Acta Appl Math https://doi.org/10.1007/s10440-018-0178-x
Global Well-posedness for the Density-Dependent Incompressible Flow of Liquid Crystals Xiaoping Zhai1
· Zhi-Min Chen1
Received: 11 November 2017 / Accepted: 27 March 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018
Abstract In the present paper, we consider the global well-posedness of the densitydependent incompressible flow of liquid crystals in R2 . The local existence and uniqueness of the system are obtained without the assumption of small density variation. The global well-posedness is proved when the initial density and liquid crystal orientation are small. However, the initial velocity field is allowed to be arbitrarily large. Keywords Global well-posedness · Liquid crystal flow · Besov space Mathematics Subject Classification (2010) 35Q35 · 35Q30 · 76D03
1 Introduction and the Main Result In this paper, we investigate the motion of incompressible inhomogeneous nematic liquid crystal flows, which are governed by the following simplified version of the Ericksen-Leslie equations ⎧ div u = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ρt + div(ρu) = 0, (1.1) ⎪ ⎪ dt + u · ∇d = ν d + |∇d|2 d , |d| = 1, ⎪ ⎪ ⎪ ⎩ (ρu)t + div(ρu ⊗ u) − μu + ∇Π = − div(∇d ∇d), with the initial conditions ρ|t=0 = ρ0 ,
u|t=0 = u0 ,
d|t=0 = d0 ,
|d0 | = 1
in R2 ,
B Z.M. Chen
[email protected] X. Zhai
[email protected]
1
School of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, China
X. Zhai, Z.M. Chen
and with far field behaviors ρ → ρ, ¯
u → 0,
d → d¯0
as |x| → ∞,
where |d¯0 | = 1, ρ is the density and u is the velocity field, d stands for the unit vector field which represents the macroscopic orientations. Without loss of generality, we fix the viscosity coefficients to be μ = ν = 1. In the homogeneous case, i.e. ρ ≡ C, Eq. (1.1) reduces to be the classical nematic liquid crystal system which has been studied by many researchers (see [8, 10, 12–19, 22]). This study stems from Lin [15] in the 1990s for the introduction of a simplification of the Ericksen-Leslie model (see [6]) without changing the basic nonlinear structure. Since then there has been a remarkable progress in liquid crystal research in both theoretical and experimental aspects. Much of the work concerns the material response to flow, surface alignment, electric and magnetic fields, particularly with regard to liquid crystals of nematic type. The interested readers can refer to the survey article [11]. For the case of inhomogeneous fluid, when d is a given constant unit vector, Eq. (1.1) is reduced to the well-known inhomogeneous incompressible Navier-Stokes system, which has been examined by many researchers (see [1, 3, 4, 9, 21, 25, 27]). For the full system (1.1), Wen and Ding [23] obtained the local existence and uniqueness of strong solutions to the Dirichlet problem in bounded domains with initial density being allowed to have vacuum. They established the global existence and uniqueness of solutions for the two dimensional case if the initial density is away from vacuum and the initial data is of small norm. Fan et al. [7] got the global well-posedness in a bounded smooth domain without the assumption that ρ0 has a positive bound from below in R2 . Li and Wang [13] considered the initial-boundary value problem for the densitydependent incompressible flow of liquid crystals in a bounded domain Ω ⊂ R3 . For the initial density away from vacuum, they got the local strong solution with large initial data and the global strong solution with small data. Moreover, they studied the continuous dependence on data and the weak-strong uniqueness of weak solutions. Liu et al. [17] studied the system (1.1) in R2 with vacuum as far field density. They proved that the 2D nonhomogeneous incompressible nematic liquid crystal flows admit a unique global strong solution provided that the initial data density and the gradient of orientation decay not too slow at the infinity, and the initial orientation satisfies a geometric condition. Furthermore, they studied the large time behavior of the solution. Using the algebraical structure of the velocity equation in (1.1), i.e., the vertical velocity component equation being a linear equation with coefficients depending on the horizontal components, Zhai et al. [26] obtained global solutions R3 under the smallness assumptions on initial density, initial orientation field and the horizontal components of the initial velocity field. Let a = 1/ρ − 1, δ = d − d¯0 , we can get from (1.1) that: ⎧ div u = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂t a + u · ∇a = 0, ⎪ ⎪ ⎨ ∂t δ + u · ∇δ = δ + |∇δ|2 δ + |∇δ|2 d¯0 , ⎪ ⎪ ⎪ ⎪ ∂t u + u · ∇u + (1 + a)(∇Π − u) = −(1 + a) div(∇δ ∇δ), ⎪ ⎪ ⎪ ⎪ ⎩ a|t=0 = a0 , u|t=0 = u0 , δ|t=0 = δ0 = d0 − d¯0 , in R2 . The main result of the present paper reads as follows:
(1.2)
Global Well-posedness for the Density-Dependent Incompressible Flow. . . 2
2 −1
p p Theorem 1.1 Let 1 < p < 4, |d¯0 | = 1, and (a0 , u0 , d0 − d¯0 ) ∈ B˙ p,1 (R2 ) × B˙ p,1 (R2 ) × 2
p ¯ Then (1.2) has a (R2 ) with div u0 = 0 and 1 + infx∈R2 a0 ≥ b > 0 for a constant b. B˙ p,1 ¯ unique local solution (a, u, d − d0 , ∇Π ) on [0, T ] such that 2 p2 2 p ˙ ∞ R2 ∩ L , a ∈ C [0, T ]; B˙ p,1 T Bp,1 R
p2 −1 2 ∇Π ∈ L1T B˙ p,1 R ,
2 2 p2 −1 2 p2 +1 2 p −1 ˙ ∞ R ∩L ∩ L1T B˙ p,1 R , u ∈ C [0, T ]; B˙ p,1 T Bp,1 R
(1.3)
2 ˙ ∞ ∩ L1T B˙ p,1 R2 . d − d¯0 ∈ C [0, T ]; B˙ p,1 R2 ∩ L T Bp,1 R 2 p
2 p
2 p +2
Moreover, there exists a positive constants ε0 such that if the initial vector filed (a0 , u0 , d0 − d¯0 ) satisfies
a0
2 p B˙ p,1
+ d0 − d¯0
2 p B˙ p,1
exp 1 + u0
2 2 −1+ p B˙ p,1
exp u0 2 −1+ 2 ≤ ε0 , p B˙ p,1
(1.4)
then the unique local solution can be extended to be global. Remark 1.2 As |d¯0 | = |d0 | = 1 and |d| = |d¯0 + δ| = 1, there exist some initial data restrictions to be chosen. In fact, by a simple computation, we have 1 = |d¯0 + δ|2 = |d¯0 |2 + 2d¯0 , δ + δ 2
(1.5)
which gives d¯0 , δ = −
δ2 2
and thus cosd¯0 , δ =
δ d¯0 , δ = − ≤ 0. 2 |d¯0 ||δ|
(1.6)
This restriction comes from the fact that if we search a solution as a perturbation of d¯0 and in the same time we demand that it verifies the constraint |d¯0 + δ| = 1, these two facts then translate into relation (1.6), which says that the perturbation that we have used are at the right of the initial vector d¯0 . This restriction is avoided by searching a solution as a bounded function with gradient in a Besov space (see [5]): d ∈ L∞ R 2 ,
2 −1
p ∇d ∈ B˙ p,1
R2 .
The restriction (1.6) implies that the initial orientations for the crystals must be at the right of the vector d¯0 . Remark 1.3 It should be mentioned that, regarding the local existence part of the above theorem, the initial density is not necessary close to a positive constant while the initial velocity field can be large. For this reason, our local and global well-posedness results considerably improve the recent result of [24] in dimension two.
X. Zhai, Z.M. Chen
Remark 1.4 As 1 < p < 4, we can choose p > 2 such that −1 + p2 < 0 which implies that one may take the initial velocity in a Besov space with a negative index of regularity, so that a highly oscillating “large” velocity may give rise to a unique global solution. Remark 1.5 Using 2ε12 (1 − |d|2 )d instead of |∇d|2 d in the system (1.1) in the study of the limit ε → 0 as in [15, 16] is also an interesting problem. We will discuss this problem in the future. Remark 1.6 We believe that the same result holds for the density-dependent incompressible flow of liquid crystals with viscosities that depend on the density as in [25]. As a by-product of Theorem 1.1, we can obtain blow-up criterion. Corollary 1.7 Let ω = ∇ × u and T ∗ be the maximal local existence time of (a, u, d − d¯0 ) in Theorem 1.1. If T ∗ < ∞, then
T∗
ω
0
2
p B˙ p,1
+ d − d¯0 2 1+ 2 dt = ∞. p B˙ p,1
(1.7)
The remainder of this paper is organized as follows. Section 2 contains some useful properties of functional spaces, basic analysis tools, technical lemmas and the estimates of the transport equation and the heat equation. In Sect. 3, we prove the local well-posedness part of Theorem 1.1. The global existence part of Theorem 1.1 is derived in Sect. 4. Corollary 1.7 is proved in Sect. 5.
2 Preliminaries Let χ and φ be two smooth radial functions 0 ≤ χ , φ ≤ 1, such that χ is supported in the ball B = {ξ ∈ Rn , |ξ | ≤ 43 } and ϕ is supported in the ring C = {ξ ∈ Rn , 34 ≤ |ξ | ≤ 83 }. Moreover, there holds ϕ 2−j ξ = 1, ∀ξ = 0. j ∈Z
We define the dyadic blocks as follows: ˙ j f = ϕ 2−j D f = 2j n
S˙j f = χ 2−j D f = 2j n
R2
R2
(F −1 ϕ) 2j y f (x − y)dy, (F −1 χ ) 2j y f (x − y)dy.
Denote by Sh (Rn ) the space of tempered distributions u such that lim S˙j u = 0
j →−∞
Then we have the formal decomposition ˙ j u, u= j ∈Z
in S .
∀u ∈ Sh Rn .
Global Well-posedness for the Density-Dependent Incompressible Flow. . .
Moreover, the Littlewood-Paley decomposition satisfies the property of almost orthogonality: ˙ j u ≡ 0 if |k − j | ≥ 2 ˙ k
˙ k (S˙j −1 u ˙ j u) ≡ 0
and
if |k − j | ≥ 5.
Now we recall the definition of homogeneous Besov spaces. Definition 2.1 Let (p, r) ∈ [1, +∞]2 , s ∈ R and u ∈ Sh (Rn ). Set ˙ j u Lp r . s u B˙ p,r = 2j s s s (Rn ) = {u ∈ Sh (Rn )| u B˙ p,r < ∞}. Then we define B˙ p,r s Remark 2.2 Let 1 ≤ p, r ≤ ∞, s ∈ R, and u ∈ Sh (Rn ). Then u belongs to B˙ p,r (Rn ) if and only if there exists {dj,r }j ∈Z such that dj,r ||r < ∞ and
˙ j u Lp ≤ dj,r 2−j s u B˙ s p,r
for all j ∈ Z.
