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L2− asymptotic stability of the mild solution to the 3D MHD equation
Applied Mathematics and Computation 269 (2015) 443–455
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
L2 − asymptotic stability of the mild solution to the 3D MHD equation Baoquan Yuan∗, Linna Bai School of Mathematics and Information Science, Henan Polytechnic University, Henan, 454000, China
a r t i c l e
i n f o
a b s t r a c t
MSC classiffication: 35B40 35D30 35Q35
In this paper, we investigate the L2 − asymptotic stability of a global-in-time mild solution of the small initial value problem to the 3D magnetohydrodynamic (MHD) equations. We construct a weak solution to the MHD equation, and then we show this weak solution converges to the mild solution in the sense of L2 energy norm.
Keywords: Mild solution 3D MHD equation Weak solution
© 2015 Elsevier Inc. All rights reserved.
1. Introduction In this paper, we consider the following 3D magnetohydrodynamic (MHD) equations
ut + u · ∇ u − b · ∇ b − u + ∇ p = 0, bt + u · ∇ b − b · ∇ u − b = 0, ∇ · u = ∇ · b = 0, u(x, 0) = u0 (x), b(x, 0) = b0 (x),
(1.1)
where u = (u1 (x, t ), u2 (x, t ), u3 (x, t )) is the velocity of the flow, b = (b1 (x, t ), b2 (x, t ), b3 (x, t )) is the magnetic field, p is the scalar 1
pressure, and (u0 , b0 ) ∈ H 2 (R3 ) is the given initial data. It is well-known that the global-in-time solution with the small initial data to the Eq. (1.1) was obtained in reference [11]. 1 They showed that for (u0 , b0 ) ∈ H˙ 2 (R3 ), there exists a positive time T such that the Eq. (1.1) has a unique solution (u, b) in L4 ([0, T ]; H˙ 1 (R3 )) which also belongs to
C ([0, T ]; H˙ 2 (R3 )) ∩ L2 ([0, T ]; H˙ 2 (R3 )). 1
3
(1.2)
Let T(u0 ,b0 ) denote the maximal time of existence of such a solution, if there exists ε 0 > 0 such that
(u0 , b0 )H˙ 12 ≤ ε0 ,
(1.3)
then T(u0 ,b0 ) = ∞. Moreover, (u, b) satisfies the energy equality
u(t )2L2 + b(t )2L2 + 2
0
t
(∇ u(t )2L2 + ∇ b(t )2L2 )dτ = u0 2L2 + b0 2L2 .
(1.4)
The purpose of this paper can be described as follows. Assume that (V(x, t), B(x, t)) is a global-in-time mild solution of the small initial value problem (1.1) with (V0 , B0 ) 1 ≤ min{ε0 , 1}, then we show that the system (1.1) has a weak solution (u(x, t), H2
∗
Corresponding author. Tel.: +863913987808. E-mail addresses: [email protected], [email protected] (B. Yuan), [email protected] (L. Bai).
http://dx.doi.org/10.1016/j.amc.2015.07.006 0096-3003/© 2015 Elsevier Inc. All rights reserved.
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B. Yuan, L. Bai / Applied Mathematics and Computation 269 (2015) 443–455
b(x, t)) in the sense of Leray corresponding to the initial datum (V0 , B0 ) perturbed by an arbitrarily large divergence free L2 −vector and magnetic fields, and this weak solution converges to the mild solution (V(x, t), B(x, t)) in the sense of L2 −norm as t → ∞. More specifically, we denote (u(x, t), b(x, t)) a weak solution to the Eq. (1.1) with the initial data u0 = V0 + w0 , b0 = B0 + h0 , where 1
(w0 , h0 ) ∈ H 2 (R3 ), then the functions w(x, t ) = u(x, t ) − V (x, t ), h(x, t ) = b(x, t ) − B(x, t ) satisfy the following perturbed initial value problem
⎧ ⎪ ⎨wt + w · ∇ w − h · ∇ h − w = −w · ∇V − V · ∇ w + h · ∇ B + B · ∇ h − ∇π , ht + w · ∇ h − h · ∇ w − h = −w · ∇ B + B · ∇ w + h · ∇ V − V · ∇ h, ∇ ⎪ ⎩ · w = ∇ · h = 0, w(x, 0) = w0 (x), h(x, 0) = h0 (x).
(1.5)
Therefore, we first construct a weak solution (w, h) of the initial problem (1.5), then we show the L2 −norms of w and h tend to 0 as t → ∞. On the topic of asymptotic stability for the solutions, there have been many classical results to the Navier–Stokes equation [1,4–8,10,12–14]. The energy decay problem of weak solutions to the Navier–Stokes equation was originally suggested by Leray in his pioneering papers [9,10]. Later, Schonbek [13] proved the algebraic decay for the Cauchy problem in R3 for large data in L1 ∩ L2 by using the approximate solutions of Caffarelli et al. [3]. Kajikiya and Miyakawa [6] improved the result of [13] by a systematic use of a modified version of the method in [13]. Wiegner [14] also showed the decay results for weak solutions to the Navier–Stokes equation. In 1997, Ogawa et al. investigated the energy decay problem of the weak solution to the Navier–Stokes equation with slowly varying external forces [12]. Later, Kozono studied the weak solutions u of the Navier–Stokes equations in Serrin’s class u ∈ Lα (0, ∞; Lq ()) for α2 + 3q = 1 with 3 < q ≤ ∞. They show that although the initial and external disturbances from u are large, the weak solution of the perturbed problem can also decay to zero as t → ∞ [5]. Recently, in [8], Karch et al. established an asymptotic stability for a global-in-time small mild solution to the Navier–Stokes equation in R3 under arbitrarily large initial L2 − perturbations. Motivated by Ogawa et al. [12] and Karch et al. [8], in our paper, we discuss the L2 − asymptotic stability of the mild solution to the MHD system in R3 . The key idea is to construct a weak solution (w(x, t), h(x, t)) of the perturbed Eq. (1.5) in the sense of Leray–Hopf by the Fridrich method, and then we show them L2 − decay to zero as t → ∞. To this end, we state our results. Theorem 1.1 (Existence of the weak solutions). Let (V = V (x, t ), B = B(x, t )) be a global-in-time solution to the small initial value problem(1.1) with V0 1 + B0 1 ≤ min{ε0 , 1}. Denote V0 = V (·, 0), B0 = B(·, 0) and let (w0 , h0 ) ∈ L2 (R3 ) be arbitrary. Then, the H2
H2
Cauchy problem (1.1) with the initial condition u0 = V0 + w0 , b0 = B0 + h0 has a global weak solution (u, b) of the form
u(x, t ) = V (x, t ) + w(x, t ), b(x, t ) = B(x, t ) + h(x, t ),
where (w(x, t), h(x, t)) is a weak solution of the corresponding perturbed problem (1.5) satisfying
(w, h) ∈ Cw ([0, T ]; L2 (R3 )) ∩ L2 ([0, T ]; H˙ 1 (R3 )) f or each T > 0. Theorem 1.2 (Asymptotic behavior of mild solutions). As t → ∞, the weak solution (u(x, t), b(x, t)) of the system (1.1) tend to the mild solution (V = V (x, t ), B = B(x, t )) in the sense of L2 −norm. That is to say
w(t )2 = u(t ) − V (t )2 → 0, h(t )2 = b(t ) − B(t )2 → 0,
where (w(x, t), h(x, t)) is constructed in Theorem 1.1. The rest of this paper is organized as follows. In Section 2, we give some definitions and Lemmas which will be used in the following proofs. Sections 3 and 4 devoted to prove Theorems 1.1 and 1.2, respectively. 2. Preliminaries In this section we introduce the definitions of Leray–Hopf weak solution and the cut-off operator, and then recall some lemmas and notations which will be used in the following parts. Definition 2.1. For (u0 (x), b0 (x)) ∈ L2 (R3 ) and T > 0. We call (u, b) the weak solution of Leray–Hopf type (Leray–Hopf weak solution) if and only if 1. (u, b) ∈ L∞ (0, T ; L2 (R3 )) ∩ L2 (0, T ; H˙ 1 (R3 )); t t t 2. Rd u(x, t ) · ϕ(x)dx − Rd u(x, 0) · ϕ(x)dx + 0 Rd u · ∇ u · ϕ(x)dxdτ − 0 Rd b · ∇ b · ϕ(x)dxdτ = 0 Rd u(x, τ ) · ϕ(x)dxdτ ; t t t Rd b(x, t ) · ϕ(x)dx − Rd b(x, 0) · ϕ(x)dx + 0 Rd u · ∇ b · ϕ(x)dxdτ − 0 Rd b · ∇ u · ϕ(x)dxdτ = 0 Rd b(x, τ ) · ϕ(x)dxdτ for any test functions ϕ ∈ C0∞ (R3 ) satisfying ∇ · ϕ = 0.
B. Yuan, L. Bai / Applied Mathematics and Computation 269 (2015) 443–455
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Definition 2.2. Let N > 0 be an integer, χ (ξ ) be a characteristic function satisfying
χ (ξ ) =
1, 0,
if if
ξ ∈ B(0; N), ξ ∈ B(0; N),
where B(0; N) denotes the ball of radius N centered at the origin. Then the cut-off operator JN is defined as
ˆ J
N f (ξ ) = χB(0;N) (ξ ) f (ξ ). Remark 2.1. We denote L2N = { f ∈ L2 (Rd )| supp fˆ ⊂ B(0; N)}. If f ∈ L2N , then f ∈ Hs for any s > 0, and JN f L2 ≤ f L2 . Lemma 2.1. (Hodge decomposition) For any vector field u ∈ L2 (Rd ) ∩ C ∞ (Rd ), there exist w and p such that u = w + ∇ p with w and p satisfying 1. w ∈ L2 (Rd ) ∩ C ∞ (Rd ), ∇ p ∈ L2 (Rd ) ∩ C ∞ (Rd ) and ∇ · w = 0; 2. Rd w · ∇ pdx = 0; 3. Dα u22 = Dα w22 + Dα ∇ p22 for any multi-index α ≥ 0. L
L
L
Now let’s introduce the operator P, an orthogonal projection operator, which maps u onto its divergence-free part w and we denote Pu = w, moreover PuL2 = wL2 ≤ uL2 . Lemma 2.2. (Picard Lemma) Let B be a Banach space and O ⊂ B be an open set. Assume F: O → B satisfies the local Lipschitz condition:
∀ X ∈ O, there exists a neighborhood of X denoted by U = U (X ) and a Lipschitz constant L = L(X ) such that F (X ) − F (X )B ≤ L(X )X − X B for any X , X ∈ U (X ). Then, for X0 ∈ U, the ODE
dX = F (X ), dt X (0) = X0 ,
(2.1)
has a unique local solution, that is , ∃ T > 0 and X ∈ C1 ([0, T]; O) such that X solves (2.1). In addition, either X = X (t ) exists for all time or ∃ T∗ > 0 such that X(t) leaves O as t → T∗ . Lemma 2.3. (Lions–Aubin Lemma) Let q1 , q2 ∈ (1, ∞) and δ 2 < δ 1 be real numbers. Assume {fm } satisfies 1. {fm } is uniformly bounded in Lq1 ([0, T ]; H δ1 (Rd )); 2. For any ρ ∈ C0∞ (Rd ), ρ ∂∂ftm in Lq2 ([0, T ]; H δ2 (Rd )) is uniformly bounded, or ρ fm (x, t1 ) − ρ fm (x, t2 )H δ2 ≤ C |t1 − t2 |. Then, there exists a subsequence of fm (still denoted by fm ) and f ∈ Lq1 ([0, T ]; H δ1 (Rd )) such that, for any ρ ∈ C0∞ (Rd ), ρ fm − ρ f → 0 in Lq1 ([0, T ]; H δ (Rd )), for any δ ∈ (δ 2 , δ 1 ). Remark 2.2. For details of the proofs of Lemma 2.3, please refer to reference [2]. Now we introduce some notations. 1. S(R3 ) is the Schwartz class of rapidly decreasing functions; 2. f, g stands for the inner-product which is defined as f, g = f (x)g(x)dx; 3. The Fourier transform of a function f(x) is defined by 3 fˆ(ξ ) = (2π )− 2
R3
e−ix·ξ f (x)dx.
Throughout this paper, C denotes a generic positive constant which may change from one line to the other line. 3. Existence of weak solution to the perturbed equations In this section, we shall employ Friedrichs’ methods to prove Theorem 1.1, the proof is divided into three steps. Step 1: Local existence of an approximate solution. Fix an integer N > 0, and consider
⎧ N N N N N N ⎪ ⎪∂t w − (PJN w ) + PJN (PJN w · ∇ PJN w ) − PJN (PJN h · ∇ PJN h ) ⎪ ⎪ N N N N ⎪ ⎪ ⎨ = −PJN (PJN w · ∇V ) − PJN (V · ∇ PJN w ) + PJN (PJN h · ∇ B) + PJN (B · ∇ PJN h ), N N N N N N ∂t h − ( JN h ) + JN ( JN w · ∇ JN h ) − JN ( JN h · ∇ JN w ) ⎪ ⎪ ⎪ = −JN ( JN wN · ∇ B) − JN (V · ∇ JN hN ) + JN ( JN hN · ∇ V ) + JN (B · ∇ JN wN ), ⎪ ⎪ ⎪ ⎩ N w (x, 0) = JN w0 (x), hN (x, 0) = JN h0 (x).
(3.1)
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Now we show the existence and uniqueness of an approximate solution. We apply the Picard Lemma with B = O = L2N .
⎧ N ∂t w = (PJN wN ) − PJN (PJN wN · ∇ PJN wN ) + PJN (PJN hN · ∇ PJN hN ) ⎪ ⎪ ⎪ ⎪ ⎨ −PJN (PJN wN · ∇V ) − PJN (V · ∇ PJN wN ) + PJN (PJN hN · ∇ B) + PJN (B · ∇ PJN hN ) F1 (wN , hN ), ⎪ ∂t hN = ( JN hN ) − JN ( JN wN · ∇ JN hN ) + JN ( JN hN · ∇ JN wN ) − JN ( JN wN · ∇ B) ⎪ ⎪ ⎪ ⎩ −JN (V · ∇ JN hN ) + JN ( JN hN · ∇ V ) + JN (B · ∇ JN wN ) F2 (wN , hN ).
(3.2)
First, we verify that if (wN , hN ) ∈ L2N , then F1 (wN , hN ), F2 (wN , hN ) ∈ L2N .
F1 (wN , hN )L2 ≤ PJN wN · ∇ PJN wN L2 + PJN hN · ∇ PJN hN L2 + (PJN wN )L2 + PJN wN · ∇ V L2 + V · ∇ PJN wN L2 + PJN hN · ∇ BL2 + B · ∇ PJN hN L2 I1 + I2 + I3 + I4 + I5 + I6 + I7 .
(3.3)
¨ We successively estimate I1 to I7 . Combining Holder inequality, Plancherel identity and properties of the Fourier transform, we obtain
I1 ≤ PJN wN L∞ ∇ PJN wN L2 ≤ CN 2 wN 2L2 . 5
Similarly, it can be shown that
I2 ≤ CN 2 hN 2L2 . 5
By the Plancherel identity we have
I3 ≤ ( JN wN )L2 = From (1.4), we know
t 0
|ξ |≤N
2 N ||ξ |2 J N w (ξ )| dξ
12
≤ N2 wN L2 .
∇V 2L2 dτ < ∞, it can be deduced that for almost every t, ∇V L2 < ∞, hence
I4 ≤ PJN wN L∞ ∇ V L2 ≤ CN 2 wN L2 ∇ V L2 3
for a.e. t.
Arguing similarly to I4 it can be derived, for a.e. t, that
I6 ≤ CN 2 hN L2 ∇ BL2 , 3
and
I5 ≤ V L2 ∇ PJN wN L∞ ≤ CN 2 V L2 wN L2 , 5
I7 ≤ CN 2 BL2 hN L2 . 5
Therefore, inserting the above estimates into (3.3), one has
F1 (wN , hN )L2 ≤ CN 2 (wN 2L2 + hN 2L2 + wN L2 + hN L2 ) + N 2 (wN L2 + hN L2 )(∇V L2 + ∇ BL2 ) < ∞. 5
3
Use an argument similar to that used in evaluating F1 (wN , hN ) to obtain that
F2 (wN , hN )L2 ≤ CN 2 (wN 2L2 + hN 2L2 + wN L2 + hN L2 ) + N 2 (wN L2 + hN L2 )(∇V L2 + ∇ BL2 ) < ∞. 5
3
N N N 2 Next we verify F1 , F2 satisfy the Lipschitz condition. Let (wN 1 , h1 ) and (w2 , h2 ) ∈ LN ,
F1 (wN1 , hN1 ) − F1 (wN2 , hN2 )L2 ≤ PJN wN1 · ∇ PJN wN1 − PJN wN2 · ∇ PJN wN2 L2 + PJN hN1 · ∇ PJN hN1 − PJN hN2 · ∇ PJN hN2 L2 +PJN (wN1 − wN2 )L2 + PJN wN1 · ∇ V − PJN wN2 · ∇ V L2 +V · ∇ PJN wN1 − V · ∇ PJN wN2 L2 + PJN hN1 · ∇ B − PJN hN2 · ∇ BL2 +B · ∇ PJN hN1 − B · ∇ PJN hN2 L2 Q1 + Q2 + Q3 + Q4 + Q5 + Q6 + Q7 . ¨ Holder inequality, Plancherel identity and properties of the Fourier transform together give
Q1 ≤ PJN (wN1 − wN2 ) · ∇ PJN wN1 L2 + PJN wN2 · ∇ PJN (wN1 − wN2 )L2 ≤ N 2 (wN1 L2 + wN2 L2 )wN1 − wN2 L2 . 5
By an analogous argument with Q1 , we have
Q2 ≤ N 2 (hN1 L2 + hN2 L2 )hN1 − hN2 L2 , 5
Q3 ≤ PJN (wN1 − wN2 )L2 ≤ N2 wN1 − wN2 L2 ,
Q4 ≤ N 2 wN1 − wN2 L2 ∇ V L2 , and Q6 ≤ N 2 hN1 − hN2 L2 ∇ BL2 for a.e. t, 3
Q5 ≤ N 2 V L2 wN1 − wN2 L2 , 5
3
Q7 ≤ N 2 BL2 hN1 − hN2 L2 . 5
(3.4)
B. Yuan, L. Bai / Applied Mathematics and Computation 269 (2015) 443–455
447
Combining the above estimates, we obtain, for almost every t, that
F1 (wN1 , hN1 ) − F1 (wN2 , hN2 )L2 ≤ CN 2 (V H1 + wN1 L2 + wN2 L2 )wN1 − wN2 L2 5
+ CN 2 (BH 1 + hN1 L2 + hN2 L2 )hN1 − hN2 L2 . 5
Using a similar argument of the estimate to
F1 (wN1 , hN1 ) − F1 (wN2 , hN2 )L2
(3.5)
yields
F2 (wN1 , hN1 ) − F2 (wN2 , hN2 )L2 ≤ CN (BH1 + hN1 L2 + hN2 L2 )wN1 − wN2 L2 5 2
+ CN 2 (V H 1 + wN1 L2 + wN2 L2 )hN1 − hN2 L2 . 5
(wN , hN )
C 1 ([0, T ]; L2N )
∈ such By Picard’s Lemma, ∃ T > 0 and Note that Eq. (3.1) is equivalent to the following system
that(wN ,
hN )
(3.6)
solves (3.1).
⎧ N ∂t w − wN + PJN (wN · ∇ wN ) − PJN (hN · ∇ hN ) = −PJN (wN · ∇V ) ⎪ ⎪ ⎪ ⎪ ⎪ −PJN (V · ∇ wN ) + PJN (hN · ∇ B) + PJN (B · ∇ hN ), ⎪ ⎨ ∂t hN − hN + JN (wN · ∇ hN ) − JN (hN · ∇ wN ) = −JN (wN · ∇ B) ⎪ ⎪ ⎪ −JN (V · ∇ hN ) + JN (hN · ∇V ) + JN (B · ∇ wN ), ⎪ ⎪ ⎪ ⎩ ∇ · wN = 0, ∇ · hN = 0.
(3.7)
In fact, since (wN , hN ) is a solution of (3.1) and JN2 = JN , P2 = P, taking P and JN on both sides of (3.1) respectively, and by the uniqueness of solutions, we have wN = PwN = JN wN , hN = JN hN . Multiplying the two equations of (3.7) by wN and hN , respectively, integrating and adding the resulting equations together it follows that
1 d (wN 2L2 + hN 2L2 ) + ∇ wN 2L2 + ∇ hN 2L2 = − PJN (wN · ∇V ) · wN dx 2 dt R3 N N N N + PJN (h · ∇ B) · w dx − JN (w · ∇ B) · h dx + JN (hN · ∇ V ) · hN dx R3
R3
R3
J1 + J2 + J3 + J4 .
(3.8)
¨ Using the Holder inequality and Sobolev embedding one can deduce
N N
|J1 | = (w ⊗ V ) : PJN ∇ w dx
≤ C wN L6 V L3 ∇ wN L2 3 R ≤ C wN H˙ 1 V ˙
1
H2
∇ wN L2 = C V H˙ 12 ∇ wN 2L2 .
Similarly, it can be shown that
|J2 | ≤ BH˙ 12 hN H˙ 1 ∇ wN L2 , |J3 | ≤ BH˙ 12 wN H˙ 1 ∇ hN L2 , |J4 | ≤ V H˙ 12 ∇ hN 2L2 . Inserting the above estimates into (3.8), and taking the initial value small enough so that V inequality implies
wN 2L2 + hN 2L2 +
t 0
1
H2
+ B
1
H2
< min{ε0 , 1}, Gronwall
(∇ wN 2L2 + ∇ hN 2L2 )dτ ≤ w0 2L2 + h0 2L2 .
(3.9)
By Picard Lemma, (wN , hN ) exists for all time. There exists a subsequence of (wN , hN ) (still denoted by (wN , hN )) and (w, h) in L∞ ([0, ∞); L2 ) ∩ L2 ([0, ∞); H 1 ) such that wN w, hN h in the weak sense of L∞ ([0, T ); L2 ) ∩ L2 ([0, T ); H 1 ) for any T > 0. Step 2: Strong convergence. 4
For any ρ ∈ C0∞ (Rd ), we show (ρ∂t wN , ρ∂t hN ) ∈ L 3 ([0, T ]; H −1 (R3 )). By the definition of H −1
ρ∂t wN
= H −1
sup
g∈H 1 ,gH 1
< ρ∂ wN , g > .
=1
Multiply both sides of the first equation of (3.7) by ρ g and integrate it over R3 to get
R3
ρ∂t wN · g(x)dx = − + +
R3
R
3
R3
ρ PJN (wN · ∇ wN ) · g(x)dx + ρ wN · g(x)dx −
R3
R3
ρ PJN (hN · ∇ hN ) · g(x)dx
ρ PJN (V · ∇ wN ) · g(x)dx
ρ PJN (hN · ∇ B) · g(x)dx −
L1 + L2 + L3 + L4 + L5 + L6 + L7 .
R3
ρ PJN (wN · ∇V ) · g(x)dx +
R3
ρ PJN (B · ∇ hN ) · g(x)dx (3.10)
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¨ Applying the Holder and Gagliardo–Nirenberg inequalities, one has
1 3 |L1 | =
− wN ⊗ wN : PJN ∇(ρ g)
≤ wN 2L4 ∇(ρ g)L2 ≤ C wN L22 wN 2˙ 1 gH 1 . H R3
Similarly, it can be shown that 1
3
|L2 | ≤ C hN L22 hN H2˙ 1 . ¨ The estimate of L3 can also be obtained by Holder inequality
|L3 | =
− ∇ρ · ∇ wN · gdx − ρ · ∇ wN · ∇ gdx
≤ C wN H˙ 1 . R3
R3
Arguing similarly as the estimate of L1 we derive that 1
3
1
3
|L4 | ≤ V L4 wN L4 ∇(ρ g)L2 ≤ C V L42 V H4˙ 1 wN L42 wN H4˙ 1 , and 1
3
1
3
1
3
1
3
1
3
1
3
|L5 | ≤ C BL42 BH4˙ 1 hN L42 hN H4˙ 1 , |L6 | ≤ C V L42 V H4˙ 1 wN L42 wN H4˙ 1 , |L7 | ≤ C BL42 BH4˙ 1 hN L42 hN H4˙ 1 . Thus it follows that
ρ∂t wN
3
3
3
3
H
H
H
H
≤ C 1 + wN 2˙ 1 + hN 2˙ 1 + V 2˙ 1 + B 2˙ 1 . H −1
(3.11)
Consequently, one has
ρ∂t wN
≤ C (T, V0 L2 , B0 L2 , w0 L2 , h0 L2 ).
4
(3.12)
L 3 ([0,T ];H −1 )
Arguing similarly to that used in deriving (3.12) it gives
ρ∂t hN
≤ C (T, V0 L2 , B0 L2 , w0 L2 , h0 L2 ).
4
(3.13)
L 3 ([0,T ];H −1 )
By the Lions–Aubin Lemma, there exists (wN , hN ) such that, for any ρ ∈ C0∞ (R3 ),
(ρ wN , ρ hN ) → (ρ w, ρ h) in L2 ([0, T ]; L2 (R3 )).
(3.14)
Step 3: Passing to the limit for the weak solution (w, h) of (1.5).
Since (wN , hN ) is the smooth solution of the Eq. (3.7), then for any test function ϕ ∈ C0∞ (R3 ) and ∇ · ϕ = 0, multiplying the first equation of (3.7) by ϕ , and integrating it over R3 × [0, t ), we have the identity (3.15). Let (w, h) be the limit of the approximate solution (wN , hN ) as N → ∞. Our aim is to prove that the limit function (w, h) is a weak solution to (1.5). First, we pass the limit for the first equation
R3
wN (x, t )ϕ(x)dx − − + −
t 0
t 0
t 0
R3
R3
R3
R3
wN0 (x, t )ϕ(x)dx =
hN ⊗ hN : PJN ∇ϕ(x)dxdτ + wN ⊗ V : PJN ∇ϕ(x)dxdτ + hN ⊗ B : PJN ∇ϕ(x)dxdτ −
As N → ∞, from Step 2, we obtain
R3
and
t 0 t
R3
R3 0 t 0
R3
R3
(wN − w)ϕ dxdτ ≤
Obviously,
(
wN0
− w0 )∇ϕ dx ≤ C
wN ⊗ wN : PJN ∇ϕ(x)dxdτ
wN · ϕ(x)dxdτ V ⊗ wN : PJN ∇ϕ(x)dxdτ
B ⊗ hN : PJN ∇ϕ(x)dxdτ .
(3.15)
ρ wN − ρ w2L2 dτ → 0, so ρ wN − ρ wL2 → 0 for a.e. t. Hence, we can easily get
0
1
0
R3
R3
0
(wN − w)ϕ dx ≤ (wN − w)ϕ 2 L2 ϕ 2 L2 → 0
t
T
t
0
t
1
as N → ∞,
(3.16)
(wN − w)(ϕ) 2 L2 (ϕ) 2 L2 dτ → 0 as N → ∞.
(3.17)
1
|w0 | dx 2
|ξ |≥N
We now estimate the nonlinear terms of (3.15).
1
12 →0
as N → ∞.
(3.18)
B. Yuan, L. Bai / Applied Mathematics and Computation 269 (2015) 443–455
t
[wN ⊗ wN : PJN ∇ϕ(x) − w ⊗ w : ∇ϕ(x)]dxdτ
R3
0
t
≤
449
R3
0
(wN ⊗ wN ) : ( JN ∇ϕ − ∇ϕ)dxdτ +
t R3
0
(wN ⊗ wN : ∇ϕ − w ⊗ w : ∇ϕ)dxdτ
M1 + M2 , ¨ By Holder and Gagliardo–Nirenberg inequalities M1 can be estimated as
M1 ≤
t
0
wN 2L4 dτ · JN ∇ϕ − ∇ϕL2 ≤ C (T, w0 L2 )JN ∇ϕ − ∇ϕL2 → 0
as N → ∞,
¨ Holder inequality and (3.14) together give
t
M2 =
R3
0
(wNj wNk − w j wk )∂ j ϕk dxdτ
√ T w0 L2 ∂ j ϕk (wN − w)L2 → 0
≤
as N → ∞.
Hence,
t R3
0
[wN ⊗ wN : PJN ∇ϕ(x) − w ⊗ w : ∇ϕ(x)]dxdτ → 0
as N → ∞.
(3.19)
Using an argument similar to that used in deriving (3.19) it yields
t
R3
0
[hN ⊗ hN : PJN ∇ϕ(x) − h ⊗ h : ∇ϕ(x)]dxdτ → 0 as N → ∞.
(3.20)
The estimate of the fourth to seventh terms on the right hand-side of (3.15) can be handled by precisely the same process as (3.19).
t
R3
0
t R3
0
t R3
0
and
t R3
0
[wN ⊗ V : PJN ∇ϕ(x) − w ⊗ V : ∇ϕ(x)]dxdτ → 0 as N → ∞,
[V ⊗ wN : PJN ∇ϕ(x) − V ⊗ w : ∇ϕ(x)]dxdτ → 0
R3
as N → ∞,
(3.23)
[B ⊗ hN : PJN ∇ϕ(x) − B ⊗ h : ∇ϕ(x)]dxdτ → 0
as N → ∞.
(3.24)
w(x, t )∇ϕ(x)dx − − + −
t 0
R3
0
R3
0
R3
t t
R3
w0 (x, t )∇ϕ(x)dx =
h ⊗ h : ∇ϕ(x)dxdτ +
t
w ⊗ V : ∇ϕ(x)dxdτ + h ⊗ B : ∇ϕ(x)dxdτ −
R3
0
R3
0
t
R3
0
R3
h(x, t )∇ϕ(x)dx −
− + −
t 0
R3
0
R3
0
R3
t t
R3
V ⊗ h : ∇ϕ(x)dxdτ + h ⊗ V : ∇ϕ(x)dxdτ −
t 0
R3
0
R3
0
R3
t t
R3
0
w ⊗ w : ∇ϕ(x)dxdτ
V ⊗ w : ∇ϕ(x)dxdτ
B ⊗ h : ∇ϕ(x)dxdτ .
h0 (x, t )∇ϕ(x)dx =
h ⊗ w : ∇ϕ(x)dxdτ +
t
w · ϕ(x)dxdτ
t
Using an identical process we can deduce that
(3.22)
[hN ⊗ B : PJN ∇ϕ(x) − h ⊗ B : ∇ϕ(x)]dxdτ → 0
Combing (3.16)–(3.24), one has
as N → ∞,
(3.21)
t 0
R3
(3.25)
w ⊗ h : ∇ϕ(x)dxdτ
h · ϕ(x)dxdτ w ⊗ B : ∇ϕ(x)dxdτ B ⊗ w : ∇ϕ(x)dxdτ .
Therefore, (w(x, t), h(x, t)) is a weak solution to the Eq. (1.5).
(3.26)
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B. Yuan, L. Bai / Applied Mathematics and Computation 269 (2015) 443–455
4. Decay for the weak solution This section is devoted to the proof of decay for the weak solution of Theorem 1.2. We need a generalized energy inequality and some corollaries. 4.1. The generalized energy inequality For the purpose of proving the generalized energy inequality, we give the following Lemma. Lemma 4.1. For every T > 0, the approximate solution (wN , hN ) constructed in Theorem 1.1 strongly converges to (w, h) in L2 ([0, T ]; L2 (R3 )). Proof. For every R > 0, define the cut-off function ϕR (x) = ϕ( Rx ), where
ϕ(x) ∈
t
0
C0∞
(R ) and ϕ(x) = 3
wN − w2L2 dt =
0
0
≤
wN − wN ϕR 2L2 =
t
wϕR − w2L2 ≤ 5
|wN − wN ϕR |2 dx ≤
R3
|wN |2 dx → 0
|x|≥R
|x|≥R
|x| ≥ 2, |x| ≤ 1.
wN − wN ϕR 2L2 dt +
Similarly, we have
for for
wN − wN ϕR + wN ϕR − wϕR + wϕR − w2L2 dt
t
≤5
0, 1,
|w|2 dx → 0
By the Lions–Aubin Lemma, we have proof is thus complete.
t 0
t 0
wN ϕR − wϕR 2L2 dt +
|x|≥2R
|wN |2 dx + 4
R≤|x|≤2R
t
0
wϕR − w2L2 dt.
|wN |2 dx
as R → ∞.
(4.1)
as R → ∞.
(4.2)
wN − w2L2 ds → 0 as N → ∞. Similarly, one has
t 0
hN − h2L2 ds → 0 as N → ∞. The
Lemma 4.2. (Generalized energy inequality) Let E(t) ∈ C1 [0, ∞) with E(t) ≥ 0 and ψ(t, x) ∈ C 1 ([0, ∞); S(R3 )). Then the weak solution (w, h) in L∞ ([0, ∞); L2 ) ∩ L2 ([0, ∞); H1 ) of the problem (1.5) satisfies the following generalized energy inequality
E (t )ψ(t ) ∗ w(t )22 + 2
E (τ )ψ(τ ) ∗ ∇ w(τ )22 ≤ E (s)ψ(s) ∗ w(s)22
s
+
t
t
E s
(τ )ψ(τ ) ∗ w(τ )22 dτ + 2
s
t
E (τ )
ψ (τ ) ∗ w(τ ), ψ(τ ) ∗ w(τ )
|w · ∇ w(τ ), ψ ∗ ψ ∗ w(τ )| + |h · ∇ h(τ ), ψ ∗ ψ ∗ w(τ )| |w · ∇V, ψ ∗ ψ ∗ w(τ )| + |V · ∇ w, ψ ∗ ψ ∗ w(τ )| + |h · ∇ B, ψ ∗ ψ ∗ w(τ )| + |B · ∇ h, ψ ∗ ψ ∗ w(τ )| dτ , + +
E (t )ψ(t ) ∗ h(t )22 + 2
+
t
s t
E s
(4.3)
E (τ )ψ(τ ) ∗ ∇ h(τ )22 ≤ E (s)ψ(s) ∗ h(s)22
(τ )ψ(τ ) ∗ h(τ ) τ + 2 2 2d
t s
E (τ ) ψ (τ ) ∗ h(τ ), ψ(τ ) ∗ h(τ )
|w · ∇ h(τ ), ψ ∗ ψ ∗ h(τ )| + |h · ∇ w(τ ), ψ ∗ ψ ∗ h(τ )| + |h · ∇ V, ψ ∗ ψ ∗ h(τ )| + |V · ∇ h, ψ ∗ ψ ∗ h(τ )| + |w · ∇ B, ψ ∗ ψ ∗ h(τ )| + |B · ∇ w, ψ ∗ ψ ∗ h(τ )| dτ . +
(4.4)
In reference [12], Ogawa et al. proved the Generalized Energy inequality for Navier–Stokes equation. For the sake of completeness, we give the detailed proof. Proof. We first recall the equation
∂t wN = −PJN (wN · ∇ wN ) + PJN (hN · ∇ hN ) + wN − PJN (wN · ∇V ) − PJN (V · ∇ wN ) + PJN (hN · ∇ B) + PJN (B · ∇ hN ).
(4.5)
B. Yuan, L. Bai / Applied Mathematics and Computation 269 (2015) 443–455
451
To deduce the term E (t )ψ ∗ wN (t )22 , we multiply (4.5) by E(t)ψ ∗ ψ ∗ wN and integrate it over R3 × [s, t], by the Leibniz d (E (t )ψ dt
formula
∂t wN (τ ), one has
L
∗ wN (t )2L2 ) = E (t )ψ ∗ wN (t )2L2 + 2E (t )ψ (τ ) ∗ wN (τ ), ψ(τ ) ∗ wN (τ ) + 2E (t )ψ(τ ) ∗ wN (τ ), ψ(τ ) ∗
E (t )ψ ∗ wN (t )2L2 + 2
+2
t s t s
E (τ )∇ψ ∗ wN 2L2 dτ ≤ E (s)ψ ∗ wN (τ )2L2 +
E (τ )
t s
E (τ )ψ ∗ wN 2L2 dτ
|ψ (τ ) ∗ wN (τ ), ψ(τ ) ∗ wN (τ )| + |PJN (wN · ∇ wN ), ψ ∗ ψ ∗ wN |
|PJN (hN · ∇ hN ), ψ ∗ ψ ∗ wN | + |PJN (wN · ∇V ), ψ ∗ ψ ∗ wN | + |PJN (V · ∇ wN ), ψ ∗ ψ ∗ wN | + |PJN (hN · ∇ B), ψ ∗ ψ ∗ wN | + |PJN (B · ∇ hN ), ψ ∗ ψ ∗ wN | dτ . +
(4.6)
For the purpose of proving the generalized energy inequality (4.3), we aim to take the limit as N → ∞ in (4.6), so we estimate each term of them. According to Lemma 4.1, we have wN → w in L2 (0, T ; L2 (R3 )). Therefore, one can easily get
ψ ∗ wN (t )L2 → ψ ∗ w(t )L2 , ∇ψ ∗ wN (t )L2 → ∇ψ ∗ w(t )L2 , for a.e. t. Now we estimate the nonlinear terms in (4.6).
PJN (wN · ∇ wN ), ψ ∗ ψ ∗ wN − w · ∇ w, ψ ∗ ψ ∗ w dτ s t = E (τ ) JN (wN · ∇ wN ), ψ ∗ ψ ∗ wN − (wN · ∇ wN ), ψ ∗ ψ ∗ wN dτ s t + E (τ ) wN · ∇ wN , ψ ∗ ψ ∗ wN − w · ∇ w, ψ ∗ ψ ∗ wN dτ s t N + E (τ ) w · ∇ w, ψ ∗ ψ ∗ w − w · ∇ w, ψ ∗ ψ ∗ w dτ t
E (τ )
s
R11 + R12 + R13 . Using the facts that wN ⊗ wN ∈ L2 (0, T ; L2 (R3 )) and ∇ψ ∗ ψ ∗ wN ∈ L2 (R3 ), we obtain
R11 ≤ C (T, w0 L2 )JN ∇(ψ ∗ ψ ∗ wN ) − ∇(ψ ∗ ψ ∗ wN )L2 → 0 as N → ∞. To estimate the term R12 , we use the Hölder and Young’s inequalities and Lemma 4.1 to obtain
R12 =
t
s
E (τ )
≤C
t s
R3
N N N N (w · ∇ w )ψ ∗ ψ ∗ w − (w · ∇ w)ψ ∗ ψ ∗ w dxdτ
E (τ )wN − wL2 dτ → 0 as N → ∞.
Similar to R12 one has
R13 ≤
t s
E (τ )w2L2 ∇(ψ ∗ ψ)L2 wN − wL2 dτ → 0 as N → ∞.
Hence, one has
t s
E (τ )PJN (wN · ∇ wN ), ψ ∗ ψ ∗ wN − w · ∇ w, ψ ∗ ψ ∗ w dτ → 0 as N → ∞.
Using a same process as the above it gives that
t s
E (τ )PJN (hN · ∇ hN ), ψ ∗ ψ ∗ wN − h · ∇ h, ψ ∗ ψ ∗ w dτ → 0 as N → ∞,
E (τ )PJN (w · ∇ V ), ψ ∗ ψ ∗ w − w · ∇ V, ψ ∗ ψ ∗ w dτ → 0 as N → ∞, s t N N E (τ )PJN (V · ∇ w ), ψ ∗ ψ ∗ w − V · ∇ w, ψ ∗ ψ ∗ w dτ → 0 as N → ∞, s t N N E (τ )PJN (h · ∇ B), ψ ∗ ψ ∗ w − h · ∇ B, ψ ∗ ψ ∗ w dτ → 0 as N → ∞,
t
s
N
N
(4.7)
452
B. Yuan, L. Bai / Applied Mathematics and Computation 269 (2015) 443–455
t
E (τ )PJN (B · ∇ hN ), ψ ∗ ψ ∗ wN − B · ∇ h, ψ ∗ ψ ∗ w dτ → 0 as N → ∞.
s
Combining the above estimates, we obtain the inequality (4.3). By an argument similar to that used in deriving the inequality (4.3) we can produce (4.4). The proof is thus complete. From the generalized energy inequality, we have the following corollaries. Note that the detailed proof is stated in references [8] and [12] and we sketch it for the completeness. Corollary 4.1. Let (w,h) be a weak solution to (3.1) satisfying the generalized energy inequality (4.3) and (4.4). Then for every ϕ ∈ S(R3 ), we have
ϕ ∗ w
2 L2
≤ e(t−s) ϕ ∗ w(s)2L2 + 2
t
|w · ∇ w, e2(t−τ ) (ϕ ∗ ϕ ∗ w)(τ )|
s
+ |w · ∇ V, e2(t−τ ) (ϕ ∗ ϕ ∗ w)(τ )| + |V · ∇ w, e2(t−τ ) (ϕ ∗ ϕ ∗ w)(τ )| + |h · ∇ h, e2(t−τ ) (ϕ ∗ ϕ ∗ w)(τ )| + |h · ∇ B, e2(t−τ ) (ϕ ∗ ϕ ∗ w)(τ )|
+ |B · ∇ h, e (ϕ ∗ ϕ ∗ w)(τ )| dτ , t ϕ ∗ h2L2 ≤ e(t−s) ϕ ∗ h(s)2L2 + 2 |w · ∇ h, e2(t−τ ) (ϕ ∗ ϕ ∗ h)(τ )| 2(t−τ )
(4.8)
s
2(t−τ )
(ϕ ∗ ϕ ∗ h)(τ )| + |V · ∇ h, e2(t−τ ) (ϕ ∗ ϕ ∗ h)(τ )| + |w · ∇ B, e (ϕ ∗ ϕ ∗ h)(τ )| + |h · ∇V, e2(t−τ ) (ϕ ∗ ϕ ∗ h)(τ )| + |B · ∇ w, e2(t−τ ) (ϕ ∗ ϕ ∗ h)(τ )| dτ . + |h · ∇ w, e
2(t−τ )
(4.9)
Proof. Take E(t) ≡ 1 and ψ(τ ) = e(t−τ ) ϕ in (4.3). Then ψ(τ ) ∗ w(τ ) = e(t−τ ) ϕ ∗ w(τ ) and
ψ (τ ) ∗ w(τ ), ψ(τ ) ∗ w(τ ) − ∇ψ(τ ) ∗ w(τ )2L2 = −e(t−τ ) ϕ ∗ w(τ ), e(t−τ ) ϕ ∗ w(τ ) − ∇ e(t−τ ) ϕ ∗ w(τ )2L2 = 0. Hence, we have
ϕ ∗ w(t )
2 L2
≤ e(t−s) ϕ ∗ w(s)2L2 + 2
t
|w · ∇ w, e2(t−τ ) (ϕ ∗ ϕ ∗ w)(τ )|
s
+ |w · ∇ V, e2(t−τ ) (ϕ ∗ ