Nonlinear Analysis 164 (2017) 54–66
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Nonlinear Analysis www.elsevier.com/locate/na
Global well-posedness of smooth solution to the supercritical SQG equation with large dispersive forcing and small viscosity Renhui Wana, *, Jiecheng Chenb a b
School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
article
info
Article history: Received 27 April 2017 Accepted 29 August 2017 Communicated by Enzo Mitidieri MSC: 35Q35 76B03
abstract It is known that global well-posedness for the supercritical SQG equation remains open, even the smooth data. In this paper, motivated by the recent work Cannone et al. (2013), we provide a simpler approach to the proof of global well-posedness for the supercritical SQG equation with large dispersive forcing, even with small viscosity. As an application, we prove global well-posedness for the 2D dissipative Boussinesq equations with small viscosity and large background data. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Supercritical SQG equation Large dispersive forcing Small viscosity
1. Introduction In this paper, we study the supercritical SQG equation with dispersive term given by ⎧ ⎨∂t θ + u · ∇θ + A−β Λα θ + Au2 = 0, u = (u1 , u2 ) = (−R2 θ, R1 θ), ⎩ θ(0, x) = θ0 (x), (t, x) ∈ R+ × R2 ,
(1.1)
where 0 < α < 1, β > 0, the scalar function θ stands for the potential temperature of the fluid, u represents the velocity field and A is the amplitude parameter. It is obvious that the viscosity is sufficiently small provided A is large enough. Ri is the Riesz transform which can be defined by iξi ˆ ˆ R f (ξ). i f (ξ) := |ξ| When neglecting the dispersive term Au2 , and setting A = 1, we call (1.1) with 0 < α ≤ 2 the dissipative SQG equation. Based on the scaling transform and the conservation of L∞ , i.e., ∥θ(t)∥L∞ ≤ ∥θ0 ∥L∞ ,
*
Corresponding author. E-mail addresses:
[email protected] (R. Wan),
[email protected] (J. Chen).
http://dx.doi.org/10.1016/j.na.2017.08.008 0362-546X/© 2017 Elsevier Ltd. All rights reserved.
R. Wan, J. Chen / Nonlinear Analysis 164 (2017) 54–66
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one can get three kinds of cases: subcritical case (α > 1), critical case (α = 1) and supercritical case (α < 1). It is not difficult to obtain the global well-posedness for the subcritical case. For the critical case, the authors in [16] and [1] obtained the global well-posedness via using two different ways: the non-local maximal principle method and the DeGiorgi’s iteration method, respectively. Later, by a new method called nonlinear maximal principles, Constantin and Vicol [7] also obtained the global regularity for the critical case. However, it is an open problem for the supercritical case, see the works [6,22] for the small solutions. By reducing the singularity of the velocity in the inviscid and supercritical SQG equation, the authors proved the global regularity for the corresponding modified systems in [4,5]. In the presence of the dispersive term Au2 , Kiselev and Nazarnov [15] studied the critical case and proved the global well-posedness by applying the method in [16]. But the dispersive term plays a negative role in [15]. By establishing some Strichartz-type estimates, Cannone–Miao–Xue [2] obtained the global regularity for the supercritical case. Very recently, for the inviscid case, it is the paper [8] that showed long-time existence of solution to (1.1) with A = 1 by using the decay estimate 1
∥eR1 t f ∥L∞ ≤ Ct− 2 ∥f ∥B˙ 2 . 1,1
(1.2)
Moreover, in our recent work [21], we obtained the global well-posedness for the inviscid case with large A, which is consistent with the work [8] by setting A = 1. Motivated by the above studies, we consider the dispersive and supercritical SQG equation and get the following result: Theorem 1.1. Let α ∈ (0, 1), β ∈ (0, 13 ), 0 < η ≪ 1, and β1 =
3β − 1 + 5βη + η < 0. 2η + 2
Let us consider (1.1) with the initial data θ0 (x) ∈ H s (R2 ), s ≥ 4. If ) ( 1+2η 1 2 α 2 1+η ∥θ ∥ + ∥θ ∥ ≤ 1, p = CAβ1 ∥θ0 ∥B1+η , σ = 3 − η − (1 + 3η), 0 H 2+η 0 Hs ˙σ 2−η 2 p,1
(1.3)
for some positive constant C, then (1.1) admits a unique global solution θ(t, x) satisfying ∫ t 2 2 2 ∥θ∥L∞ H s + A−β ∥Λα θ(τ )∥H s dτ ≤ 4∥θ0 ∥H s . t
0
Remark 1.2. (a) In fact, for (1.1), global well-posedness for supercritical SQG equation is known only for data of size A−β (β > 0), which is small if A is large. However, thanks to the dispersive term Au2 , we can get the global result for the data of size Aγ (γ > 0), which is large if A is large. (b) [2] showed that there exists A0 > 0 sufficiently large such that if A ≥ A0 then the supercritical SQG equation has a unique global solution, where the relation among A0 , viscosity and the norm of θ0 is very complicated and was not explicitly spelled out. However, our approach gives the relation clearly. Here we do not pay much attention to the regularity of the initial data. (c) One can see the authors in [2] made great efforts in some Strichartz-type estimates, to avoid these, here we introduce the initial data in some Besov space and make use of the decay estimate of a new semi-group, see Section 2. As an application of our idea, we will consider 2D Boussinesq equations: ⎧ ⎪ ⎪∂t u + u · ∇u + ∇p − µ1 ∆u = κθe2 , ⎨ ∂t θ + u · ∇θ − µ2 ∆θ = 0, divu = 0, ⎪ ⎪ ⎩ u(0, x) = u0 (x), θ(0, x) = θ0 (x),
(1.4)
56
R. Wan, J. Chen / Nonlinear Analysis 164 (2017) 54–66
where u and θ stand for the fluid velocity and the temperature in thermal convection or the density in geophysical flows, respectively, p represents the pressure, κ is a gravitational constant and e2 is the unit vector in the vertical direction. µ1 and µ2 are positive parameters. One can get the global well-posedness for the case κ = µ1 = µ2 = 1, however, the following content devotes to the case: κ is large enough while µ1 and µ2 are small enough. Taking the operator “curl” to the both sides of the first equation in (1.4), and setting µ1 = µ2 = κ−β , then we have a new equivalent system which can be given by ⎧ ∂t ω + u · ∇ω − κ−β ∆ω = κ∂1 θ, ⎪ ⎪ ⎨ ∂t θ + u · ∇θ − κ−β ∆θ = 0, (1.5) ⎪ u = ∇⊥ (−∆)−1 ω, ⎪ ⎩ ω(0, x) = ω0 (x), θ(0, x) = θ0 (x), where ∇⊥ := (−∂2 , ∂1 ). Next, let us consider the initial data near a nontrivial equilibrium, θ0 (x) = ρ0 (x) − κx2 , and we shall seek the solution of (1.5) with the form θ(t, x) = ρ(t, x) − κx2 which implies that we shall regard the following new system ⎧ ∂t ω + u · ∇ω − κ−β ∆u = κ∂1 ρ, ⎪ ⎪ ⎨ ∂t ρ + u · ∇ρ − κ−β ∆ρ = κu2 , ⎪u = ∇⊥ (−∆)−1 ω, ⎪ ⎩ ω(0, x) = ω0 (x), ρ(0, x) = ρ0 (x).
(1.6)
This equilibrium called large background data has been studied by [8] and [21]. Our second result can be stated as follows: Theorem 1.3. Let β ∈ (0, 31 ), 0 < η ≪ 1, and β2 =
3β − 1 + 5βη + η < 0. 2η + 2
Let us consider (1.6) with the initial data ω0 ∈ H˙ −1 ∩ H˙ s−1 (R2 ) and ρ0 ∈ H s (R2 ), s ≥ 4. If C(B1 + B2 ) ≤ 1
(1.7)
for some positive constant C, where 1+2η
1
1+2η
1
2+2η 2+2η −1 B1 = κβ2 {∥(ω0 , Λρ0 )∥ 2+2η ∥(ω0 , Λρ0 )∥H ω0 , ρ0 )∥ 2+2η ∥(Λ−1 ω0 , ρ0 )∥H 1+η + ∥(Λ 1+η }, ˙ 1−4η ˙ 1−4η
B 2 2−η ,1
B 2 2−η ,1
2
2
B2 = κ2β2 (∥Λ−1 ω0 ∥H s + ∥ρ0 ∥H s ), then (1.6) admits a unique global solution (ω, ρ) satisfying ∫ t 2 2 2 2 −1 −β ∥Λ ω∥L∞ H s−1 + ∥ρ∥L∞ H s + κ (∥Λ−1 ∇ω∥H s−1 + ∥∇ρ∥H s )dτ t
t
0
2
2
≤ 4(∥Λ−1 ω0 ∥H s−1 + ∥ρ0 ∥H s ).
R. Wan, J. Chen / Nonlinear Analysis 164 (2017) 54–66
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Remark 1.4. We point out that one can get similar result for the case with fractional diffusions, i.e., ⎧ −β α ⎪ ⎪∂t ω + u · ∇ω + κ−β Λα u = κ∂1 ρ, ⎨ ∂t ρ + u · ∇ρ + κ Λ ρ = κu2 , ⎪ u = ∇⊥ (−∆)−1 ω, ⎪ ⎩ ω(0, x) = ω0 (x), ρ(0, x) = ρ0 (x), where 0 < α < 1 and 0 < β < 31 . We refer to [3,10–12,18,19] and references therein for some related results. The present paper is structured as follows: In Section 2, we provide some definitions of spaces and several lemmas. Sections 3 and 4 devote to the proof of Theorem 1.1 and Theorem 1.3, respectively. The Appendix provides the proof of Lemma 4.1. Let us complete this section by describing the notations we shall use in this paper. Notations. For A, B two operators, we denote [A, B] = AB − BA, the commutator between A and B. The uniform constant C is different on different lines. In some places of this paper, we may use Lp , H˙ s (H s ) and s s s s (R2 )), respectively. We shall denote (R2 ) (Bp,r ) to stand for Lp (R2 ), H˙ s (R2 ) (H s (R2 )) and B˙ p,r (Bp,r B˙ p,r 2 by (a|b) the L inner product of a and b, and (a|b)H˙ s stands for the standard H˙ s inner product of a and b, more precisely, (a|b)H˙ s = (Λs a|Λs b).
2. Preliminaries In this section, we give some necessary definitions, propositions and lemmas. α The fractional Laplacian operator Λα = (−∆) 2 is defined through the Fourier transform, namely, α f (ξ) = |ξ|α fˆ(ξ), Λˆ
where the Fourier transform is given by ∫ fˆ(ξ) =
e−ix·ξ f (x)dx.
R2
Let B = {ξ ∈ R2 , |ξ| ≤ 43 } and C = {ξ ∈ R2 , 34 ≤ |ξ| ≤ 38 }. Choose two nonnegative smooth radial functions χ, φ supported, respectively, in B and C such that ∑ χ(ξ) + φ(2−j ξ) = 1, ξ ∈ R2 , j≥0
∑
φ(2−j ξ) = 1, ξ ∈ R2 \ {0}.
j∈Z
˜ = F−1 χ, where In particular, φ = 1 in {ξ ∈ R2 : ≤ |ξ| ≤ 73 }. We denote φj = φ(2−j ξ), h = F−1 φ and h −1 F stands for the inverse Fourier transform. Then the dyadic blocks ∆j and Sj can be defined as follows ∫ ∆j f = φ(2−j D)f = 22j h(2j y)f (x − y)dy, 6 7
R2
Sj f =
∑
∆k f = χ(2−j D)f = 22j
k≤j−1
∫
˜ j y)f (x − y)dy. h(2
R2
Formally, ∆j = Sj − Sj−1 is a frequency projection to the annulus {ξ : C1 2j ≤ |ξ| ≤ C2 2j }, and Sj is a frequency projection to the ball {ξ : |ξ| ≤ C2j }. One easily verifies that with our choice of φ ∆j ∆k f = 0 if |j − k| ≥ 2 and ∆j (Sk−1 f ∆k f ) = 0 if |j − k| ≥ 5. Let us recall the definition of the Besov space.
R. Wan, J. Chen / Nonlinear Analysis 164 (2017) 54–66
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s Definition 2.1. Let s ∈ R, (p, q) ∈ [1, ∞]2 , the homogeneous Besov space B˙ p,q (R2 ) is defined by s B˙ p,q (R2 ) = {f ∈ S′ (R2 ); ∥f ∥B˙ p,q s (R2 ) < ∞},
where
∥f ∥B˙ p,q s (R2 ) =
⎧∑ 1 q ⎪ 2sqj ∥∆j f ∥Lp (R2 ) ) q , ⎨(
for 1 ≤ q < ∞,
j∈Z
sup 2sj ∥∆j f ∥Lp (R2 ) ,
⎪ ⎩
for q = ∞,
j∈Z
and S′ (R2 ) denotes the dual space of S(R2 ) = {f ∈ S(R2 ); ∂ α fˆ(0) = 0; ∀ α ∈ N2 {multi-index} and can be identified by the quotient space of S ′ /P with the polynomials space P. s Definition 2.2. Let s > 0, and (p, q) ∈ [1, ∞]2 , the inhomogeneous Besov space Bp,q (R2 ) is defined by s Bp,q (R2 ) = {f ∈ S ′ (R2 ); ∥f ∥Bp,q s (R2 ) < ∞},
where ∥f ∥Bp,q s (R2 ) = ∥f ∥Lp (R2 ) + ∥f ∥B ˙ s (R2 ) . p,q For the special case p = q = 2, we have ∥f ∥H˙ s (R2 ) ≈ ∥f ∥B˙ s
2,2
(R2 ) ,
where a ≈ b means C −1 b ≤ a ≤ Cb for some positive constant C, and the H˙ s (R2 ) and H s (R2 ) (s > 0) norm of f can be also defined as follows: def
∥f ∥H˙ s (R2 ) : = ∥Λs f ∥L2 (R2 ) and def
∥f ∥H s (R2 ) : = ∥f ∥L2 (R2 ) + ∥Λs f ∥L2 (R2 ) . Let us recall some product estimate and commutator estimates, see [13] and [14], respectively. Lemma 2.3. (i) Let s > 0, 1 ≤ p, r ≤ ∞, then { ∥f g∥B˙ p,r ∥f ∥Lp1 (R2 ) ∥g∥B˙ ps s (R2 ) ≤ C
(R2 ) 2 ,r
+ ∥g∥Lr1 (R2 ) ∥g∥B˙ rs
(R2 ) 2 ,r
}
,
where 1 ≤ p1 , r1 ≤ ∞ such that p1 = p11 + p12 = r11 + r12 . (ii)Let s > 0, and 1 < p < ∞, then { } ∥[Λs , f ]g∥Lp (R2 ) ≤ C ∥∇f ∥Lp1 (R2 ) ∥Λs−1 g∥Lp2 (R2 ) + ∥Λs f ∥Lp3 (R2 ) ∥g∥Lp4 (R2 ) where 1 < p2 , p3 < ∞ such that
1 p
=
1 p1
+
1 p2
=
1 p3
+
(2.1)
(2.2)
1 p4 .
The following proposition provides Bernstein type inequalities. For more details about Besov space such as some useful embedding relations, see [9,20]. Proposition 2.4. Let 1 ≤ p ≤ q ≤ ∞. Then for any β, γ ∈ (N∪{0})2 , there exists a constant C independent of f, j such that
R. Wan, J. Chen / Nonlinear Analysis 164 (2017) 54–66
59
(1) If f satisfies supp fˆ ⊂ {ξ ∈ R2 : |ξ| ≤ K2j }, then 1
1
∥∂ γ f ∥Lq (R2 ) ≤ C2j|γ|+jd( p − q ) ∥f ∥Lp (R2 ) . (2) If f satisfies supp fˆ ⊂ {ξ ∈ R2 : K1 2j ≤ |ξ| ≤ K2 2j } then ∥f ∥Lp (R2 ) ≤ C2−j|γ| sup ∥∂ β f ∥Lp (R2 ) . |β|=|γ|
We provided some lemmas. Lemma 2.5. Let 1 < p < ∞. Then 1
2
∥eR1 t f ∥Lp ≤ Ct− 2 (1− p ) ∥f ∥
2− 4
p B˙ p p−1 ,2
.
0 Proof . Thanks to the embedding relation B˙ r,2 ↪→ Lr for 1 < r < ∞. We suffice to prove ∑ 2 4 1 ∥∆j ′ f ∥ p . ∥eR1 t ∆j f ∥Lp ≤ Ct− 2 (1− p ) 2j(2− p ) L p−1
|j ′ −j|≤1
From Theorem 1.4 in [21], one can get 1
∥eR1 t ∆0 f ∥L∞ ≤ Ct− 2 ∥f ∥L1 . Combining with the L2 bound and using the Riesz–Thorin interpolation theorem, we obtain 1
2
∥eR1 t ∆0 f ∥Lp ≤ Ct− 2 (1− p ) ∥f ∥
p
L p−1
.
Following the same procedure as [21], we have ∥eR1 t ∆j f ∥Lp
g=(∆j ′ f )(2−j x)
∑
=
=
∥(e−R1 t ∆0 g)(2j x)∥Lp
|j ′ −j|≤1 ∑ 2j −p
∥e−R1 t ∆0 g∥Lp
2
|j ′ −j|≤1 1 (1− 2 ) − 2 j −2 p p
≤ Ct =
2
∑
∥g∥
|j ′ −j|≤1 2 ) j(2− 4 ) −1 (1− p p 2
Ct
2
∑
p
L p−1
∥∆j ′ f ∥
|j ′ −j|≤1
Thus, we have proved this lemma.
p
L p−1
.
□
Lemma 2.6. Let A > 0, 0 < β < 1, and η be a sufficiently small positive constant. Then we have ⎧ ⎫ ⎨ β−1+(3β+1)η ⎬ −β α 2 ∥e−A Λ t−AR1 t f ∥L∞ ≤ C min A t−1−η ∥f ∥ 2−η− α2 (1+3η) , ∥f ∥H 1+η . ⎩ ⎭ B˙ 2 2−η ,1
Proof . It is easy to get the second bound if one use the embedding inequality −β
∥e−A
Λα t−AR1 t
−β
f ∥L∞ ≤ C∥e−A
Λα t−AR1 t
f ∥H 1+η ≤ ∥f ∥H 1+η .
R. Wan, J. Chen / Nonlinear Analysis 164 (2017) 54–66
60
2
p ↪→ L∞ , thanks to Lemma 2.5, we obtain Let p = η2 . Using the embedding relation B˙ p,1
∥e−A
−β
Λα t−AR1 t
f ∥L∞ ≤ C
∑
2
−β
2
−β jα
2j p ∥∆j e−A
Λα t−AR1 t
f ∥Lp
j∈Z
≤ C
∑
2j p e−cA
j∈Z 1+3η β −1 2
≤ C(A t
2
∑
)
t
∥∆j eAR1 t f ∥Lp α
2
2j( p − 2 (1+3η)) ∥∆j eAR1 t f ∥p
j∈Z
≤ CA
β−1+(3β+1)η −1−η 2
t
∑
2
α
2j( p − 2 (1+3η)) ∥∆j f ∥
≤ CA
β−1+(3β+1)η −1−η 2
t
∥f ∥
2− 4
p B˙ p p−1 ,1
j∈Z 2−η− α (1+3η)
2 B˙ 2 2−η ,1
where we have used α
∥∆j eΛ t f ∥Lp ≤ Ce−c2
jα
t
∥∆j f ∥Lp , p ∈ [2, ∞),
the proof of which comes from the generalized Bernstein inequality, see [6]. Collecting with the upper two estimates, we can get the desired result. □
3. Proof of Theorem 1.1 Local well-posedness can be easily obtained by using the energy method [17], here it suffices to show global a priori bound. Let us keep the following equalities in mind: (AR1 θ|θ) = (AR1 θ|θ)H˙ s = 0, s > 0. Taking the L2 inner product with θ, we have 2 α 1 d 2 ∥θ∥L2 + A−β ∥Λ 2 θ∥L2 = 0. 2 dt
(3.1)
By using the product estimate and with (3.1), using the cancellation (u · ∇Λs θ|Λs θ) = 0, and applying the Young’s inequality, one can get 2 α 1 d 2 ∥θ∥H s + A−β ∥Λ 2 θ∥H s ≤ C(u · ∇θ|θ)H˙ s ≤ |([Λs , u · ∇]θ|Λs θ)| 2 dt ≤ C(∥∇u∥L∞ + ∥∇θ∥L∞ )∥θ∥H s ∥θ∥H˙ s 2 α A−β 2 2 2 ≤ CAβ (∥∇u∥L∞ + ∥∇θ∥L∞ )∥θ∥H s + ∥Λ 2 θ∥H s . 2
Applying the Gronwall’s lemma yields ∫ t ∫ t 2 α 2 2 2 2 −β β 2 (∥∇u∥L∞ + ∥∇θ∥L∞ )dτ }, ∥θ(t)∥H s + A ∥Λ θ∥L2 dτ ≤ ∥θ0 ∥H s exp{CA 0
0
so it suffices to bound the right hand side of (3.2). Denote −β
Tt f := e−A
Λα t−AR1 t
f.
(3.2)
R. Wan, J. Chen / Nonlinear Analysis 164 (2017) 54–66
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Thanks to Lemma 2.6, we have ∥∇Tt f ∥L∞ ≤ CA
β−1+(3β+1)η −1−η 2
t
∥f ∥
3−η− α (1+3η)
2 B˙ 2 2−η ,1
,
(3.3)
∥∇Tt f ∥L∞ ≤ C∥f ∥H 2+η .
(3.4)
By Duhamel principle, we have t
∫
Tt−τ (u · ∇θ)dτ.
θ(t) = Tt θ0 + 0
This yields ≤ L1 + L2 , ∥∇θ∥L2 L∞ x T
where ∫ L1 = ∥∇Tt θ0 ∥L2 L∞ , L2 = ∥∇ x T
For L1 , by (3.3) and (3.4), ∫ 2 L1 ≤ C 0
a
2 ∥θ0 ∥H 2+η dτ
T
∫
0
t
Tt−τ (u · ∇θ)dτ ∥L2 L∞ . x T
β−1+(3β+1)η −1−η 2
)2 dt 3−η− α 2 (1+3η) B˙ 2 2−η ,1 2 2η)−1 a−1−2η Aβ−1+(3β+1)η ∥θ0 ∥ 3−η− α2 (1+3η) . B˙
+
(A
t
∥θ0 ∥
a
2
≤ Ca∥θ0 ∥H 2+η + (1 +
2 2−η ,1
Choosing ⎛
2 3−η− α 2 (1+3η) B˙
(1 + 2η)−1 A(3β+1)η+β−1 ∥θ0 ∥
2 2−η ,1
⎜ ⎜ a=⎜ ⎝
1 ⎞ 2+2η
⎟ ⎟ ⎟ ⎠
2
∥θ0 ∥H 2+η
,
we have 1
L21 ≤ C(1 + 2η)− 2+2η A
β−1+(3β+1)η 2+2η
2 B˙ 2 2−η ,1
For L2 . Thanks to (3.3) and (3.4), we have ∫ t L2 ≤ C∥ ∥∇Tt−τ (u · ∇θ)∥L∞ dτ ∥L2 T ∫0 t β−1+(3β+1)η 2 ≤ C∥ min{A (t − τ )−1−η ∥u · ∇θ∥ ≤ C∥
3−η− α (1+3η)
2 B˙ 2 2−η ,1
0
∫
1+2η
1
∥θ0 ∥H1+η ∥θ0 ∥ 1+η 2+η . 3−η− α (1+3η)
t
min{A
β−1+(3β+1)η 2
, ∥u · ∇θ∥H 2+η }dτ ∥L2
T
α
(t − τ )−1−η , 1}∥θ∥H s ∥Λ 2 θ∥H s dτ ∥L2 , T
0
where we have used the following estimate ∥u · ∇θ∥H 2+η + ∥u · ∇θ∥
α
3−η− α (1+3η)
2 B˙ 2 2−η ,1
≤ ∥θ∥H s ∥Λ 2 θ∥H s ,
see the proof of (3.5) in the Appendix. Let g1 = min{A
β−1+(2β+1)η −1−η 2
t
α
, 1}, g2 = ∥θ∥H s ∥Λ 2 θ∥H s χ[0,T ] ,
(3.5)
R. Wan, J. Chen / Nonlinear Analysis 164 (2017) 54–66
62
where χ stands for the characteristic function, by following the idea of the estimate of L1 , we can get 1
∥g1 ∥L1 ≤ Cη − η+1 A
β−1+(3β+1)η 2(η+1)
.
By Young’s inequality, we infer that L2 ≤ ∥g1 ⋆ g2 ∥L2 ≤ C∥g1 ∥L1 ∥g2 ∥L2 1
≤ Cη − η+1 A
β−1+(3β+1)η 2(η+1)
α
∥Λ 2 θ∥L2 H s ∥θ∥L∞ Hs . T T
It is easy to see
2 ∥∇θ∥L2 L∞ T
can be bounded similarly. Thus combining with the above estimates, we get 2
∥(∇θ, ∇u)∥L2 L∞ ≤ CA
β−1+(3β+1)η 2+2η
1+2η
1
∥θ0 ∥ 1+η ∥θ0 ∥H1+η 2+η 3−η− α (1+3η) 2 B˙ 2 2−η ,1
T
+ CA
β−1+(3β+1)η η+1
α
2
2
∥Λ 2 θ∥L2 H s ∥θ∥L∞ H s . T
T
Denote T¯ := sup{t > 0 : ∥θ(t)∥2H s + A−β
∫
t 0
α
2
2
∥Λ 2 θ∥L2 dτ ≤ 4∥θ0 ∥H s }.
If T¯ < ∞, denote β1 =
3β − 1 + 5βη + η , 2η + 2
one obtains from (3.2) that 2
∥θ(t)∥H s + A−β ≤
2 ∥θ0 ∥H s
∫
t
0
β1
α
2
∥Λ 2 θ∥L2 dτ 1+2η
1
4
2β1 exp{CA ∥θ0 ∥ 1+η ∥θ0 ∥H1+η ∥θ0 ∥H s }. 2+η + CA 3−η− α (1+3η) 2 B˙ 2 2−η ,1
If 0 < β < 31 , we can get β1 < 0 for small η. Condition (1.3) yields 1
1+2η
4
2β1 CAβ1 ∥θ0 ∥ 1+η ∥θ0 ∥H1+η ∥θ0 ∥H s ≤ 1. 2+η + CA 3−η− α (1+3η) 2 B˙ 2 2−η ,1
Thus we can get a contradiction with the previous assumption T¯ < ∞ by the continuous arguments. This concludes the proof of Theorem 1.1. 4. Proof of Theorem 1.3 Like the previous section, we only show the global priori bound. Firstly, let us diagonalize system (1.6), denote I+ := w + Λρ, I− := w − Λρ, and then one can get {
∂t I+ + u · ∇I+ + [Λ, u · ∇]ρ − κ−β ∆I+ = κR1 I+ , ∂t I− + u · ∇I− − [Λ, u · ∇]ρ − κ−β ∆I− = −κR1 I− .
R. Wan, J. Chen / Nonlinear Analysis 164 (2017) 54–66
63
Thanks to the energy estimate in [21], we have d 2 2 2 2 (∥ω∥H˙ −1 ∩H˙ s−1 + ∥ρ∥H s ) + κ−β (∥∇w∥H˙ −1 ∩H˙ s−1 + ∥∇ρ∥H s ) dt 2 2 2 2 2 ≤ Cκβ (∥u∥L∞ + ∥ω∥L∞ + ∥ρ∥L∞ )(∥ω∥H˙ −1 ∩H˙ s−1 + ∥ρ∥H s ) κ−β 2 2 (∥∇w∥H˙ −1 ∩H˙ s−1 + ∥∇ρ∥H s ) + 2 2 2 2 ≤ Cκβ ∥(I± , Λ−1 RI± , Λ−1 I± )∥L∞ (∥ω∥H˙ −1 ∩H˙ s−1 + ∥ρ∥H s ) κ−β 2 2 (∥∇w∥H˙ −1 ∩H˙ s−1 + ∥∇ρ∥H s ). + 2 Denote Tt f := eκ
−β
∆t±κR1 t
f,
by Lemma 2.6, we have ∥Tt f ∥L∞ ≤ Cκ
β−1+(3β+1)η −1−η 2
t
∥f ∥B˙ 1−4η
(4.1)
2 2−η ,1
and ∥Tt f ∥L∞ ≤ C∥f ∥H 1+η .
(4.2)
By Duhamel principle, one gets ∫
t
Tt−τ G± (τ )dτ,
I± = Tt I± (0) + 0
where G± =: −u · ∇J± ∓ [Λ, u · ∇]ρ. Then ∥I± ∥L2 L∞ ≤ L3 + L4 , T
where ∫
t
L3 = ∥Tt I± (0)∥L2 L∞ , L4 = ∥ T
Tt−τ G± (τ )dτ ∥L2 L∞ . T
0
Next, we need a lemma. Lemma 4.1. Let 0 < η ≪ 1, s ≥ 4, then (a) ∥G± ∥B˙ 1−4η + ∥G± ∥H 1+η ≤ C∥u∥H s (∥∇ω∥H s−1 + ∥∇ρ∥H s ); 2 2−η ,1
(b) ∥Λ−1 G± ∥B˙ 1−4η + ∥Λ−1 G± ∥H 1+η ≤ C∥u∥H s (∥∇ω∥H˙ −1 ∩H s−1 + ∥∇ρ∥H s ). 2 2−η ,1
The proof of Lemma 4.1 will be given in the Appendix. Now, let us continue the proof. For L3 . Following the idea as the previous estimate of L1 , we can get L3 ≤ Cκ
β−1+(3β+1)η 4+4η
1
1+2η
2+2η ∥I± (0)∥ 2+2η ∥I± (0)∥H 1+η . ˙ 1−4η
B 2 2−η ,1
R. Wan, J. Chen / Nonlinear Analysis 164 (2017) 54–66
64
For L4 . By Lemma 4.1, (4.1) and (4.2), one can infer that ∫ t β−1+(3β+1)η 2 (t − τ )−1−η , 1}∥u∥H s (∥∇ω∥H s−1 + ∥∇ρ∥H s )dτ ∥L2 L4 ≤ C∥ min{κ
T
0
≤ C∥g3 ⋆ g4 ∥L2 , where β−1+(3β+1)η −1−η 2
t
g3 = min{κ
, 1}, g4 = ∥u∥H s (∥∇ω∥H s−1 + ∥∇ρ∥H s )χ[0,T ] .
By Young’s inequality, we get L4 ≤ C∥g3 ∥L1 (0,∞) ∥g4 ∥L2 [0,T ] ≤ Cκ
β−1+(3β+1)η 2+2η
∥u∥L∞ H s ∥∥∇ω∥H s−1 + ∥∇ρ∥H s ∥ L2 . T T
Thus we have ∥I± ∥L2 L∞ ≤ Cκ
β−1+(3β+1)η 4+4η
B 2 2−η ,1
T
+κ
1+2η
1
2+2η ∥I± (0)∥H ∥I± (0)∥ 2+2η 1+η ˙ 1−4η
β−1+(3β+1)η 2+2η
∥u∥L∞ H s ∥∥∇ω∥H s−1 + ∥∇ρ∥H s ∥L2 . T T
−1
For the estimate of ∥(Λ
I± , Λ
−1
RI± )∥L2 L∞ , using Lemma 4.1 again, one have T
∥(Λ−1 I± , Λ−1 RI± )∥L2 L∞ ≤ Cκ
β−1+(3β+1)η 4+4η
1+2η
B 2 2−η ,1
T
+κ
1
2+2η ∥I± (0)∥ 2+2η ∥Λ−1 I± (0)∥H 1+η ˙ −4η
β−1+(3β+1)η 2+2η
∥u∥L∞ H s ∥∥∇ω∥H s−1 + ∥∇ρ∥H s ∥L2 . T T
Denote { } 2 T¯ := sup t > 0 : A(t) ≤ 4(∥ω0 ∥2H˙ −1 ∩H˙ s−1 + ∥ρ0 ∥H s ) , where 2
2
A(t) = ∥ω(t)∥H˙ −1 ∩H˙ s−1 + ∥ρ(t)∥H s + κ−β
∫ 0
t
2
2
∥∇ω∥H˙ −1 ∩H˙ s−1 + ∥∇ρ∥H s dτ.
If T¯ < ∞, let β2 =
3β − 1 + 5βη + η , 2 + 2η
thanks to the above estimates, we have 2
2
A(t) ≤ (∥Λ−1 ω0 ∥H s + ∥ρ0 ∥H s ) exp{C(B1 + B2 )}, where 1
1+2η
1
1+2η
2+2η 2+2η 2+2η ∥I± (0)∥H ∥Λ−1 I± (0)∥H B1 = κβ2 (∥I± (0)∥ 2+2η 1+η + ∥I± (0)∥ ˙ −4η 1+η ), ˙ 1−4η
B 2 2−η ,1
B 2 2−η ,1
2
2
B2 = κ2β2 (∥Λ−1 ω0 ∥H s + ∥ρ0 ∥H s ). Thanks to (1.7), we can get C(B1 + B2 ) ≤ 1, which yields a contradiction with the assumption T¯ < ∞ by the continuous arguments. Therefore, T¯ = ∞. This completes the proof of Theorem 1.3.
R. Wan, J. Chen / Nonlinear Analysis 164 (2017) 54–66
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Acknowledgment This paper is partially supported by NSF of China under Grant 11671363. Appendix In this Appendix, we give the proof of (3.5) and Lemma 4.1. Proof of 3.5. By using (2.1), we have ∥u · ∇θ∥H 2+η ≤ C(∥u∥L∞ ∥∇θ∥H 2+η + ∥∇θ∥L∞ ∥u∥H 2+η ) α ≤ C∥u∥H s ∥∇θ∥H 3 ≤ C∥θ∥H s ∥Λ 2 θ∥H s . Similar arguments yield ∥u · ∇θ∥
α
3−η− α (1+3η)
2 B˙ 2 2−η ,1
≤ C∥θ∥H s ∥Λ 2 θ∥H s .
So we can get the desired inequality (3.5). □ Proof of Lemma 4.1. Due to the similarity, we only show the proof of (b). Due to the fact that divu = 0, we can get one additional derivative from G± . We have ∥Λ−1 (u · ∇I± )∥H 1+η ≤ C∥u ⊗ I± ∥H 1+η ≤ C∥u∥H 1+η ∥(ω, ∇ρ)∥H 1+η . By product estimate and Bernstein’s inequality, then ∥Λ−1 (u · ∇I± )∥B˙ 1−4η ≤ C∥u ⊗ I± ∥B˙ 1−4η 2 2−η ,1
2 2−η ,1
≤ C∥u∥L2 ∥(ω, Λρ)∥B˙ 1−2η 2,1
+ C∥u∥B˙ 1−3η ∥(ω, Λρ)∥L2 . 2,1
One can easily see ∥Λ−1 (u · ∇I± )∥H 1+η + ∥Λ−1 (u · ∇I± )∥B˙ 1−4η ≤ ∥u ⊗ I± ∥H 1+η + ∥u ⊗ I± ∥B˙ 1−4η . 2 2−η ,1
2 2−η ,1
Following the proof of (3.5), one can easily get ∥u ⊗ I± ∥H 1+η + ∥u ⊗ I± ∥B˙ 1−4η ≤ C∥u∥H s ∥(Λ−1 ω, ∇ρ)∥H s . 2 2−η ,1
For the second term, we can split the commutator into two terms, and following the same idea can lead to the desired result. So we can complete the proof.
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