is the solution of system (1.6). Then the following is always true: det(I + E) = 1, (2.1) for all time t ≥ 0, where the usual strain tensor E = F − I. Lemma 2.2. Assume that ∇ · E0⊤ = 0 is satisfied, then the solution (v, F ) of system (1.6) satisfies the following identities: ∇ · F ⊤ = 0, and ∇ · E ⊤ = 0, (2.2) for all time t ≥ 0. Lemma 2.3. Assume that (1.4) is satisfied and (v, F ) is the solution of system (1.6). Then the following is always true: ∂m Eij − ∂j Eim = Elj ∂l Eim − Elm ∂l Eij , (2.3) for all time t ≥ 0. 5
3
Littlewood–Paley theory and Besov spaces
The proof of most of the results presented in this paper requires a dyadic decomposition of Fourier variable (Littlewood–Paley composition). Let us briefly explain how it may be built in the case x ∈ RN , N ≥ 2, (see [12, 13, 14]). Let S(RN ) be the Schwarz class. ϕ(ξ) is a smooth function valued in [0,1] such that suppϕ ⊂ {
X 3 8 ≤ |ξ| ≤ } and ϕ(2−q ξ) = 1, |ξ| 6= 0. 4 3 q∈Z
Let h(x) = (F −1 ϕ)(x). For f ∈ S ′ (denote the set of temperate distributes, which is the dual one of S), we can define the homogeneous dyadic blocks as follows: Z −q Nq ∆q f (x) := ϕ(2 D)f (x) = 2 h(2q y)f (x − y)dy, if q ∈ Z, RN
where F −1 represents the inverse Fourier transform. Define the low frequency cut-off by X Sq f (x) := ∆p f (x) = χ(2−q D)f (x). p≤q−1
The Littlewood–Paley decomposition has nice properties of quasi-orthogonality, ∆p ∆q f1 ≡ 0, if |p − q| ≥ 2, and ∆q (Sp−1 f1 ∆p f2 ) ≡ 0, if |p − q| ≥ 5. Lemma 3.1 (Bernstein). Let k ∈ N and 0 < R1 < R2 . There exists a constant C depending only on R1 , R2 and N such that for all 1 6 a 6 b 6 ∞ and f ∈ La , we have 1
1
Supp F f ⊂ B(0, R1 λ) ⇒ sup k∂ α f kLb 6 C k+1 λk+N ( a − b ) kf kLa ; |α|=k
Supp F f ⊂ C(0, R1 λ, R2 λ) ⇒ C −k−1 λk kf kLa 6 sup k∂ α f kLa 6 C k+1 λk kf kLa . |α|=k
Here, F represents the Fourier transform. The Besov space can be characterized in virtue of the Littlewood–Paley decomposition. s Definition 3.1. Let 1 ≤ p ≤ ∞ and s ∈ R. For 1 ≤ r ≤ ∞, the Besov spaces B˙ p,r (RN ), N ≥ 2, are defined by
s f ∈ B˙ p,r (RN ) ⇔ 2qs k∆q f kLp (RN ) lr < ∞ q
s and B (R ) = B˙ 2,1 (RN ). s
N
s The definition of B˙ p,r (RN ) does not depend on the choice of the Littlewood–Paley decomposition. Let us recall some classical estimates in Sobolev spaces for the product of two functions [6, 15]. Formally, Bony’s decomposition is defined by
uv = Tu v + Tv u + R(u, v) = Tu v + Tv′ u, with Tu v =
X
Sq−1 u∆q v, R(u, v) =
q∈Z
X q∈Z
e q v, ∆ e qv = ∆q u∆
X
∆q′ v.
|q′ −q|≤1
Proposition 3.1. For any (s, p, r) ∈ R × [1, ∞]2 and t < 0, there exists a constant C such that kTu vkB˙ s ≤ CkukL∞ kvkB˙ s p,r
p,r
and kTu vkB˙ p,r s+t ≤ Ckuk ˙ t kvk ˙ s . B B p,r
p,r
For any (s1 , p1 , r1 ) and (s2 , p2 , r2 ) in R × [1, ∞]2 and t < 0, there exists a constant C such that 6
1 p
• if s1 + s2 > 0,
=
1 p1
+
1 p2
≤ 1 and
1 r
=
1 r1
1 r2
+
≤ 1, then
s s1 +s2 ≤ Ckuk ˙ s1 kR(u, v)kB˙ p,r Bp ,r kvkB˙ p2 ,r ; 1
1 p
• if s1 + s2 = 0,
=
1 p1
+
1 p2
≤ 1 and
1 r
=
1 r1
1 r2
+
1
2
2
≥ 1, then
kR(u, v)kB˙ 0 ≤ CkukB˙ ps1 ,r kvkB˙ ps2 ,r . p,r
1
1
2
2
Proposition 3.2. Let 1 ≤ r, p, p1 , p2 ≤ ∞. Then following inequalities hold true: kuvkB˙ s . kukL∞ kvkB˙ s + kvkL∞ kukB˙ s , if s > 0, p,r
kuvk
s +s2 − N p
1 ˙ p,r B
p,r
s1 kvk ˙ s2 , . kukB˙ p,r if s1 , s2 < Bp,∞
kuvkB˙ s . kukB˙ s kvk p,r
kuvk
−N p ˙ p,∞ B
(3.1)
p,r
p,r
N
p ∩L∞ B˙ p,∞
, if |s|
0, p N , p
(3.2) (3.3)
N N , ], p ≥ 2. p p
(3.4)
s e q (0, T ; Bp,r Finally, we state the definition of L ), which is first introduced by J.-Y. Chemin and N. Lerner in [8].