We now define the Chemin-Lerner space (see [2, 21]), a refinement of the space s (Rn )). LλT (B˙ p,r s λT (B˙ p,r Definition 2.3 Let s ≤ pn , (r, λ, p) ∈ [1, +∞]3 and T ∈ (0, +∞]. We define L (Rn )) n as the completion of C([0, T ]; S (R )) by the norm
f Lλ (B˙ p,r s ) =
T
T
˙ q f (t)λ p dt
2rqs
L
0
q∈Z
λr 1r
< ∞,
s λT (B˙ p,r with the usual change if r = ∞. For short, we just denote this space by L ).
Remark 2.4 It is easy to observe that for 0 < s1 < s2 , θ ∈ [0, 1], p, r, λ, λ1 , λ2 ∈ [1, +∞], we have the following interpolation inequality in the Chemin-Lerner space (see [2]): θ u Lλ (B˙ p,r s ) ≤ u λ 1
with
1 λ
=
θ λ1
+
1−θ λ2
s
1 ) LT (B˙ p,r
T
u (1−θ) λ2 ˙ s2
LT (Bp,r )
and s = θ s1 + (1 − θ )s2 .
Let us emphasize that, according to the Minkowski inequality, we have f Lλ (B˙ p,r s ) ≤ f Lλ (B˙ s ) p,r T
T
if λ ≤ r,
f Lλ (B˙ p,r s ) ≥ f Lλ (B˙ s ) , p,r T
T
if λ ≥ r.
In order to prove main Theorem 1.1, we introduce the following weighted Chemin-Lerner type norm from [2, 21]: Definition 2.5 Let f ∈ L1loc (0, ∞), f (t) ≥ 0. We define u Lq
˙s T ,f (Bp,r )
=
j ∈Z
T
2rj s 0
˙ j u(t)q p dt f (t) L
qr 1r
for s ∈ R, p ∈ [1, ∞], q, r ∈ [1, ∞), and with the standard modification for q = ∞ or r = ∞.
X. Zhai, Z.M. Chen
The following Bernstein’s lemma will be repeatedly used throughout this paper. Lemma 2.6 Let B be a ball and C a ring of Rn . For any positive real number λ, nonnegative integer k, smooth homogeneous function σ of degree m, and real numbers a and b with 1 ≤ a ≤ b, there exists a cosntant C such that Supp uˆ ⊂ λB ⇒ sup ∂ α uLb ≤ C k+1 λk+n(1/a−1/b) u La , |α|=k
Supp uˆ ⊂ λC ⇒ C −k−1 λk u La ≤ sup ∂ α uLa ≤ C k+1 λk u La , |α|=k
Supp uˆ ⊂ λC ⇒ σ (D)uLb ≤ Cσ,m λm+n(1/a−1/b) u La . As an application of the above basic facts on Littlewood-Paley theory, we present the following product laws in Besov spaces. Lemma 2.7 (See [25]) Let 1 ≤ p, q ≤ ∞, s1 ≤ qn , s2 ≤ n min{ p1 , q1 } and s1 + s2 > s1 s2 n max{0, p1 + q1 − 1}. For ∀(a, b) ∈ B˙ q,1 (Rn ) × B˙ p,1 (Rn ), we have ab
n
s1 +s2 − q B˙ p,1
a B˙ s1 b B˙ s2 . q,1
(2.1)
p,1
s Lemma 2.8 (See [2]) Let 1 ≤ p, q ≤ ∞, s ≤ 1 + n min{ p1 , q1 }, v ∈ B˙ q,1 (Rn ) and u ∈ n +1
p (Rn ). Assume that B˙ p,1
s > −n min
1 1 ,1 − , p q
Then there holds
or
s > −1 − n min
[u · ∇, ˙ j ]v q dj 2−j s u L
1 1 ,1 − p q
n +1
p B˙ p,1
if div u = 0.
v B˙ s . q,1
In order to estimate the transport equation, we need the following proposition. Proposition 2.9 (See [27]) Let (p, q) ∈ [1, ∞]2 , q1 − n
n
n
1 p
q ≤ n1 , m ∈ Z, a0 ∈ B˙ q,1 (Rn ), ∇u ∈
p q L1T (B˙ p,1 ) with div u = 0, and a ∈ C([0, T ]; B˙ q,1 (Rn )) such that (a, u) solves
∂t a + u · ∇a = 0,
(2.2)
a(x, 0) = a0 . Then there hold for ∀t ≤ T a
n
˙q ∞ L t (Bq,1 )
a − S˙m a with V (t) ∇u
n p
≤ a0
n ˙q ∞ L t (Bq,1 )
L1t (B˙ p,1 )
.
n
q B˙ q,1
≤
eCV (t) ,
j ≥m
˙ j a0 Lq + a0 2 q j
(2.3)
n
n q B˙ q,1
eCV (t) − 1 ,
(2.4)
Global Well-posedness for the Density-Dependent Incompressible Flow. . .
In the following, we recall the solvability of the following Cauchy problem of the heat equation in the Chemin-Lerner space: ∂t u − σ u = f (x, t),
u(x, 0) = u0 .
(2.5)
Proposition 2.10 (See [2]) Let s ∈ R, 1 ≤ p, r, ρ ≤ ∞ and 0 < T ≤ ∞. Assume that u0 ∈ s−2+ 2 s ρT (B˙ p,r ρ (Rn )). Then the heat equation (2.5) has a unique solution B˙ p,r (Rn ) and f ∈ L s+ 2
n ˙s ∞ ρ ˙ ρ n u∈L T (Bp,r (R )) ∩ LT (Bp,r (R )). In addition, there exists a constant C > 0 such that for all ρ1 ∈ [ρ, ∞], we have 1
σ ρ1 u
s+ 2 ρ1 (B˙ p,r ρ1 ) L T
and u L1 (B˙ s+2 ) ≤ T
1 −1 s +σ ρ ≤ C u0 B˙ p,r f
s−2+ 2 ρ (B˙ p,r ρ ) L T
,
2q ˙ q f L1 (Lp ) . ˙ q u0 Lp + 2qs 1 − e−Cσ 2 T T
p,1
q∈Z s (Rn )). If furthermore r is finite then u ∈ C([0, T ], B˙ p,r
3 Local Well-posedness Part of Theorem 1.1 Before presenting the main details of the local well-posedness part of Theorem 1.1, we first give the following proposition which deals with the linear estimates of the momentum equation. The proof of this proposition can be similarly obtained from Proposition 3.2 in [25]. Here we omit the proof. 2 −1
2
p 2 ˙p Proposition 3.1 Let 1 < p < 4, u0 ∈ B˙ p,1 (R2 ) and a ∈ L∞ T (Bp,1 (R )) with 1 + a ≥ b > 0. 2 −1
2
2 −1
p p p (R2 )), g ∈ L1T (B˙ p,1 (R2 )) and ∂t g = div R with R ∈ L1T (B˙ p,1 (R2 )). Let Let f ∈ L1T (B˙ p,1 2 −1
2 +1
2 −1
p p p (R2 )) ∩ L1T (B˙ p,1 (R2 )) × L1T (B˙ p,1 (R2 )) solve (u, ∇Π ) ∈ C([0, T ]; B˙ p,1 ⎧ ∂t u − (1 + a)(u − ∇Π ) = f, ⎪ ⎪ ⎨ div u = g, ⎪ ⎪ ⎩ u|t=0 = u0 .
(3.1)
Then there holds for t ∈ [0, T ] u
2 −1 ˙p ∞ L t (Bp,1 )
+ u
2 p +1 L1t (B˙ p,1 )
u0
2 p −1 B˙ p,1
+ 1 + a
2 ˙p L∞ t (Bp,1 )
∞ g + 1 + a L∞ t (L )
3 f
2 p L1t (B˙ p,1 )
2 p −1
L1t (B˙ p,1 )
+ 2m a
+ R
2 ˙p L∞ t (Bp,1 )
u
2 p −1
L1t (B˙ p,1 )
2 p L1t (B˙ p,1 )
,
(3.2) provided that a − S˙m a
2
˙p L∞ T (Bp,1 )
≤ c0
for some sufficiently small positive constant c0 and some integer m ∈ Z.
(3.3)
X. Zhai, Z.M. Chen
3.1 Existence Part of Theorem 1.1 The proof of the existence is performed in a standard manner. We begin by solving an approximate problem, then prove the solutions being uniformly bounded, and finally study the convergence to a solution of the initial equation by a compactness argument. Step 1. Construction of a regular approximate solution. Similar to [25, 27], there exists a sequence a0n , un0 , δ0n ∈ H ∞ (R2 ) with div un0 = 0 that approximate the initial data in the corresponding spaces. Theorem 1.1 of [20] (up to make some modification) ensures that system (1.2) with the initial vector field (a0n , un0 , δ0n ) admits a unique local in time solution (a n , un , δ n , ∇Π n ) verifying ∇Π n ∈ L1 0, T n ; H s R2 a n ∈ C 0, T n ; H s R2 , un ∈ C 0, T n ; H s−1 R2 ∩ L1loc 0, T n ; H s+1 R2 , δ n ∈ C 0, T n ; H s R2 ∩ L1loc 0, T n ; H s+2 R2 , with s > 1. Step 2. Uniform estimates to the approximate solutions. We will prove that there exists a positive time 0 < T < inf T n , such that (a n , un , δ n , ∇Π n ) is uniformly bounded in the space 2 p2 −1+ p2 1+ 2 p2 −1+ 2 1 ˙ 2+ p ˙ ˙ ∞ ∞ ˙ ∞ ∩ L1T B˙ p,1 p × L × L1T B˙ p,1 p . ET = L T Bp,1 × LT Bp,1 T Bp,1 ∩ LT Bp,1
Let et be the heat flow, un = unF + u¯ n , δ n = δFn + δ¯n with unF = et un0 , δFn = et δ0n , then (a n , u¯ n , δ¯n , ∇Π n ) solves ⎧ n n ∂t a + uF + u¯ n · ∇a n = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂t u¯ n + 1 + a n ∇Π n − u¯ n = Vn , ⎪ ⎪ ⎨ (3.4) ∂t δ¯n − δ¯n + unF · ∇ δ¯n = Mn , ⎪ ⎪ ⎪ div u¯ n = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ a n , u¯ n , δ¯n = (a , 0, 0), 0 t=0 with Vn = −unF · ∇ u¯ n − unF · ∇unF − u¯ n · ∇unF − u¯ n · ∇ u¯ n + a n unF + 1 + a n ∇δFn · δFn + ∇δFn · δ¯n + ∇ δ¯n · δ¯n + ∇ δ¯n · δFn , 2 Mn = −unF · ∇δFn − u¯ n · ∇δFn − u¯ n · ∇ δ¯ n + |∇δFn |2 δFn + d¯0 + ∇δFn δ¯n 2 2 + ∇ δ¯n δFn + d¯0 + ∇ δ¯ n δ¯n + 2∇δFn · ∇ δ¯ n δFn + d¯0 + 2∇δFn · ∇ δ¯n δ¯n . From Proposition 2.10, we can deduce that n u 2 u0 −1+ 2 , −1+ p F p ˙ ∞ L t (Bp,1
n δ F
2 2+ p L1t (B˙ p,1 )
n u F
2 1+ p L1t (B˙ p,1 )
B˙ p,1
)
n δ F
2
˙p ∞ L t (Bp,1 )
δ0
2
p B˙ p,1
,
(3.5)
2j 2j ˙ j δ0 Lp , 2 p 1 − e−ct2
(3.6)
2 2j ˙ j u0 Lp . 2(−1+ p )j 1 − e−ct2
(3.7)
j ∈Z
j ∈Z
Global Well-posedness for the Density-Dependent Incompressible Flow. . .
Denote
Z n (t) u¯ n
2
−1+ p ˙ ∞ ) L t (Bp,1
+ δ¯n
2
˙p ∞ L t (Bp,1 )
+ ∇Π n
2
−1+ p L1t (B˙ p,1 )
+ u¯ n
2 1+ p L1t (B˙ p,1 )
+ δ¯n
2+ 2
p L1t (B˙ p,1 )
.
Firstly, we show the estimate of δ¯n . ˙ q to the third equation of (3.4), and then using a standard commutator process Applying give ˙ q δ¯n + u¯ n · ∇ ˙ q δ¯n − ˙ q δ¯n ˙ q δ¯n + unF · ∇ ∂t ˙ q δ¯n + u¯ n · ∇, ˙ q δ¯n + ˙ q ∇δFn 2 δFn + d¯0 = unF · ∇, ˙ q ∇δFn 2 δ¯n + ˙ q ∇ δ¯ n 2 δFn + d¯0 + ˙ q ∇ δ¯n 2 δ¯n + ˙ q 2∇δFn · ∇ δ¯n δFn + d¯0 + ˙ q 2∇δFn · ∇ δ¯n δ¯n + ˙ q unF · ∇δFn + ˙ q u¯ n · ∇δFn . + (3.8) ˙ q δ¯n |p−2 ˙ q δ¯n and inteTaking the L2 inner product of the above resulting equation with | grating with respect to t , we can get n n δ¯ δ¯ + 2 2 2+ p p ˙ ∞ L t (Bp,1 )
L1t (B˙ p,1 )
˙ q δ¯n 1 p 22q/p unF · ∇, L (L ) t
q∈Z
+
˙ q δ¯n 1 p + unF · ∇δFn 22q/p u¯ n · ∇, L (L )
q∈Z
2 + ∇δFn δFn + d¯0
2 p L1t (B˙ p,1 )
2 + ∇ δ¯n δ¯n
2 p L1t (B˙ p,1 )
2 p
L1t (B˙ p,1 )
t
2 + ∇δFn δ¯n
2 p L1t (B˙ p,1 )
+ ∇δFn · ∇ δ¯n δFn + d¯0
+ u¯ n · ∇δFn
2 p
L1t (B˙ p,1 )
2 + ∇ δ¯n δFn + d¯0
2 p L1t (B˙ p,1 )
+ ∇δFn · ∇ δ¯ n δ¯n
2
p L1t (B˙ p,1 ) 2
p L1t (B˙ p,1 )
.
(3.9) Applying Lemmas 2.6, 2.7 and 2.8 yields that t ˙ q δ¯n 1 p u¯ n 2 δ¯n 22q/p u¯ n ·∇, 1+
Lt (L )
0
q∈Z
n u · ∇δ n F F
2 p L1t (B˙ p,1 )
p B˙ p,1
t
un F
2 p B˙ p,1
0
0
2
p B˙ p,1
n δ F
1+ 2
2 dτ Z n (t) ,
p B˙ p,1
(3.10)
dτ
t
un 1/2 2 un 1/2 2 δ n 1/22 δ n 1/2 2 dτ F F F F −1+ 1+ 2+ p B˙ p,1
t
un F
0
u0
2 −1+ p B˙ p,1
2 −1+ p B˙ p,1
p B˙ p,1
n u F
+ δ0
2 1+ p B˙ p,1
2 p B˙ p,1
p B˙ p,1
t
δ n F
dτ +
n u F
p B˙ p,1
0
2 1+ p L1t (B˙ p,1 )
2
p B˙ p,1
n δ F
+ δFn
2+ 2
p B˙ p,1
dτ
2 2+ p L1t (B˙ p,1 )
,
(3.11)
X. Zhai, Z.M. Chen
˙ q δ¯n 1 p 22q/p unF ·∇, L (L ) t
q∈Z
t
un F
2 1+ p B˙ p,1
0
n δ¯
2 p B˙ p,1
t
un F
dτ
1+ 2
p B˙ p,1
0
Z n (τ )dτ, (3.12)
n u¯ · ∇δ n F
2 p L1t (B˙ p,1 )
t
u¯ n
0
n δ F
2 p B˙ p,1
u¯ n 1/2 2 u¯ n 1/2 2 δ n 1/22 δ n 1/2 2 dτ F F −1+ 1+ 2+ p B˙ p,1
2 Z n (t) + δ0
2 p L1t (B˙ p,1 )
t
δ n F
n δ¯
p B˙ p,1
t
δ n F
2 p B˙ p,1
δ n F
1+ 2
n δ F
2 1+ p B˙ p,1
2 2+ p B˙ p,1
2 p B˙ p,1
n 2 n ∇δ δ¯ F
2
p L1t (B˙ p,1 )
dτ
p B˙ p,1
p B˙ p,1
t
δ¯n 3
dτ +
2 p B˙ p,1
0
n δ F
2 2+ p
L1t (B˙ p,1 )
n δ¯
2 1+ p B˙ p,1
p B˙ p,1
n δ F
2 p B˙ p,1
t
δ¯n
2 p B˙ p,1
p B˙ p,1
n δ¯
2 2+ p B˙ p,1
2 Z n (t) + δ0 3
2 p L1t (B˙ p,1 )
2
p B˙ p,1
2 p B˙ p,1
t
δ¯n
0
2 p B˙ p,1
p B˙ p,1
t
δ¯n
0
(3.14)
, + 1 dτ
+1 dτ
B˙ p,1
+ 1 δFn
2 2+ p
L1t (B˙ p,1 )
t
δ¯n 2
2 p B˙ p,1
0
2+ 2
p B˙ p,1
dτ
(3.15)
,
n δ¯
2+ 2
p B˙ p,1
dτ (3.16)
B˙ p,1
n δ F
2
2
n δ F
2+ 2
p B˙ p,1
2
p B˙ p,1
δ n 3 2 +1 δ n F F p
0
B˙ p,1
2
p B˙ p,1
t
δ0 0
dτ
2+ 2
p B˙ p,1
t
dτ +
n 2 δ¯ 1+ 2 dτ p
B˙ p,1
0
n δ¯
p B˙ p,1
3 Z n (t) , t 2 δ¯n p2 δFn 1+ p2 dτ
(3.13)
,
δ¯n 1/22 δ¯n 1/2 2 δ n 1/22 δ n 1/2 2 δ n F F F 2+ 2+
0
n 2 n ∇ δ¯ δ¯
n δ¯
p B˙ p,1
t
0
2 2+ p
L1t (B˙ p,1 )
p B˙ p,1
t
0
n δ F
p B˙ p,1
4 Z n (t) + δ0
2 p L1t (B˙ p,1 )
2 p B˙ p,1
p B˙ p,1
δ n 1/22 δ n 1/2 2 δ¯n 3/22 δ¯n 1/2 2 dτ F F 2+ 2+
0
n ∇δ · ∇ δ¯ n δ n + d¯0 F F
p B˙ p,1
t
0
1+ 2
p B˙ p,1
0
dτ
t
0
n ∇δ · ∇ δ¯n δ¯n F
1+ 2
p B˙ p,1
p B˙ p,1
n δ F
p B˙ p,1
2+ 2
p B˙ p,1
dτ
Z n (τ )dτ,
(3.17)
Global Well-posedness for the Density-Dependent Incompressible Flow. . .
n 2 n ∇δ δ + d¯0 F F
2 p L1t (B˙ p,1 )
δ n F
2 p B˙ p,1
0
δ n 2 2 + 1 δ n F F p
0
B˙ p,1
δ0 2
δ n F
2
δ n F
2
+ 1 δ¯n
p B˙ p,1
0
d0
2
p B˙ p,1
(3.18)
,
B˙ p,1
t
dτ
2 + 1 δ¯n 1+ p2 dτ
p B˙ p,1
0
2+ 2
p L1t (B˙ p,1 )
t
2+ 2
p B˙ p,1
+ 1 δFn
2 p B˙ p,1
2 p L1t (B˙ p,1 )
B˙ p,1
t
n 2 n ∇ δ¯ δ + d¯0 F
2 + 1 δFn 1+ p2 dτ
t
2
p B˙ p,1
n δ¯
2+ 2
p B˙ p,1
2 + 1 Z n (t) .
dτ (3.19)
Inserting the estimates (3.10)–(3.19) into (3.9), we have n δ¯ + δ¯n 2 2 2+ p p ˙ ∞ L t (Bp,1 )
1 + δ0 +
L1t (B˙ p,1 )
2
p B˙ p,1
un F
2 4 Z n (t) + C Z n (t) + 1 + δ0
t
0
+ u0
2 1+ p B˙ p,1
+ 1 + δ0
2 −1+ p B˙ p,1
2 p B˙ p,1
2 p B˙ p,1
n δ F
2 2+ p B˙ p,1
Z n (τ )dτ
3 n u
2 1+ p L1t (B˙ p,1 )
F
+ δFn
2 2+ p L1t (B˙ p,1 )
(3.20)
.
Secondly, we show the estimates of u¯ n and ∇Π n from the momentum equation in (3.4). In fact, by Proposition 3.1, one has n u¯ + u¯ n + ∇Π n 2 2 2 p −1 p +1 p −1 ˙ ∞ L t (Bp,1 )
1 + a n
L1t (B˙ p,1 )
2 ˙p L∞ t (Bp,1 )
3 Vn
provided that
L1t (B˙ p,1 )
2 p −1 L1t (B˙ p,1 )
+ 2m a n
n a − S˙m a n
2 p
˙ L∞ T (Bp,1 )
)
B˙ p,1
0
+ u¯ n × 0
2 −1+ p B˙ p,1
un F
n u F
t
+ ∇ δ¯ n
2 1+ p B˙ p,1
2 p B˙ p,1
B˙ p,1
2 1+ p B˙ p,1
2
−1+ p B˙ p,1
n u¯
dτ + 1 + a n
2 −1+ p B˙ p,1
(3.21)
2 p L1t (B˙ p,1 )
(3.22)
1+ 2
p B˙ p,1
2 ˙p ∞ L t (Bp,1 )
+ ∇δFn
n δ¯
n u¯
≤ c0 .
By the product laws in Besov spaces, we have t n n n u Vn ¯ 2 2 uF 1+ 2 + u −1+ p −1+ p F p L1t (B˙ p,1
2 ˙p L∞ t (Bp,1 )
2 p B˙ p,1
n δ F
+ ∇ δ¯ n
2 −1+ p B˙ p,1
2 p B˙ p,1
+ ∇δFn
n δ F
2 −1+ p B˙ p,1
2 p B˙ p,1
dτ
n δ¯
2
−1+ p B˙ p,1
X. Zhai, Z.M. Chen
t
un F
0
+ u¯ n ×
2 −1+ p B˙ p,1
n u
n u
2
−1+ p B˙ p,1
F
un F
t
2 1+ p B˙ p,1
0
1 + a n
2 p B˙ p,1
u0
un F
1+ 2
p B˙ p,1
4 2
˙p ∞ L t (Bp,1 )
+ 1 + a n
3
+ 1 + a n
u0
2
˙p ∞ L t (Bp,1 )
+ δ0
3 2
2
p B˙ p,1
n δ¯
n u F
2 2+ p B˙ p,1
2 1+ p L1t (B˙ p,1 )
dτ
+ δFn
2 Z (t) +
2 2+ p L1t (B˙ p,1 )
(3.23)
2 p −1
+ δ0
2
−1+ p B˙ p,1
2 p B˙ p,1
L1t (B˙ p,1 )
n u
2
F
p B˙ p,1
un F
2 1+ p L1t (B˙ p,1 )
+ δFn
t
n
1+ 2
p B˙ p,1
0
˙p ∞ L t (Bp,1 )
2
Z n (τ )dτ.
L1t (B˙ p,1 )
1 + a n
1+ 2
p B˙ p,1
+ δ¯n
2 2+ p B˙ p,1
F
2
Therefore, we can get the desired estimate n n u¯ ¯ + ∇Π n 2 −1 + u 2 p p +1 ˙ ∞ L t (Bp,1 )
n δ
−1+ p B˙ p,1
t 0
n u¯
˙p ∞ L t (Bp,1 )
+ δFn
2
2 −1+ p B˙ p,1
dτ + 1 + a n
1+ 2
p B˙ p,1
˙p ∞ L t (Bp,1 )
2 + Z n (t) +
+ u¯ n
2 1+ p B˙ p,1
F
2 2+ p L1t (B˙ p,1 )
n
Z (τ )dτ
2m tZ n (t).
(3.24)
Here Z n is bounded as, by combining (3.20) with (3.24), Z n (t)
un F
t
0
2 1+ p B˙ p,1
+ 1 + δ0
+ 1 + a n
3
+ 1 + a n
4
2 ˙p ∞ L t (Bp,1 )
× unF
2 ˙p ∞ L t (Bp,1 )
2 1+ p L1t (B˙ p,1 )
2 p B˙ p,1
n δ F
1 + δ0 u0
+ δFn
2 2+ p B˙ p,1
2 p B˙ p,1
2 −1+ p B˙ p,1
2 2+ p L1t (B˙ p,1 )
Z n (τ )dτ
3 Z n (t) + Z n (t) + 2m t Z n (t)
+ 1 + δ0
3 2 p B˙ p,1
(3.25)
.
By Gronwall’s inequality, we have Z n (t) exp u0 ×
2 −1+ p B˙ p,1
+ 1 + δ0
1 + a n
+ 1 + a n × unF
3
2 ˙p ∞ L t (Bp,1 )
4 2 ˙p ∞ L t (Bp,1 )
2 1+ p L1t (B˙ p,1 )
2 2 p B˙ p,1
1 + δ0
u0
+ δFn
2 p B˙ p,1
2 −1+ p B˙ p,1
2 2+ p L1t (B˙ p,1 )
3 Z n (t) + Z n (t) + 2m t Z n (t)
+ 1 + δ0 3
.
2 p B˙ p,1
(3.26)
Global Well-posedness for the Density-Dependent Incompressible Flow. . .
Thirdly, we consider the uniform eatimate of a n . Applying Proposition 2.9 to the transport equation of (3.4), we have for t ∈ [0, T n ) n n a a 2 exp C un + u¯ n ≤ 2 2 0 F p p p +1 ˙ ∞ L t (Bp,1 )
L1t (B˙ p,1 )
B˙ p,1
≤ C a0
2 p B˙ p,1
exp C Z n (t) + u0
2 p −1 B˙ p,1
(3.27)
.
Fourthly, we consider a lower bound of T n . For any function χ ∈ D(R) with χ (0) = 0, the composite function χ (a n ) with initial function χ (a0n ) solves the renormalized transport equation ∂t χ (a n ) + (unF + u¯ n ) · ∇χ (a n ) = 0. Hence a similar argument as (2.4) leads to n χ a − S˙m χ a n 2 ˙p ∞ L t (Bp,1 )
≤
2j ˙ j χ a0n p + χ a0n 2 p L
j ≥m
≤
2 p B˙ p,1
2j ˙ j χ (a0 ) p + C 1 + a0 2 p L
j ≥m
+ C a0
2 p B˙ p,1
exp CZ n (t) + C unF
2 p +1 L1t (B˙ p,1 )
2 p B˙ p,1
n a − a0 0
exp CZ n (t) + C unF
−1
2
p B˙ p,1
2 p +1 L1t (B˙ p,1 )
−1 ,
where we have used n χ a − χ (a0 ) 0
2
p B˙ p,1
n 1 n = a0 − a0 χ τ a0 + (1 − τ )a0 dτ ˙ p2 0
≤ C 1 + a0
2
p B˙ p,1
Bp,1
n a − a0 0
2
p B˙ p,1
.
As a consequence, we obtain for t ∈ [0, T n ) (Id − S˙m )a n
2 ˙p ∞ L t (Bp,1 )
≤
2j ˙ j a0n p + a0n 2 p L
2 p B˙ p,1
j ≥m
≤
exp C unF + u¯ n
2j ˙ j a0 Lp + C 1 + a0 2 p
j ≥m
+ C a0
2 p B˙ p,1
2 p +1 L1t (B˙ p,1 )
2 p B˙ p,1
n a − a0 0
−1
2
p B˙ p,1
n exp CZ (t) + C unF
2 p +1 L1t (B˙ p,1 )
−1 .
(3.28)
Now, for any n ∈ N, we define
T∗n = sup t ∈ 0, T n : Z n (t) ≤ 2ε0 , with ε0 ∈ (0, 12 ) to be determined. We shall prove infn∈N T∗n > 0. We deduce from (3.27) for t ≤ T∗n that n a ≤ C a0 p2 exp C 1 + u0 p2 −1 . 2 p ˙ ∞ L t (Bp,1 )
B˙ p,1
B˙ p,1
(3.29)
(3.30)
X. Zhai, Z.M. Chen 2
p Notice that a0 ∈ B˙ p,1 (R2 ), there exist m = m(c0 ) ∈ Z and n0 = n0 (c0 ) ∈ N such that
C 1 + a0 ×
exp 1 + u0
2 p B˙ p,1
2 2 p −1 B˙ p,1
2j ˙ j a0 Lp + sup 1 + a0 2 p
n≥n0
j ≥m
2 p B˙ p,1
n a − a0 0
2 p B˙ p,1
1 ≤ c0 . 2
(3.31)
Thanks to (3.28), taking ε0 and T0 small enough and n1 ≥ n0 large enough ensure C 1 + a0
2
p B˙ p,1
exp 1 + u0
2 2 −1
p B˙ p,1
a0
2
exp 2Cε0 + C unF
2 p +1 L1T (B˙ p,1 ) 0
p B˙ p,1
1 − 1 ≤ c0 2 (3.32)
for any n ≥ n1 . Combining (3.30)–(3.32) implies that (3.22) with T = min(T∗n , T0 ) is fulfilled for any n ≥ n1 . Without loss of generality, we may assume T∗n ≤ T0 . Then for any t ≤ T∗n , we deduce from (3.26) that (3.33) Z n (t) A1 (a0 , u0 , δ0 ) 2ε0 + 8ε03 + 2m t Z n (t) + A2 (a0 , u0 , δ0 )θ (t) with Ai (a0 , u0 , δ0 ) (i = 1, 2), θ (t) being determined by 2
def 1 + a0 A1 (a0 , u0 , δ0 ) = exp u0 −1+ p2 + 1 + δ0 p2 B˙ p,1
× 1 + δ0 A2 (a0 , u0 , δ0 ) = exp u0
2
p B˙ p,1
θ (t) =
2
−1+ p B˙ p,1 2
−1+ p B˙ p,1
1 − e−ct2
exp 1 + u0
+ 1 + δ0
2 −1
p B˙ p,1
(3.34)
+ 1 + δ0 2j
2
p B˙ p,1
3
,
def
× u0
B˙ p,1
2
2
p B˙ p,1
3 2
p B˙ p,1
1 + a0
2
p B˙ p,1
exp 1 + u0
4
2 −1
p B˙ p,1
(3.35)
,
2j 2 ˙ j u0 Lp + 2 p ˙ j δ0 Lp . 2(−1+ p )j
(3.36)
j ∈Z
Finally, taking ε0 and T1 small enough and n2 ≥ n1 large enough ensures for any n ≥ n2
A1 (a0 , u0 , δ0 ) 2ε0 + 8ε03 + 2m T1 ≤
1 − e−cT1 2
2j
2j (−1+ 2 )j p ˙ j u0 Lp + 2 p ˙ j δ0 Lp ≤ 2
j ∈Z
1 , 2 ε0 , 2A2 (a0 , u0 , δ0 )
which together with (3.33) implies Z n (t) ≤ ε0 ,
∀t ≤ min T∗n , T1 , n ≥ n2 .
However, by the definition of T∗n , we eventually infer T∗n ≥ T1 and supn≥n2 Z n (T1 ) ≤ ε0 , which along with (3.5) and (3.27) ensures that
n n n a , u , δ , ∇Π n n∈N is uniformly bounded in ET1 .
Global Well-posedness for the Density-Dependent Incompressible Flow. . .
Step 3. Convergence. We can use the definition of Besov spaces to get the convergence of (unF , δFn ) to (uF , δF ) ¯ More details and use Ascoli’s theorem to get the convergence of (a n , u¯ n , δ¯n ) to (a, u, ¯ δ). about this convergence can be similarly obtained to Sect. 3 in [24]. Moreover, we can verify that a − S˙m a
2 p
˙ L∞ t (Bp,1 )
≤ c0 b
(3.37)
with the constant c0 being sufficiently small.
3.2 Uniqueness We shall apply a Lagrangian approach as in [3, 27] to solve the uniqueness part of Theorem 1.1. Let us first derive algebraic relations involving changes of coordinates. For a matrix A = (Aij ) : R2 → R2×2 , we denote AT its transpose matrix, Tr(A) its trace, det(A) its determinant and (div A)j =∂i Aij . For a C 1 vector field u : R2 → R2 , denote (Du)ij =∂j ui and ∇u=(Du)T . By using the Cauchy-Lipschitz theorem, the unique trajectory X(t, y) of u(t, x) is determined by t u τ, X(τ, y) dτ, X(t, y) = y + 0
such that X(t, ·) is a C -diffeomorphism over R2 . Denote A(t, y)=(Dy X(t, y))−1 . Then the divergence free condition for u is equivalent to det(A) ≡ 1. While applying Lemma A.1 in the appendix of [3] gives 1
∇x u(t, x) = AT (t, x)∇y v(t, y), divx u t, X(t, y) = Tr(Dy v · A)(t, y) = divy (Av)(t, y).
(3.38)
Let (a, u, δ, ∇Π ) be a solution to (1.2) and satisfy 2 p2 2 p ˙ ∞ a ∈ C [0, T ]; B˙ p,1 R2 ∩ L , T Bp,1 R
p2 −1 2 ∇Π ∈ L1T B˙ p,1 R ,
2 2 p2 −1 2 p2 +1 2 p −1 ˙ ∞ R ∩L ∩ L1T B˙ p,1 R , u ∈ C [0, T ]; B˙ p,1 T Bp,1 R
(3.39)
2 p2 2 p2 +2 2 p ˙ ∞ R2 ∩ L ∩ L1T B˙ p,1 R . δ ∈ C [0, T ]; B˙ p,1 T Bp,1 R
To obtain the Lagrangian formulation of (1.2), we define d¯0 (y) = w0 (X(t, y)) and (η, v, w, h, P )(t, y)=(a, u, δ, ∇δ, Π ) t, X(t, y) .
(3.40)
Thanks to the transport equation of (1.2), we infer η(t, ·) ≡ a0 . By the chain rule, we have ∇x Π t, X(t, y) = AT ∇y P (t, y),
(∂t u + u · ∇x u) t, X(t, y) = ∂t v(t, y).
X. Zhai, Z.M. Chen
We infer that (v, w, h, ∇Π ) solves ⎧ T
T ⎪ ⎪ ∂t v − (1 + a0 ) div A A∇v + (1 + a0 )A ∇Π = −(1 + a0 ) div{Ah h}, ⎪ ⎪ ⎪ ⎪ ⎪ ∂t w − AT : ∇h = |h|2 (w + w0 ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂t h + AT ∇v · h − div(A∇h) = 2h · ∇h(w + w 0 ) + |h|2 h, (3.41) ⎪ div(Av) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ h = AT ∇w, ⎪ ⎪ ⎪ ⎪ ⎩ (v, w)|t=0 = (u0 , w0 ). Now, let (ai , ui , δi , ∇Πi ), i = 1, 2, be two solutions to (3.41) and let (ηi , vi , wi , Pi ) def be determined by (3.40). Denote (δv, δw, δh, δP , δA) = (v2 − v1 , w2 − w1 , h2 − h1 , P2 − P1 , A2 − A1 ), where −1 t def Ai (t, y) = Id + Dvi (τ, y)dτ ,
for i = 1, 2.
(3.42)
0
Then the system for (δv, δw, δh, ∇δP ) reads ⎧ ∂t δv − (1 + a0 )δv + (1 + a0 )∇δP = (1 + a0 )(δF1 + δF2 + δF3 ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂t δw = δF4 + δF5 , ⎪ ⎪ ⎨ ∂t δh − δh = δF6 + δF7 + δF8 , ⎪ ⎪ ⎪ ⎪ div δv = δg, ∂t δg = δR, ⎪ ⎪ ⎪ ⎪ ⎩ (δv, δw, δh)|t=0 = (0, 0, 0), where
Id − AT2 ∇δP − δA∇P1 ,
def δF2 = div A2 AT2 − Id ∇δv + A2 AT2 − A1 AT1 ∇v1 , def
δF1 =
def
δF3 = δAT : ∇h2 + AT1 : ∇δh,
def δF4 = div AT2 − Id ∇δh + δAT ∇h1 , def
δF5 = −δAT ∇v2 · h2 − AT1 ∇δv · h2 − AT1 ∇v1 · δh, def
δF6 = δh · h2 (w2 + w0 ) + h1 · δh(w2 + w 0 ) + |h1 |2 δw,
def δF7 = div δA(h2 h2 ) + A1 (δh h2 ) + A1 (h1 δh) , def
δF8 = 2δh · ∇h2 (w2 + w 0 ) + 2h1 · ∇δh(w2 + w0 ) + h1 · ∇h1 δw, def δg = div (Id − A2 )δv − δAv1 , def
δR = −∂t A2 δv + (Id − A2 )∂t δv − ∂t δAv1 − δA∂t v1 .
(3.43)
Global Well-posedness for the Density-Dependent Incompressible Flow. . .
In the sequel, we shall take T to be so small that T Dvi (t) 2 dt ≤ c, p
i = 1, 2,
B˙ p,1
0
(3.44)
for a small enough constant c. Moreover, as proved in the appendix of [3], we have the following estimates: δA
2 p
Dδv
2 p
Dvi
2 p
Dvi
˙ L∞ t (Bp,1 )
∂t Ai Ai − Id ∂t δA
L1t (B˙ p,1 )
˙ L∞ t (Bp,1 )
2 p −1 L2t (B˙ p,1 )
v1
2 p
L1t (B˙ p,1 ) 2 p
L1t (B˙ p,1 ) 2 p
L1t (B˙ p,1 )
(3.45)
,
,
i = 1, 2,
(3.46)
,
i = 1, 2,
(3.47)
+ v2
2 p L2t (B˙ p,1 )
2 p L2t (B˙ p,1 )
Dδv
2
p L1t (B˙ p,1 )
+ δv
2
p L2t (B˙ p,1 )
(3.48)
.
By Proposition 3.1, we have δv
2 −1
˙p ∞ L t (Bp,1 )
+ δv
1 + a0 + δg
6 2 p B˙ p,1
2 p L1t (B˙ p,1 )
2 p +1
L1t (B˙ p,1 )
δF1
+ ∇δP 2 p −1
L1t (B˙ p,1 )
+ 2m δv
+ δF2
2 p L1t (B˙ p,1 )
+ ∂t δv
2 p −1
L1t (B˙ p,1 )
2 p −1
L1t (B˙ p,1 )
2 p −1
L1t (B˙ p,1 )
+ δF3
2 p −1
L1t (B˙ p,1 )
+ δR
2 p −1
L1t (B˙ p,1 )
(3.49)
.
Similarly, one has δw δh
2
˙p ∞ L t (Bp,1 ) 2 −1
+ δw
˙p ∞ L t (Bp,1 )
2 p +2
δF4
2 +1
δF6
L1t (B˙ p,1 )
+ δh
p L1t (B˙ p,1 )
2 p
L1t (B˙ p,1 )
+ δF5
2 −1
p L1t (B˙ p,1 )
2 p
L1t (B˙ p,1 )
+ δF7
(3.50)
,
2 −1
p L1t (B˙ p,1 )
+ δF8
2 −1
p L1t (B˙ p,1 )
.
(3.51) Denote def
δE(t) = δv
2 −1
˙p ∞ L t (Bp,1 )
+ δh
+ δw
2 +1
p L1t (B˙ p,1 )
2
˙p ∞ L t (Bp,1 )
+ δh
2 −1
˙p ∞ L t (Bp,1 )
+ ∇δP
2 p −1
L1t (B˙ p,1 )
+ δv
2 p +1
L1t (B˙ p,1 )
.
Thus, by estimates (3.45)–(3.48) and the product law in Besov spaces, we have δF1
δF2
2 −1+ p
L1t (B˙ p,1
∇v2 )
v2 2
−1+ p L1t (B˙ p,1 )
∇δP
2 1+ p L1t (B˙ p,1 )
∇v2 v2
2 p
L1t (B˙ p,1 )
2
p L1t (B˙ p,1 )
+ ∇P1
∇δv
2 1+ p L1t (B˙ p,1 )
2 −1+ p
L1t (B˙ p,1
2 −1+ p L1t (B˙ p,1 ) 2
p L1t (B˙ p,1 )
+ v1
+ ∇P1 )
δv )
2 1+ p
L1t (B˙ p,1 )
(3.52)
δE(t),
+ ∇v1
2 1+ p L1t (B˙ p,1 )
2 −1+ p
L1t (B˙ p,1
2
p L1t (B˙ p,1 )
δE(t),
∇δv
2
p L1t (B˙ p,1 )
(3.53)
X. Zhai, Z.M. Chen
δF3
δF4
2 p
L1t (B˙ p,1 )
2
−1+ p L1t (B˙ p,1 )
∇δv h2
2 1+ p
∇v2
δF5
2 −1+ p L1t (B˙ p,1 )
2
δh
2 1+ p L1t (B˙ p,1 )
v2
2 p L1t (B˙ p,1 )
δh 0
2 p B˙ p,1
h2
δw
+ h1 h2 2
2 p
L2t (B˙ p,1 )
ε δh
2 p B˙ p,1
1+ 2
p L1t (B˙ p,1 )
(3.55)
δE(t),
2 1+ p L1t (B˙ p,1 )
w2
2 p
2 p L1t (B˙ p,1 )
h2
2
−1+ p ˙ ∞ ) L t (Bp,1
h2
+ v1 2
2 −1+ p ˙ ∞ ) L t (Bp,1
2 1+ p L1t (B˙ p,1 )
δE(t),
+ 1 + δh
2 p B˙ p,1
h1
2 p B˙ p,1
2 p B˙ p,1
w2
2 p B˙ p,1
+ 1 dτ
2 p
L1t (B˙ p,1 )
+ ∇v1 2
2 p L1t (B˙ p,1 )
2
2
p L1t (B˙ p,1 )
δh
h2
2 p
L1t (B˙ p,1 )
−1+ p ˙ ∞ ) L t (Bp,1
δh
h1 2
2
δh
h2
h2 2
∇h2
2 p L1t (B˙ p,1 )
∇h1
2
δh
2 1+ p
L1t (B˙ p,1 )
+ 2 δE(t), 2
p L1t (B˙ p,1 )
h2
(3.57) 2
˙p ∞ L t (Bp,1 )
δv
2 1+ p
L1t (B˙ p,1 )
2
−1+ p ˙ ∞ ) L t (Bp,1 2
−1+ p ˙ ∞ ) L t (Bp,1 2 1+ p
L1t (B˙ p,1 )
δE(t)
+ h1 2 w2
p L1t (B˙ p,1 )
2
h2
2
˙p ∞ L t (Bp,1 )
2
p L1t (B˙ p,1 )
+1
˙p ∞ L t (Bp,1 )
h2 2
˙p ∞ L t (Bp,1 )
2 −1+ p ˙ ∞ ) L t (Bp,1
2 −1+ p ˙ ∞ ) L t (Bp,1
h1 2
w2 2
w2 2
+ ∇v1
2 ˙p ∞ L t (Bp,1 )
˙p ∞ L t (Bp,1 )
2 1+ p L1t (B˙ p,1 )
2 1+ p L1t (B˙ p,1 )
2
2 −1+ p ˙ ∞ ) L t (Bp,1
+ δw
h1
−1+ p ˙ ∞ ) L t (Bp,1
2 p L1t (B˙ p,1 )
δh
2
+ v1 2
dτ
−1+ p ˙ ∞ ) L t (Bp,1
+ h2
2 1+ p
L1t (B˙ p,1 )
εδE(t) + h2
2
p B˙ p,1
∇δv
L1t (B˙ p,1 )
+ ∇v1 2
2 −1+ p L1t (B˙ p,1 )
2 −1+ p ˙ ∞ ) L t (Bp,1
+ ∇v1
δF8
δv
1+ 2
p L1t (B˙ p,1 )
2
h1 2
2
p B˙ p,1
εδE(t) + h2
2
1+ 2
p L1t (B˙ p,1 )
h1
t
0
−1+ p L1t (B˙ p,1 )
2
−1+ p ˙ ∞ ) L t (Bp,1
+ v1
2 1+ p L1t (B˙ p,1 )
t
+
δF7
δh
(3.54)
p L1t (B˙ p,1 )
+ ∇v1
2 p L1t (B˙ p,1 )
2 1+ p
L1t (B˙ p,1 )
(3.56)
δF6
2
2 1+ p L1t (B˙ p,1 )
δh
δE(t),
+ h1
1+ 2
p L1t (B˙ p,1 )
2 p
L1t (B˙ p,1 )
+ ∇δv
2
∇v2
p L1t (B˙ p,1 )
2 1+ p
L1t (B˙ p,1 )
p L1t (B˙ p,1 )
+ h1
2 p L1t (B˙ p,1 )
+ ∇v1 2
+ v1
∇δh
p L1t (B˙ p,1 )
+ ∇v1
2 1+ p
L1t (B˙ p,1 )
2
p L1t (B˙ p,1 )
∇δv
h2
L1t (B˙ p,1 )
∇v2
v2
2 p
L1t (B˙ p,1 )
h1
2 ˙p ∞ L t (Bp,1 )
2 ˙p ∞ L t (Bp,1 ) 2
δE(t),
+1
−1+ p ˙ ∞ ) L t (Bp,1
(3.58)
Global Well-posedness for the Density-Dependent Incompressible Flow. . .
t
+
h1 0
h2
2 p B˙ p,1
2 1+ p L1t (B˙ p,1 )
+ h1 + h1
δh
w2
+ h1
2
h1
2
h1
−1+ p ˙ ∞ ) L t (Bp,1
2 1+ p
( w2
h1
+ 1 dτ
+ 1 δh
L1t (B˙ p,1 )
2 1+ p
2
2 p B˙ p,1
2 1+ p L1t (B˙ p,1 )
L1t (B˙ p,1 )
−1+ p ˙ ∞ ) L t (Bp,1
w2
˙p ∞ L t (Bp,1 )
2
−1+ p ˙ ∞ ) L t (Bp,1
εδE(t) + h2
2 p B˙ p,1
2
−1+ p ˙ ∞ ) L t (Bp,1
+ 1 δh
w2 2
2 ˙p ∞ L t (Bp,1 )
δw
2
˙p ∞ L t (Bp,1 )
2 1+ p
L1t (B˙ p,1 )
+ 1)δE(t)
2
˙p ∞ L t (Bp,1 )
2 1+ p L1t (B˙ p,1 )
+ ε δh
2
−1+ p ˙ ∞ ) L t (Bp,1
+ 1 δE(t).
w2 2
2 ˙p ∞ L t (Bp,1 )
(3.59)
Using the same argument, we deduce from (3.47) and (3.48) that δR
v2
2 p −1 L1t (B˙ p,1 )
2 p +1 L1t (B˙ p,1 )
+ ∂t v1
2 p −1 L1t (B˙ p,1 )
+ v1
2 p L2t (B˙ p,1 )
(3.60)
δE(t).
2 p ), we apply (3.38) to rewrite δg as follows: In order to bound δg in L1t (Bp,1
δg = Tr Dδv(Id − A2 ) − Dv1 δA , from which, we readily get that δg
2 p L1t (B˙ p,1 )
v1
2 p +1 L1t (B˙ p,1 )
+ v2
2 p +1 L1t (B˙ p,1 )
(3.61)
δE(t).
Taking estimates (3.52)–(3.61) into (3.49), (3.50) and (3.51) leads to 1 δE(t) W (t) + 2m t 2 δE(t)
(3.62)
with
def
W (t) = v1
2 p +1 L1t (B˙ p,1 )
+ v2
1 + v1
2 p +1
L1t (B˙ p,1 )
+ ∇P1 + h1
+ h1
2 −1+ p L1t (B˙ p,1 )
2 −1+ p ˙ ∞ ) L t (Bp,1
+ v1 2
2 p +1 L1t (B˙ p,1 )
2 p +1 L1t (B˙ p,1 ) 2 p +1
L1t (B˙ p,1 )
+ h2
h1
h2 2
˙p ∞ L t (Bp,1 )
2
−1+ p ˙ ∞ ) L t (Bp,1
+ h2
2 −1+ p ˙ ∞ ) L t (Bp,1
2 1+ p L1t (B˙ p,1 )
2
+ h2
2 p +1
L1t (B˙ p,1 )
h2
+ ∂t v1
1 + w2
2 1+ p L1t (B˙ p,1 )
1 + w2 2
2 −1
p L1t (B˙ p,1 )
2
˙p ∞ L t (Bp,1 )
1 + w2 2
2 −1+ p ˙ ∞ ) L t (Bp,1
2 −1+ p ˙ ∞ ) L t (Bp,1
+ v1 2
2 p +1 L1t (B˙ p,1 )
h1 2
2
˙p ∞ L t (Bp,1 )
,
(3.63)
which together with the smallness of t implies that δE(t) = 0. Hence the uniqueness on [0, T ] is obtained by a standard argument.
X. Zhai, Z.M. Chen
4 Global Well-posedness Part of Theorem 1.1 The strategy of the proof of Theorem 1.1 is to seek a solution of (1.2) in the perutbation form of u = v + w and Π = π + p with (w, p) solving the classical Navier-Stokes system ⎧ ∂t w − w + w · ∇w + ∇p = 0, ⎪ ⎪ ⎨ div w = 0, (4.1) ⎪ ⎪ ⎩ w|t=0 = u0 , and (v, π) solving ⎧ ∂ a + (v + w) · ∇a = 0, ⎪ ⎪ t ⎪ ⎪ ⎪ ⎪ ∂t v + v · ∇v + w · ∇v + v · ∇w − (1 + a)v + (1 + a)∇π ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ − aw + a∇p + (1 + a) div(∇δ ∇δ) = 0, ⎪ ⎪ ∂t δ − δ + (v + w) · ∇δ = |∇δ|2 δ + |∇δ|2 d¯0 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ div v = 0, ⎪ ⎪ ⎪ ⎩ (a, v, δ)|t=0 = (a0 , 0, δ0 ).
(4.2)
Before presenting our estimates, we need the following two propositions whose details can be founded in [9]: The first proposition is to investigate the following transport equation: ⎧ ∂t a + (v + w) · ∇a = 0, ⎪ ⎪ ⎨ div v = div w = 0, (4.3) ⎪ ⎪ ⎩ a|t=0 = a0 . 1+ 2
Proposition 4.1 (See [9]) Let v, w ∈ L1 ((0, T ), B˙ p,1 p (R2 )) be divergence-free vector fields, 2 t p (R2 ). We denote g(t) = w(t) 1+ p2 and aλ = a exp{−λ 0 g(t )dt }. Then and a0 ∈ B˙ p,1 B˙ p,1
2 p
(4.3) has a unique solution a ∈ C([0, T ]; B˙ p,1 (R2 )) so that aλ
2 ˙p ∞ L t (Bp,1 )
λ + aλ ≤ a0 p2 + C v 2 aλ 2 2 1+ p p ˙p ∞ 2 L1t,g (B˙ p,1 ) L1t (B˙ p,1 ) L B˙ p,1 t (Bp,1 )
(4.4)
for any t ∈ (0, T ] and λ large enough. The second proposition assures the existence and uniqueness of a global solution for the classical Navier-Stokes system (4.1). More precisely, we have: −1+ p2
Proposition 4.2 (See [9]) For 1 < p < 4 and u0 ∈ B˙ p,1 solution w so that
(R2 ), Eq. (4.1) has a unique
−1+ 2 −1+ 2 1+ 2 ∞ (0, +∞); B˙ p,1 p ∩ L1 (0, +∞); B˙ p,1 p w ∈ C [0, +∞); B˙ p,1 p ∩ L
Global Well-posedness for the Density-Dependent Incompressible Flow. . .
and w
2
−1+ p ˙ ∞ ) L t (Bp,1
≤ C u0
+ w
2 −1+ p B˙ p,1
2 1+ p
L1t (B˙ p,1 )
1 + u0
+ ∇p
2 −1+ p B˙ p,1
2 −1+ p
L1t (B˙ p,1
)
exp u0 2 −1+ 2 .
(4.5)
p B˙ p,1
4.1 The Estimate of the Pressure The goal of this section is to get the pressure estimate in the framework of weighted CheminLerner type space. We first get by taking divergence to the second equation of (4.2) that −π = div(a∇π) + div(a∇p) − div(av) − div(aw) + div(v · ∇v + w · ∇v + v · ∇w) + div (1 + a) div(∇δ ∇δ) . We denote
f (t) = w(t)
and
2 1+ p B˙ p,1
+ ∇p(t)
2
−1+ p B˙ p,1
+ w(t) 2
(4.6)
2
p B˙ p,1
t πλ = π exp − λf t dt ,
(4.7)
0
where λ > 0, similar notations for t aλ , vλ and δλ . Multiplying (4.6) by exp{− 0 λf (t )dt }, we obtain ∇πλ = ∇(−)−1 div(a∇πλ ) + div(aλ ∇p) − div(avλ ) − div(aλ w) + div(v · ∇vλ + w · ∇vλ + vλ · ∇w) + div (1 + a) div(∇δ ∇δλ ) . ˙ j to the above equation and using Lemma 2.6, we have Applying ∇πλ 2 ≤ C a∇πλ 2 + aλ ∇p 2 + avλ −1+ p −1+ p −1+ p L1t (B˙ p,1
L1t (B˙ p,1
)
+ aλ w
2 −1+ p
L1t (B˙ p,1
+ v · ∇vλ
L1t (B˙ p,1
)
)
+ w · ∇vλ + vλ · ∇w )
2 −1+ p L1t (B˙ p,1 )
2 −1+ p
L1t (B˙ p,1
2 −1+ p
L1t (B˙ p,1
)
L1t (B˙ p,1
)
2 −1+ p L1t (B˙ p,1 )
B˙ p,1
0
≤ C aλ
B˙ p,1
2 p
L1t,f (B˙ p,1 )
.
2
−1+ p B˙ p,1
.
According to Lemma 2.7 and standard product laws in Besov spaces, there hold avλ
2 −1+ p
L1t (B˙ p,1
≤ C a )
2
˙p ∞ L t (Bp,1 )
vλ
2 1+ p
L1t (B˙ p,1 )
,
)
)
+ (1 + a) div(∇δ ∇δλ )
We continue to estimate each term on the right-hand side of (4.9). By (4.7) and Lemma 2.7, one has t aλ w aλ p2 w 1+ p2 + ∇p 2 + aλ ∇p 2 ≤ −1+ p −1+ p L1t (B˙ p,1
(4.8)
(4.9)
dτ
X. Zhai, Z.M. Chen
a∇πλ v · ∇vλ w · ∇vλ + vλ · ∇w (1 + a) div(∇δ ∇δλ ) Thus, if C a ∇πλ
2
˙p ∞ L t (Bp,1 )
2 −1+ p L1t (B˙ p,1 )
2 −1+ p
L1t (B˙ p,1
≤ C a )
2
−1+ p L1t (B˙ p,1 ) 2 −1+ p
L1t (B˙ p,1
≤ C v
2 −1+ p L1t (B˙ p,1 )
∇πλ 2
−1+ p ˙ ∞ ) L t (Bp,1
≤ ε vλ )
2
˙p ∞ L t (Bp,1 )
2 1+ p
L1t (B˙ p,1 )
≤ C 1 + a
2 −1+ p
L1t (B˙ p,1
vλ
1+ 2
p L1t (B˙ p,1 )
+ C vλ
2 ˙p ∞ L t (Bp,1 )
δ
, )
, 2 −1+ p
L1t,f (B˙ p,1 2
˙p ∞ L t (Bp,1 )
, )
δλ
2 2+ p
L1t (B˙ p,1 )
.
≤ 12 , we finally get from (4.9) that
≤ C aλ + vλ
2
p L1t,f (B˙ p,1 )
+ v
2 −1+ p
L1t,f (B˙ p,1
+ 1 + a
2
−1+ p ˙ ∞ ) L t (Bp,1
+ a )
2 ˙p ∞ L t (Bp,1 )
δ
2
˙p ∞ L t (Bp,1 )
2 ˙p ∞ L t (Bp,1 )
vλ
vλ
δλ
1+ 2
p L1t (B˙ p,1 )
+ ε vλ
1+ 2
p L1t (B˙ p,1 )
2 1+ p
L1t (B˙ p,1 )
2 2+ p L1t (B˙ p,1 )
(4.10)
.
4.2 The Proof of Theorem 1.1 2
−1+ p2
p In fact, given a0 ∈ B˙ p,1 (R2 ), u0 ∈ B˙ p,1
2
p (R2 ), δ0 ∈ B˙ p,1 (R2 ) with a0
2
p B˙ p,1
being suffi-
ciently small, it follows by the arguments in Sect. 3 that there exists a positive time T so that (1.2) has a unique local solution (a, u, δ) satisfying 2 2 p p ∞ 0, T ; B˙ p,1 a ∈ C [0, T ]; B˙ p,1 R2 ∩ L R2 ,
−1+ 2 −1+ 2 1+ 2 ∞ 0, T ; B˙ p,1 p R2 ∩ L1 0, T ; B˙ p,1 p R2 , u ∈ C [0, T ]; B˙ p,1 p R2 ∩ L
(4.11)
2 2 2+ 2 p p ∞ 0, T ; B˙ p,1 R2 ∩ L R2 ∩ L1 0, T ; B˙ p,1 p R2 . δ ∈ C [0, T ]; B˙ p,1
We denote T ∗ to be the largest possible time such that (4.11) holds true. Then, the proof of Theorem 1.1 is reduced to show that T ∗ = ∞ under the assumption of (1.4). Therefore, we split the velocity u and the pressure Π as u = v + w, Π = π + p, where (w, p) solves (4.1) and (v, ∇π) together with (a, δ) solves (4.2). Then thanks to Proposition 4.2, it remains to solve (4.2) globally.
4.3 The Estimates of v and δ Thanks to the definition of weighted Chemin-Lerner type norm, it is readily to get from (4.2) that ∂t vλ + λf (t)vλ − vλ + v · ∇vλ + w · ∇vλ + vλ · ∇w − avλ + (1 + a)∇πλ − aλ w + aλ ∇p + (1 + a) div(∇δ ∇δλ ) = 0, ∂t δλ + λf (t)δλ − δλ + vλ · ∇δ + w · ∇δλ = |∇δ|2 δλ + ∇δ : ∇δλ d¯0 .
(4.12) (4.13)
Global Well-posedness for the Density-Dependent Incompressible Flow. . .
˙ j to Eq. (4.12) and taking the L2 inner product of the resultant equation with Applying p−2 ˙ j vλ , we obtain ˙ j vλ | | d ˙ j vλ Lp + λf (t) ˙ j vλ Lp + C1 22j ˙ j vλ Lp dt ˙ j (aλ w) p + ˙ j (aλ ∇p) p + ˙ j (1 + a)∇πλ p + ˙ j (avλ ) p ≤ L L L L ˙ ˙ + j (v · ∇vλ ) Lp + j (w · ∇vλ + vλ · ∇w) Lp ˙ j (1 + a) div(∇δ ∇δλ ) p . + (4.14) L Integrating from 0 to t and by a product laws in Besov spaces, we have vλ
2
−1+ p ˙ ∞ ) L t (Bp,1
+ λ vλ
≤ C aλ w + avλ
2 −1+ p L1t (B˙ p,1 ) 2 −1+ p
L1t (B˙ p,1
+ (1 + a)∇πλ
+ v · ∇vλ )
+ w · ∇vλ + vλ · ∇w ≤ C aλ
2 p
L1t,f (B˙ p,1 )
+ C a
+ C v
2
˙p ∞ L t (Bp,1 )
+ C1 vλ
2
−1+ p L1t,f (B˙ p,1 )
vλ
2
2 −1+ p
2 1+ p
2
˙p ∞ L t (Bp,1 )
2 −1+ p
L1t (B˙ p,1
)
)
2 −1+ p L1t (B˙ p,1 )
vλ
2 1+ p
L1t (B˙ p,1 )
+ C 1 + a
under the assumption that C a
+ aλ ∇p )
+ (1 + a) div(∇δ ∇δλ )
−1+ p ˙ ∞ ) L t (Bp,1
L1t (B˙ p,1 )
2 −1+ p
L1t (B˙ p,1
L1t (B˙ p,1
2 −1+ p L1t (B˙ p,1 )
1+ 2
p L1t (B˙ p,1 )
+ ε vλ 2
˙p ∞ L t (Bp,1 )
δ
2 1+ p
L1t (B˙ p,1 ) 2
˙p ∞ L t (Bp,1 )
+ C vλ
δλ
2 −1+ p
L1t,f (B˙ p,1
2 2+ p
L1t (B˙ p,1 )
)
, (4.15)
≤ 12 .
Similar arguments as in deriving (4.14) can be used to conclude from d ˙ j δλ Lp + λf (t) j δλ Lp + C2 22j ˙ j δλ Lp dt ˙ j (w · ∇δλ + vλ · ∇δ) p + ˙ j |∇δ|2 δλ ≤
Lp
L
˙ j (∇δ : ∇δλ d¯0 ) p + L
that δλ
2
˙p ∞ L t (Bp,1 )
δ0
+ λ δλ
2 p B˙ p,1
2 p
L1t,f (B˙ p,1 )
+ w · ∇δλ
+ ∇δ : ∇δλ d¯0
2
+ C2 δλ
2 p L1t (B˙ p,1 )
p L1t (B˙ p,1 )
2 2+ p
L1t (B˙ p,1 )
+ vλ · ∇δ
2 p L1t (B˙ p,1 )
+ |∇δ|2 δλ
2 p
L1t (B˙ p,1 )
(4.16)
.
By the product laws in Besov spaces and Young’s inequality, we have t t 1 1 ≤ C w δλ 2 2 δλ 2 2+ 2 w w · ∇δλ 2 2 δλ 1+ 2 dτ ≤ C p p p L1t (B˙ p,1 )
0
≤ ε δλ
B˙ p,1
2 2+ p
L1t (B˙ p,1 )
B˙ p,1
+ C δλ
p B˙ p,1
0
2 −1+ p
L1t,f (B˙ p,1
, )
p B˙ p,1
2
p B˙ p,1
dτ
(4.17)
X. Zhai, Z.M. Chen
∇δ : ∇δλ d¯0 |∇δ|2 δλ
t
2 p L1t (B˙ p,1 )
δ 0
p B˙ p,1
δ 0
vλ · ∇δ
2
p B˙ p,1
2
˙p ∞ L t (Bp,1 )
δλ
2 2+ p
L1t (B˙ p,1 )
(4.18)
,
2
p B˙ p,1
dτ
vλ 0
t
≤C
δ
2+ 2
p B˙ p,1
δλ
2 2+ p
L1t (B˙ p,1 )
2
p B˙ p,1
δλ
dτ 2 2+ p
L1t (B˙ p,1 )
(4.19)
,
2
p B˙ p,1
δ
1+ 2
p B˙ p,1
dτ
1
1
1
1
p B˙ p,1
p B˙ p,1
p B˙ p,1
p B˙ p,1
2+ 2
+ δ
vλ 2 −1+ 2 vλ 2 1+ 2 δ 2 2 δ 2 2+ 2 dτ
0
≤ v
δ
t
≤C
2 p L1t (B˙ p,1 )
2
˙p ∞ L t (Bp,1 )
t
≤C
dτ δ
2+ 2
p B˙ p,1
δ 2 1+ 2 δλ
0
≤ δ
δλ
t
≤C
2 p L1t (B˙ p,1 )
2
p B˙ p,1
2
−1+ p ˙ ∞ ) L t (Bp,1
δλ
p L1t (B˙ p,1 )
2
˙p ∞ L t (Bp,1 )
vλ
1+ 2
p L1t (B˙ p,1 )
(4.20)
.
Inserting estimates (4.17)–(4.20) into (4.16), we have δλ
+ λ δλ
2
˙p ∞ L t (Bp,1 )
δ0
2
p B˙ p,1
+ v
+ C2 δλ
2 p
L1t,f (B˙ p,1 )
+ ε δλ
2 2+ p
L1t (B˙ p,1 )
2
−1+ p ˙ ∞ ) L t (Bp,1
+ δ
+ δλ 2
˙p ∞ L t (Bp,1 )
2 2+ p
L1t (B˙ p,1 ) 2 −1+ p
L1t,f (B˙ p,1
δ
)
2 2+ p L1t (B˙ p,1 )
δλ
2+ 2
p L1t (B˙ p,1 )
+ δ
2
˙p ∞ L t (Bp,1 )
vλ
1+ 2
p L1t (B˙ p,1 )
.
(4.21) By the combination of (4.15) and (4.21), letting c¯¯ = min{C1 , C2 } and using the definition of weighted Chemin-Lerner type norm, we show that vλ
2
−1+ p ˙ ∞ ) L t (Bp,1
¯¯ λ + c v δ0
2
p B˙ p,1
+ a
+ δλ 2 1+ p
L1t (B˙ p,1 )
+ aλ 2
˙p ∞ L t (Bp,1 )
+ 1 + a
2
˙p ∞ L t (Bp,1 )
+ λ vλ
¯¯ δλ + cσ 2 p
L1t,f (B˙ p,1 )
+ v 2
˙p ∞ L t (Bp,1 )
+ δλ
2 p L1t,f (B˙ p,1 )
2 2+ p
L1t (B˙ p,1 )
+ vλ 2
−1+ p ˙ ∞ ) L t (Bp,1
+ v
2 −1+ p L1t,f (B˙ p,1 )
2 −1+ p
L1t,f (B˙ p,1
+ δ 2
−1+ p ˙ ∞ ) L t (Bp,1
+ δλ )
2
˙p ∞ L t (Bp,1 )
+ δ
2 −1+ p
L1t,f (B˙ p,1
vλ
1+ 2
p L1t (B˙ p,1 )
2 2+ p
L1t (B˙ p,1 )
)
δ
2
˙p ∞ L t (Bp,1 )
δλ
2 2+ p
L1t (B˙ p,1 )
. (4.22)
By Proposition 4.1, we obtain aλ
2
˙p ∞ L t (Bp,1 )
+ λ aλ
2 p
L1t,f (B˙ p,1 )
a0
2
p B˙ p,1
+ v
2 1+ p
L1t (B˙ p,1 )
aλ
2
˙p ∞ L t (Bp,1 )
.
(4.23)
Global Well-posedness for the Density-Dependent Incompressible Flow. . .
Choosing λ large enough, we get from (4.22) and (4.23) that aλ
2
˙p ∞ L t (Bp,1 )
a0
+ vλ
2
p B˙ p,1
+ a
2
−1+ p ˙ ∞ ) L t (Bp,1
+ δ0
2 ˙p ∞ L t (Bp,1 )
+ 1 + a
2
p B˙ p,1
+ δλ
+ v
+ v
1+ 2
p L1t (B˙ p,1 )
2 −1+ p ˙ ∞ ) L t (Bp,1
+ v
2
˙p ∞ L t (Bp,1 )
2
˙p ∞ L t (Bp,1 )
aλ
+ δ
¯¯ λ + c v
¯¯ λ + c δ
2 2+ p
L1t (B˙ p,1 )
2
˙p ∞ L t (Bp,1 )
2 ˙p ∞ L t (Bp,1 )
+ δ
2
−1+ p ˙ ∞ ) L t (Bp,1
2 1+ p
L1t (B˙ p,1 )
vλ
2 2+ p L1t (B˙ p,1 )
2 1+ p
L1t (B˙ p,1 )
δ
2
˙p ∞ L t (Bp,1 )
δλ
2+ 2
p L1t (B˙ p,1 )
. (4.24)
Now let c2 be a small enough positive constant, which will be determined later on. We define T ∗∗ by T ∗∗ = sup t < T ∗ : a + v
+ δ
2 1+ p L1t (B˙ p,1 )
+ v
2
˙p ∞ L t (Bp,1 )
2 2+ p L1t (B˙ p,1 )
2
−1+ p ˙ ∞ ) L t (Bp,1
+ δ
≤ c2 .
2
˙p ∞ L t (Bp,1 )
(4.25)
Taking
c¯¯ , 6
1 c2 = min , 2
(4.26)
we infer from (4.24) that, for any t < T ∗∗ , aλ
2
˙p ∞ L t (Bp,1 )
a0
+ vλ
2
p B˙ p,1
2
−1+ p ˙ ∞ ) L t (Bp,1
+ δ0
2
p B˙ p,1
+ δλ
2
˙p ∞ L t (Bp,1 )
¯¯ λ + c v
2 1+ p
L1t (B˙ p,1 )
¯¯ λ + c δ
2 2+ p
L1t (B˙ p,1 )
(4.27)
.
It follows from (4.7) that
a
2
˙p ∞ L t (Bp,1 )
+ v
2
−1+ p ˙ ∞ ) L t (Bp,1
+ δ
t × exp − λf t dt
2
˙p ∞ L t (Bp,1 )
¯¯ + c v
2 1+ p
L1t (B˙ p,1 )
¯¯ + c δ
2 2+ p
L1t (B˙ p,1 )
0
≤ aλ
2
˙p ∞ L t (Bp,1 )
+ vλ
2
−1+ p ˙ ∞ ) L t (Bp,1
+ δλ
2
˙p ∞ L t (Bp,1 )
¯¯ λ + c v
1+ 2
p L1t (B˙ p,1 )
¯¯ λ + c δ
2+ 2
p L1t (B˙ p,1 )
,
(4.28) which implies for ∀t < T ∗∗ a
2
˙p ∞ L t (Bp,1 )
a0
+ v
2 p B˙ p,1
2
−1+ p ˙ ∞ ) L t (Bp,1
+ δ0
2 p B˙ p,1
+ δ
exp 0
2
˙p ∞ L t (Bp,1 )
t
+ v
λf t dt
2 1+ p
L1t (B˙ p,1 )
+ δ
2 2+ p
L1t (B˙ p,1 )
X. Zhai, Z.M. Chen
a0
2 p B˙ p,1
a0
2
p B˙ p,1
+ δ0 + δ0
2 p B˙ p,1
t exp C2 w t 0
2
p B˙ p,1
+ ∇p t
2 1+ p B˙ p,1
exp C2 1 + u0
2 −1+ p B˙ p,1
2 2
−1+ p B˙ p,1
2 + w t p2 dt
B˙ p,1
exp u0 2 −1+ 2 .
(4.29)
p B˙ p,1
If we take C0 large enough and ε0 sufficiently small in (1.4), there holds a
2 ˙p ∞ L t (Bp,1 )
+ v
2 −1+ p ˙ ∞ ) L t (Bp,1
+ δ
2 ˙p ∞ L t (Bp,1 )
+ v
2 1+ p L1t (B˙ p,1 )
+ δ
2 2+ p L1t (B˙ p,1 )
≤
c2 , 2
(4.30)
for t < T ∗∗ , which contradicts to (4.25). Whence we conclude that T ∗∗ = T ∗ = ∞. This completes the proof of Theorem 1.1.
5 The Proof of Corollary 1.7 We prove Corollary 1.7 by contradiction. Let 0 < T ∗ < ∞ be the maximum time for the existence of strong solution (a, u, δ) to the system (1.2). Assume that (1.7) is not true. Then there exists a positive constant M0 such that T∗ (5.1) ω p2 + δ 2 1+ 2 dt ≤ M0 . B˙ p,1
0
p B˙ p,1
The goal is to show that if assumption (5.1) holds, there is a bound C depending only on a0 , u0 , δ0 , T ∗ and M0 such that sup a + u ≤ C. (5.2) 2 + δ 2 2 −1+ p p p ˙ ∞ L t (Bp,1 )
0
˙ ∞ L t (Bp,1
˙ ∞ L t (Bp,1 )
)
In order to do so, we first recall the well known fact that the elliptic system, div u = 0 and ω = ∇ × u implies the relation between the gradient of velocity and the vorticity as ∇u = P (ω) + cω, where P is a singular integral operator of the Calderón-Zygmund type, and c is a constant matrix. By the boundedness of the singular integral operator, we have ∇u p2 ≤ ω p2 . B˙ p,1
Therefore, if
T∗
ω(t)
0
2
p B˙ p,1
B˙ p,1
dt < ∞,
we get from Proposition 2.9 that a
2
p B˙ p,1
≤ a0
2
p B˙ p,1
exp C ω
2 p L1t (B˙ p,1 )
< ∞.
(5.3)
By a basic energy method, we can get similarly to (3.20) that δ
2
∞∗ (B˙ p ) L p,1 T
δ0
+ δ
2 p B˙ p,1
2+ 2
p L1T ∗ (B˙ p,1 )
+ 0
T∗
∇u
2 p B˙ p,1
+ δ 2 1+ 2 δ p B˙ p,1
2 p B˙ p,1
T∗
dt + 0
δ 2 1+ 2 dt. p B˙ p,1
(5.4)
Global Well-posedness for the Density-Dependent Incompressible Flow. . .
The Gronwall inequality thus implies δ
2
∞∗ (B˙ p ) L p,1 T
δ0
+ δ
2+ 2
p L1T ∗ (B˙ p,1 )
2 p B˙ p,1
T∗
+
δ 2 1+ 2 dt exp p B˙ p,1
0
T∗
ω
0
2 p B˙ p,1
+ δ 2 1+ 2 dt < ∞.
(5.5)
p B˙ p,1
Finally, from Eq. (1.2) and Proposition 3.1, we have u
2
−1+ p ∞∗ (B˙ ) L p,1 T
u0
+ u
2 −1+ p B˙ p,1
2 1+ p
L1T ∗ (B˙ p,1 )
+ ∇Π
2 1+ p
L1T ∗ (B˙ p,1 )
T∗
+
∇u 0
2
p B˙ p,1
u
2
−1+ p B˙ p,1
T∗
22m a 2
+
2 p B˙ p,1
0
u
2 −1+ p B˙ p,1
dt
T∗
dt +
1 + a 2
2 p B˙ p,1
0
δ 2 1+ 2 dt.
(5.6)
p B˙ p,1
Using the Gronwall inequality again, we have u
2
−1+ p ∞∗ (B˙ ) L p,1 T
+ u
u0
2
−1+ p B˙ p,1
< ∞.
2 1+ p
L1T ∗ (B˙ p,1 )
+
T∗
+ ∇Π
1 + a
0
2 2
p B˙ p,1
2 1+ p
L1T ∗ (B˙ p,1 )
δ
2 1+ 2
p B˙ p,1
T∗
ω
dt exp 0
2
p B˙ p,1
+ 2 a 2m
2 2
p B˙ p,1
(5.7)
Therefore, the combination of (5.3), (5.5) and (5.7) together with a standard argument, we can extend the solution (a, u, δ) beyond T ∗ . This leads to the desired contradiction. Acknowledgement
This work is supported by NSFC under grant numbers 11601533 and 11571240.
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