s e q (0, T ; B˙ p,r Definition 3.2. We denote by L ) the space of distributions, which is the completion of S(Rd ) by the following norm:
kakLeq (0,T ;B˙ s ) = 2sk k∆k akLq (0,T ;Lp (Rd )) lr . p,r
k
e q (0, T ; B˙ s ) also hold [6, 15]. Similar results of Propositions 3.1–3.2 in space L p,r Denote
kvkLe1 (B˜ s+2,s+1 ) := 2p(s+1) min(R0 , 2p )k∆p vkL1T (L2x ) T
lrp
2,r
(3.5)
Lemma 3.2. Let s > −1 and r ∈ [1, ∞], we have
∞ kvk e 1 ˜ s+2,s+1 + CkvkL∞ (L∞ ) kuk e 1 ˜ s+2,s+1 . kuvkLe1 (B˜ s+2,s+1 ) ≤ CkukL∞ ) ) L (B L (B x T (Lx ) T T
T
2,r
Proof. We write, for p ∈ Z, ∆p Tu v =
T
2,r
X
2,r
(3.6)
∆p (Sq−1 u∆q v),
|q−p|≤3
hence, k∆p Tu vkL1T (L2 ) ≤ C
X
∞ k∆q vkL1 (L2 ) , kukL∞ T (Lx ) T
|q−p|≤3
and ∞ kvk e 1 ˜ s+2,s+1 . kTu vkLe1 (B˜ s+2,s+1 ) ≤ CkukL∞ ) L (B T (Lx )
(3.7)
∞ kuk e 1 ˜ s+2,s+1 . kTv ukLe1 (B˜ s+2,s+1 ) ≤ CkvkL∞ ) L (B T (Lx )
(3.8)
T
2,r
T
2,r
Similarly, we have T
T
2,r
We write, for p ∈ Z, ∆p R(u, v) =
X
q≥p−2
hence,
e q v), ∆p (∆q u∆
2p(s+1) min(R0 , 2p )k∆p R(u, v)kL1T (L2 ) 7
2,r
≤ C
X
2(p−q)(s+1)
q≥p−2
≤ C
X
min(R0 , 2p ) q(s+1) ∞ 2 min(R0 , 2q )k∆q ukL1T (L2 ) kvkL∞ T (Lx ) min(R0 , 2q )
∞ , 2(p−q)(s+1) cq kukLe1 (B˜ s+2,s+1 ) kvkL∞ T (Lx ) T
2,r
q≥p−2
and ∞ kuk e 1 ˜ s+2,s+1 , kR(u, v)kLe1 (B˜ s+2,s+1 ) ≤ CkvkL∞ ) L (B T (Lx ) T
T
2,r
(3.9)
2,r
where we use the fact that s > −1.
4
A linear model with convection
After the change of function c = Λ−1 ∇ · E, the system (1.6) reads (1.15). At first, we will study the following mixed linear system ∇ · v = ∇ · c = 0, x ∈ RN , N ≥ 2, vit + Tu · ∇vi + ∂i p − µ∆vi − Λci = Gi , c t + Tu · ∇c + Λv = L, (v, c)(0, x) = (v0 , c0 )(x),
(4.1)
where divu = 0 and c0 = Λ−1 ∇ · E0 . Using the similar arguments as that in [12] (Proposition 2.3), we can obtain the following proposition and omit the details. The mainR different is R that there is a pressure term ∇p in (4.1)2 . Using that fact that ∇ · v = ∇ · c = 0, we have v · ∇p = c · ∇p = 0, and obtain the following proposition. R e (t) = t k∇u(τ )kL∞ dτ . The Proposition 4.1. Let (v, c) be a solution of (4.1) on [0, T ], ρ ∈ R and U 0 following estimate holds: k(v, c)l kLe∞ (B˙ ρ t
≤
e 1 ˙ ρ+2 2,r )∩Lt (B2,r )
+ kch kLe∞ (B˙ ρ+1 )∩Le1 (B˙ ρ+1 ) + kv h kLe∞ (B˙ ρ t
2,r
t
2,r
t
e 1 ˙ ρ+2 2,r )∩Lt (B2,r )
e
CeC U(t) k(v0 , c0 )l kB˙ ρ + kch0 kB˙ ρ+1 + kv0h kB˙ ρ + k(L, G)l kLe1 (B˙ ρ ) 2,r 2,r t 2,r 2,r +kLh kLe1 (B˙ ρ+1 ) + kGh kLe1 (B˙ ρ ) , t
t
2,r
2,r
where C depends only on N and ρ.
Then, we will study the following mixed linear system ∇ · v = ∇ · c = 0, x ∈ RN , N ≥ 2, vt − µ∆v − Λc = 0, ct + Λv = 0, (v, c)(0, x) = (v0 , c0 )(x).
(4.2)
Using the similar arguments as that in [7] (Lemma 4.1–4.2, Proposition 4.4), we have the following lemma. Lemma 4.1. Let G be the Green matrix of the following system ct + Λv = 0, vt − µ∆v − Λc = 0, ˆ then, we have the following explicit expression for G: λ+ eλ− t −λ− eλ+ t eλ+ t −eλ− t I − |ξ|I N ×N N ×N λ+ −λ− λ+ −λ− , Gˆ = λ+ eλ+ t −λ− eλ− t eλ+ t −eλ− t − λ+ −λ− |ξ|IN ×N IN ×N λ+ −λ−
where
1p 2 4 1 µ |ξ| − 4|ξ|2 . λ± = − µ|ξ|2 ± 2 2 8
(4.3)
(4.4)
Lemma 4.2. There exist positive constant R and ϑ depending on µ such that 2
ˆ t)| ≤ C|α| e−ϑ|ξ| t (1 + |ξ|)α (1 + t)|α| , ∀ |ξ| ≤ R, |∂ξα G(ξ,
ˆ t) G(ξ,
2 0 0 IN ×N 0 + e−µ|ξ| t 0 IN ×N 0 0 IN ×N 0 IN ×N + Gˆ2 (ξ, t) +Gˆ1 (ξ, t) IN ×N 0 0 −1
= e−µ
t
0 IN ×N
(4.5)
, ∀ |ξ| ≥ R,
(4.6)
where Gˆ1 and Gˆ2 satisfy the estimates t
2
t
2
|∂ξα Gˆ1 | ≤ C|ξ|−|α|−1 (e− 2µ + e−ϑ|ξ| t ), |∂ξα Gˆ2 | ≤ C|ξ|−|α|−2 (e− 2µ + e−ϑ|ξ| t ).
(4.7) (4.8)
Proposition 4.2. Let C be a ring centered at 0 in RN . Then there exist positive constants R0 , C, ϑ depending on µ such that, if suppˆ u ⊂ λC, then we have (a) if λ ≤ R0 , then 2
kG ∗ ukL2 ≤ Ce−ϑλ t kukL2 ;
(4.9)
(b) if b ≤ λ ≤ R0 , then for any 1 ≤ p ≤ ∞, 2
kG ∗ ukLp ≤ C(1 + b−N −1 )e−ϑλ t kukLp ;
(4.10)
(c) if λ > R0 , then for any 1 ≤ p ≤ ∞, kG 1 ∗ ukLp ≤ Cλ−1 e−ϑt kukLp ,
(4.11)
kG 2 ∗ ukLp ≤ Cλ−2 e−ϑt kukLp .
(4.12)
From Proposition 4.2, or using the similar arguments as that in [6] (lemma 2), we can obtain the following lemma. Lemma 4.3. Let (v, c) be a solution of (4.2). There exist two constants ϑ and C such that, if 2j > R0 , then for all p ∈ [1, ∞], 2j (4.13) k∆j v(t)kLp ≤ C 2−j e−ϑt k∆j c0 kLp + e−ϑt2 + 2−2j e−ϑt k∆j v0 kLp , k∆j c(t)kLp ≤ Ce−ϑt k∆j c0 kLp + 2−j k∆j v0 kLp .
(4.14)
j
For all m ≥ 1, there exist two constants α and C such that, if 2 ≤ R0 then 2j
eϑt2 k(∆j c, ∆j v)(t)kL2 ≤ Ck(∆j c0 , ∆j v0 )kL2 .
5
(4.15)
A priori estimates
Assume that v, E be a solution of (1.6). Letting Xp,0 = Ys,0 + Zp,0 ,
(5.1)
Ys,0 = kE0l kB˙ s + kE0h kB˙ s+1 + kv0 kB˙ s , 2,r
Zp,0 = kE0h k
2,r
2,r
N
p B˙ p,1
9
+ kv0h k
N −1
p B˙ p,1
,
(5.2) (5.3)
and Xp (t) = Ys (t) + Zp (t), l
(5.4)
h
Ys (t) = kE kLe2 (B˙ s+1 )∩Le1 (B˙ s+2 ) + kEkLe∞ (B˙ s+1 ) + kE kLe1 (B˙ s+1 ) + kvkLe∞ (B˙ s t
t
2,r
t
2,r
Zp (t) = kE h k
N
e ∞ (B˙ p ) L t p,1
t
2,r
+ kE h k
N
e 1 (B˙ p ) L t p,1
+ kv h k
e 1 ˙ s+2 2,r )∩Lt (B2,r )
t
2,r
N −1
e ∞ (B˙ p L t p,1
N +1
e 1 (B˙ p )∩L t p,1
,
.
(5.5) (5.6)
)
In this section, we will prove the following Claim 1: There exist two positive constants λ1 and C0 such that, if Xp2 (t) ≤ λ1 , for all t ∈ [0, T ],
(5.7)
Xpi (t) ≤ C0 Xpi ,0 , for all i = 1, 2 t ∈ [0, T ].
(5.8)
then Lemma 5.1. Under the condition (5.7), we have Ys (t) ≤ C1 Ys,0 , for all t ∈ [0, T ].
(5.9)
Proof. Let c = Λ−1 ∇ · E, where Λs f = F −1 (|ξ|s fˆ). Then, the system (1.6) reads ∇ · v = ∇ · c = 0, x ∈ RN , N ≥ 2, v it + Tv · ∇vi + ∂i p − µ∆vi − Λci = Gi := −∂j Tv′i vj + Ejk ∂j Eik , ct + Tv · ∇c + Λv = L := −∂j Tc′ vj + Λ−1 ∇ · (∇vE) − [Λ−1 ∇·, v·]∇E, ∆Eij = Λ∂j ci + ∂k (∂k Eij − ∂j Eik ), (v, c)(0, x) = (v0 , Λ−1 ∇ · E0 )(x).
(5.10)
From Proposition 4.1, we have
k(v, c)l kLe∞ (B˙ s
e 1 ˙ s+2 2,r )∩Lt (B2,r )
t
+ kch kLe∞ (B˙ s+1 )∩Le1 (B˙ s+1 ) + kv h kLe∞ (B˙ s t
t
2,r
e 1 ˙ s+2 2,r )∩Lt (B2,r )
t
2,r
e
CeC U(t) k(v0 , c0 )l kB˙ s + kch0 kB˙ s+1 + kv0h kB˙ s + kLl kLe1 (B˙ s ) 2,r 2,r 2,r t 2,r h +kL kLe1 (B˙ s+1 ) + kGkLe1 (B˙ s ) t t 2,r 2,r l C Ys,0 + kL kLe1 (B˙ s ) + kLh kLe1 (B˙ s+1 ) + kGkLe1 (B˙ s ) ,
≤
≤
t
t
2,r
t
2,r
(5.11)
2,r
R e (t) = t k∇v(τ )kL∞ dτ ≤ CXp2 (t) ≤ C, when λ1 ≤ 1. From Proposition 3.1 and s > −1, we where U 0 have, for all t ∈ [0, T ], k∂j Tv′i vj kLe1 (B˙ s t
2,r )
≤ kT∂j vi vj kLe1 (B˙ s
2,r )
t
≤ Ckvk
N +1 p L1t (B˙ p 2,1 ) 2
+ kR(vi , vj )kLe1 (B˙ s+1 ) t
kvkLe∞ (B˙ s t
2,r )
2,r
≤ Cλ1 kvkLe∞ (B˙ s ) . t
(5.12)
2,r
From (3.1) and ∇ · E ⊤ = 0, we have under the assumption that s + 1 > 0, s + 1 < s + 1 ≤ N2 (if r = 1), kEjk ∂j Eik kLe1 (B˙ s
2,r )
t
= k∂j (Ejk Eik )kLe1 (B˙ s t
2,r )
N 2
(if r > 1) or
≤ CkEkL2t (L∞ ) kEkLe2 (B˙ s+1 ) ≤ Cλ1 kEkLe2 (B˙ s+1 ) . t
t
2,r
2,r
(5.13)
From (5.12)–(5.13), we have kGkLe1 (B˙ s t
2,r )
≤ Cλ1 (kvkLe∞ (B˙ s t
2,r )
+ kEkLe2 (B˙ s+1 ) ). t
From Proposition 3.1, under the assumption that s + 1 > 0, s + 1 < we have, for all t ∈ [0, T ], k(∂j Tc′ vj )l kLe1 (B˙ s t
2,r )
(5.14)
2,r
N 2
(if r > 1) or s + 1 ≤
≤ Ck(Tc′ v)l kLe1 (B˙ s+1 ) ≤ CkckL2t (L∞ ) kvkLe2 (B˙ s+1 ) ≤ Cλ1 kvkLe2 (B˙ s+1 ) , t
t
2,r
10
2,r
t
2,r
N 2
(if r = 1), (5.15)
∞ kvk e 1 ˙ s+2 ≤ Cλ1 kvk e 1 ˙ s+2 . k(∂j Tc′ vj )h kLe1 (B˙ s+1 ) ≤ CkckL∞ L (B ) L (B ) t (L ) t
t
2,r
t
2,r
(5.16)
2,r
where we use the following estimates l
h
l
kckL∞ ≤ kc kL∞ + kc kL∞ ≤ C
kc k
h
N
2 B˙ 2,1
+ kc k
!
N p B˙ p 2,1 2
l
≤C
kE k
h
N
2 B˙ 2,1
+ kE k
N p B˙ p 2,1 2
!
and ∞ ≤ CXp (t) ≤ Cλ1 . kckL2t (L∞ ) + kckL∞ 2 t (L )
Similarly, from Proposition 3.1, we have, k(TE ∇v)l kLe1 (B˙ s
2,r )
t
≤ Cλ1 k∇vkLe2 (B˙ s ) ,
≤ CkEkL2t (L∞ ) k∇vkLe2 (B˙ s
2,r )
t
t
2,r
∞ k∇vk e 1 ˙ s+1 ≤ Cλ1 k∇vk e 1 ˙ s+1 , k(TE ∇v)h kLe1 (B˙ s+1 ) ≤ CkEkL∞ L (B ) L (B ) t (L ) t
t
2,r
k(T∇v E)l kLe1 (B˙ s t
2,r )
k(T∇v E)h kLe1 (B˙ s+1 ) t
t
2,r
≤
−1 Ck∇vkL2 (B˙ ∞,∞ e 2 (B˙ s+1 ) ) kEkL
≤
Cλ1 kEkLe2 (B˙ s+1 ) ,
t
t
t
(5.18)
2,r
(5.19)
2,r
≤ Ck∇vkL1t (L∞ ) kEkLe∞ (B˙ s+1 ) t
2,r
≤ Cλ1 (kE kLe∞ (B˙ s
2,r )
t
k(R(Ejk , ∂j vi ))l kLe1 (B˙ s
2,r )
t
k(R(E, ∇v))h kLe1 (B˙ s+1 ) 2,r
2,r
+ kE h kLe∞ (B˙ s+1 ) ),
l
t
2,r
(5.17)
t
=
k∂j (R(Ejk , vi ))l kLe1 (B˙ s
≤
l
k(R(E, v)) k
≤
Ckvk
≤
Cλ1 kEkLe2 (B˙ s+1 ) ,
2,r )
t
N p L2t (B˙ p22,∞ )
t
≤ k(R(E, ∇v))l k ≤ Ck∇vk
e 1 (B˙ L t
L1t (B˙ p22,∞ )
kEkLe2 (B˙ s+1 ) t
2,r
(5.21)
s+ N +1 p2 ) 2p2 ,r 2+p2
kEkLe∞ (B˙ s+1 ) t
l
2,r
h
≤ Cλ1 (kE kLe∞ (B˙ s
2,r )
t
s+ N +1 p2 ) 2p2 ,r 2+p2
2,r
e 1 (B˙ L t
N p
(5.20)
2,r
+ kE kLe∞ (B˙ s+1 ) ), t
(5.22)
2,r
where we use the following estimates −1 k∇vkL2 (B˙ ∞,∞ ) ≤ Ck∇vk t
N −1
p L2t (B˙ p 2,1 )
≤ CXp2 (t) ≤ Cλ1 ,
2
k∇vkL1t (L∞ ) ≤ Ck∇vk
N p
L1t (B˙ p 2,1 )
≤ CXp2 (t) ≤ Cλ1 .
2
From (5.17)–(5.22), we obtain k(Λ−1 ∇ · (∇vE))l kLe1 (B˙ s t
2,r )
≤ Cλ1 (k∇vkLe2 (B˙ s t
2,r )
+ kEkLe2 (B˙ s+1 ) ), t
k(Λ−1 ∇ · (∇vE))h kLe1 (B˙ s+1 ) ≤ Cλ1 (k∇vkLe1 (B˙ s+1 ) + kE l kLe∞ (B˙ s t
t
2,r
t
2,r
2,r )
(5.23)
2,r
+ kE h kLe∞ (B˙ s+1 ) ). t
2,r
(5.24)
Similar, using the classical commutator estimate (Lemma 2.97 in [2]), we obtain k([Λ−1 ∇·, v·]∇E)l kLe1 (B˙ s t
2,r )
≤ Cλ1 (k∇vkLe2 (B˙ s t
11
2,r )
+ kEkLe2 (B˙ s+1 ) ), t
2,r
(5.25)
k([Λ−1 ∇·, v·]∇E)h kLe1 (B˙ s+1 ) ≤ Cλ1 (k∇vkLe1 (B˙ s+1 ) + kE l kLe∞ (B˙ s t
t
2,r
2,r )
t
2,r
+ kE h kLe∞ (B˙ s+1 ) ). t
2,r
(5.26)
From (5.15)–(5.16) and (5.23)–(5.26), we get kLl kLe1 (B˙ s
2,r )
t
≤ Cλ1 (kvkLe2 (B˙ s+1 ) + kEkLe2 (B˙ s+1 ) ), t
t
2,r
kLh kLe1 (B˙ s+1 ) ≤ Cλ1 (kvkLe1 (B˙ s+2 ) + kE l kLe∞ (B˙ s t
t
2,r
2,r )
t
2,r
(5.27)
2,r
+ kE h kLe∞ (B˙ s+1 ) ). t
(5.28)
2,r
From (5.11), (5.14) and (5.27)–(5.28), we have k(v, c)l kLe∞ (B˙ s
e 1 ˙ s+2 2,r )∩Lt (B2,r )
t
≤
+ kch kLe∞ (B˙ s+1 )∩Le1 (B˙ s+1 ) + kv h kLe∞ (B˙ s t
t
2,r
e 1 ˙ s+2 2,r )∩Lt (B2,r )
t
2,r
CYs,0 + Cλ1 Ys (t).
(5.29)
From Lemmas 2.2–2.3, we have ∂m Eij − ∂j Eim = Elj ∂l Eim − Elm ∂l Eij = ∂l (Elj Eim − Elm Eij ).
(5.30)
and from (5.10)4 , Proposition 3.2, Lemma 3.2, kEkLe∞ (B˙ s+1 ) + kE l kLe2 (B˙ s+1 )∩Le1 (B˙ s+2 ) + kE h kLe1 (B˙ s+1 ) t
t
2,r
t
2,r
t
2,r
2,r
h
l
≤ CkckLe∞ (B˙ s+1 ) + Ckc kLe2 (B˙ s+1 )∩Le1 (B˙ s+2 ) + Ckc kLe1 (B˙ s+1 ) t
t
2,r
2
t
2,r
t
2,r
2,r
2 l
+CkE kLe∞ (B˙ s+1 ) + Ck(E ) kLe2 (B˙ s+1 )∩Le1 (B˙ s+2 ) + Ck(E 2 )h kLe1 (B˙ s+1 ) t
t
2,r
t
2,r
t
2,r
2,r
h
l
≤ CkckLe∞ (B˙ s+1 ) + Ckc kLe2 (B˙ s+1 )∩Le1 (B˙ s+2 ) + Ckc kLe1 (B˙ s+1 ) t
t
2,r
+CkEk
N ˙ p2 L∞ t (Bp2 ,1 )
t
2,r
t
2,r
2,r
(kEkLe∞ (B˙ s+1 ) + kE l kLe2 (B˙ s+1 )∩Le1 (B˙ s+2 ) + kE h kLe1 (B˙ s+1 ) ) t
t
2,r
2,r
t
t
2,r
2,r
≤ CkckLe∞ (B˙ s+1 ) + Ckcl kLe2 (B˙ s+1 )∩Le1 (B˙ s+2 ) + Ckch kLe1 (B˙ s+1 ) t
t
2,r
t
2,r
t
2,r
2,r
+Cλ1 (kEkLe∞ (B˙ s+1 ) + kE l kLe2 (B˙ s+1 )∩Le1 (B˙ s+2 ) + kE h kLe1 (B˙ s+1 ) ). t
t
2,r
t
2,r
t
2,r
2,r
When λ1 is small enough, Cλ1 < 12 , we have kEkLe∞ (B˙ s+1 ) + kE l kLe2 (B˙ s+1 )∩Le1 (B˙ s+2 ) + kE h kLe1 (B˙ s+1 ) t
2,r
t
2,r
t
2,r
t
2,r
≤ CkckLe∞ (B˙ s+1 ) + Ckcl kLe2 (B˙ s+1 )∩Le1 (B˙ s+2 ) + Ckch kLe1 (B˙ s+1 ) . t
t
2,r
2,r
t
2,r
t
(5.31)
2,r
From (5.29) and (5.31), we get Ys (t) ≤ CYs,0 + Cλ1 Ys (t). When λ1 is small enough, Cλ1
R0 , then for all p ∈ [1, ∞],
≤
e q + Rq3 ) ∆j Λ−1 ∇ · (L 1 e q + R + R2 ) ∆j (G q q
k∆j veq (t)kLp C2−2j e−ϑt k∆j ∆q ∇E0 kLp 2j +C e−ϑ2 t + 2−2j e−ϑt k∆j ∆q u0 kLp Z t e q kLp + k∆j ∇R3 kLp ) 2−2j e−ϑ(t−τ ) (k∆j ∇L +C q
+(e
0 −ϑ22j (t−τ )
≤
eq kLp + k∆j Rq1 kLp + k∆j Rq2 kLp ) dτ, + 2−2j e−ϑ(t−τ ) )(k∆j G
k∆j ∇e cq k L p Ce−ϑt (k∆j ∆q ∇E0 kLp + k∆j ∆q v0 kLp ) Z t e q kLp e q kLp + k∆j ∇R3 kLp + k∆j G +C e−ϑ(t−τ ) k∆j ∇L q 0
13
!
dτ
(5.38)
+k∆j Rq1 kLp + k∆j Rq2 kLp dτ.
(5.39)
We now have to fin suitable bounds for the right-hand side of (5.38)–(5.39). In what follows we shall freely use the following estimate (Proposition 8 in [6]): kg ◦ ψq kLp = kgkLp , for all function g ∈ Lp ,
(5.40)
e
k∇ψq±1 kL∞ ≤ eC U ≤ C, k∇ψq±1 − IdkL∞ ≤ e
e CU
(5.41)
− 1 ≤ eCλ1 − 1 ≤ Cλ1 ,
(5.42)
e
k∇k ψq±1 kL∞ ≤ C2(k−1)q (eC U − 1) ≤ C2(k−1)q λ1 , for k = 2, 3,
(5.43)
Rt
e (t) = k∇v(τ )kL∞ dτ ≤ CXp2 (t) ≤ Cλ1 . where U 0 e q , we can use the fact that, owing to the chain rule, As for ∆j ∇L e q = ∆j (∂l Lq ◦ ψq ∇ψq,l ). ∆j ∇L
Then
e q kLp ≤ Ck∇Lq ◦ ψq kLp k∇ψq kL∞ ≤ Ck∇Lq kLp , k∆j ∇L
combining the estimate (5.34), we get
e q kLp ≤ Ck∇∆q LkLp + Ck∇vkL∞ k∆j ∇L
X
Using the Vishik’s trick used in [34], we can get
e q kLp ≤ C2q−j k∇∆q LkLp + k∇vkL∞ k∆j ∇L
where we use the facts that
k∇∆q′ EkLp .
q′ ∼q
X
q′ ∼q
k∇∆q′ EkLp ,
(5.44)
e q kLp ≤ C2−j k∇2 (Lq ◦ ψq )kLp , k∆j ∇L
∂k ∂l (Lq ◦ ψq ) = (∂m ∂n Lq ◦ ψq )∂k ψq,m ∂l ψq,n + (∂m Lq ◦ ψq )∂k ∂l ψq,m , and k∇2 (Lq ◦ ψq )kLp ≤ C2q k∇Lq kLp . Similar, we get
eq kLp ≤ C2q−j k∆q GkLp + k∇vkL∞ k∆j G
X
q′ ∼q
k∆q′ vkLp .
(5.45)
Let us now turn to the study of ∇Rq3 . As a preliminary step, we have
k∆j ∇Rq3 kLp ≤ C2−j k∆j ∇2 Rq3 kLp ≤ C2−j k∇2 Rq3 kLp . Using the chain rule and H¨ older inequality, we get k∇2 Rq3 kLp ≤
and
Ck∇3 veq kk∇φq ◦ ψq − IdkL∞ + Ck∇2 veq kk∇2 φq ◦ ψq kL∞ k∇ψq kL∞ +Ck∇e vq k k∇3 φq ◦ ψq kL∞ k∇ψq k2L∞ + k∇2 φq ◦ ψq kL∞ k∇2 ψq kL∞ k∆j ∇Rq3 kLp ≤ C23q−j λ1 kvq kLp .
(5.46)
k∆j Rq1 kLp ≤ C2q−j λ1 k∇Eq kLp ,
(5.47)
Similar, we have
14
k∆j Rq2 kLp ≤ C23q−j λ1 kvq kLp .
(5.48)
From (5.38)–(5.39), (5.44)–(5.48), we have for 2j > R0 ,
.
2j 1 p + 2 2j k∆j e cq k L ∞ eq kL∞ k∆j veq kL1t (Lp ) p p + k∆j v t (L ) t (L )∩Lt (L )
k(∆j ∆q ∇E0 , ∆j ∆q v0 )kLp + 2q−j k(∆q ∇L, ∆q G)kL1t (Lp ) XZ t +2q−j k∇vkL∞ k(∇Eq′ , vq′ )kLp dτ + 2q−j λ1 k∇Eq kL1t (Lp ) + 22q kvq kL1t (Lp ) . q′ ∼q
0
Since vq = Sq−N0 veq ◦ φq + cq ◦ φq + cq = Sq−N0 e
X
j≥q−N0
X
j≥q−N0
from the following estimate ([7], Lemma 2.3),
∆j veq ◦ φq ,
(5.49)
∆j e cq ◦ φq ,
(5.50)
kSj (∆q f ◦ ψq±1 )kLp ≤ C(λ1 + 2j−q )k∆q f kLp ,
(5.51)
we have for all 2q > R0 2N0 , 2q p + 2 2q kcq kL∞ kvq kL1t (Lp ) (Lp )∩L1t (Lp ) + kvq kL∞ t (L ) t
C23N0 k(∆q ∇E0 , ∆q v0 )kLp + k(∆q ∇L, ∆q G)kL1t (Lp ) +
≤
XZ
0
q′ ∼q
t
k∇vkL∞ k(∇Eq′ , vq′ )kLp dτ
2q 1 (Lp ) kv k +C23N0 λ1 2q kEq kL∞ p )∩L1 (Lp ) + kvq kL∞ (Lp ) + 2 q (L L t t t t q −N0 2q 1 2 kcq kL∞ +C2 kvq kL1t (Lp ) . p p + kvq kL∞ (Lp ) + 2 t t (L )∩Lt (L )
Choosing N0 to be an integer such that C2−N0 ∈ ( 14 , 12 ), we have for all 2q > R0 2N0 ,
≤
2q 1 kvq kL1t (Lp ) 2q kcq kL∞ p p + kvq kL∞ (Lp ) + 2 t t (L )∩Lt (L )
C k(∆q ∇E0 , ∆q v0 )kLp + k(∆q ∇L, ∆q G)kL1t (Lp ) +
XZ
q′ ∼q
0
t
k∇vkL∞ k(∇Eq′ , vq′ )kLp dτ
2q 1 kvq kL1t (Lp ) . +Cλ1 2q kEq kL∞ p p + kvq kL∞ (Lp ) + 2 t t (L )∩Lt (L )
From Lemma 5.1, we have for R0 2N0 −4 ≤ 2q ≤ R0 2N0 ,
p ≤ CYs (t) ≤ CYs,0 , k(∇Eq , vq )kL∞ t (L )
and X
2q >R0 2N0
≤
N 2q 1 2q( p −1) 2q kcq kL∞ kvq kL1t (Lp ) p p + kvq kL∞ (Lp ) + 2 t t (L )∩Lt (L )
CYs,0 + C +C
Z
X
2q >R0 2N0 t
k∇vkL∞
0
+Cλ1
X
2q >R0 2N0
N 2 p −1 k(∆q ∇E0 , ∆q v0 )kLp + k(∆q ∇L, ∆q G)kL1t (Lp )
X
N
2 p −1 k(∇Eq , vq )kLp dτ
2q >R0 2N0
N 2q ∞ (Lp ) + 2 1 1 kv k 2 p −1 2q kEq kL∞ + kv k p p p q q L Lt (L ) . t t (L )∩Lt (L ) 15
(5.52)
From (5.10)4 and (5.30), we have kE h k ≤
kch k
≤
kch k kch k
≤
N
N
e 1 (B˙ p ) e ∞ (B˙ p )∩L L t t p,1 p,1 N
N
+ Ck(E 2 )h k
N
N
+ CkEk
e 1 (B˙ p ) e ∞ (B˙ p )∩L L t t p,1 p,1 e 1 (B˙ p ) e ∞ (B˙ p )∩L L t t p,1 p,1 N
N
e 1 (B˙ p ) e ∞ (B˙ p )∩L L t t p,1 p,1
N
N
e 1 (B˙ p ) e ∞ (B˙ p )∩L L t t p,1 p,1 N
kEk
N
e ∞ (B˙ p2 )∩L e 1 (B˙ p2 ) L t t p ,1 p ,1 2
2
+ Cλ1 (kE l kLe∞ (B˙ s+1 ) + kE h k t
2,r
N
˙ p L∞ t (Bp,1 ) N
˙ p L∞ t (Bp,1 )
).
When λ1 is small enough, Cλ1 < 21 , we have kE h k
N
≤ kch k
N
e ∞ (B˙ p )∩L e 1 (B˙ p ) L t t p,1 p,1
N
+ CYs,0 .
N
e ∞ (B˙ p )∩L e 1 (B˙ p ) L t t p,1 p,1
(5.53)
From (5.52)–(5.53) and X
N
1 2q p kcq kL∞ p p ≤ CYs (t) ≤ CYs,0 , t (L )∩Lt (L )
R0 Tn∗ , N N ×N N N N eT (B˙ 2 ) eT (B˙ 2 −1 ) ∩ L1 (B˙ 2 +1 ) and En ∈ C . vn ∈ C T 2,1 2,1 2,1
˜n ) ∈ Vps ,r (T ). This stands in contradiction with the definition of And one can easily obtain that (˜ vn , E 1 ∗ ∗ Tn . Thus, Tn = ∞ and (6.3) holds true globally. Third step: passing to the limit. Using the Aubin-Lions type argument [6], we easily have that, up s to extraction, (vn , En ) converges to (v, E), and (v, E) ∈ Vp,r (∞), p = p1 or p2 , (v, E) is the solution of (1.6). N
Then, we can easily obtain that (v, E) ∈ ETp1 . From the following local result in [30], we can obtain the uniqueness of the solution when p1 ≤ 2N . This finishes the proof of Theorem 1.2. Theorem 6.1 ([30]). Suppose that the initial data satisfies the incompressible constraints (1.3), v0 ∈ N N p −1 N p ) and E0 ∈ (B˙ p,1 )N ×N , p ∈ [1, 2N ]. Then there exist T > 0 and a unique local solution for system (B˙ p,1 N ×N N N s+1 s−1 s (1.6) that satisfies (v, E) ∈ ETp , where ETs = L1 ([0, T ]; B˙ p,1 . ) ) ∩ C([0, T ]; B˙ p,1 ) × C([0, T ]; B˙ p,1
Acknowledgements This work is supported partially by NSFC No.10871175, 10931007, 10901137, Zhejiang Provincial Natural Science Foundation of China Z6100217, and SRFDP No.20090101120005.
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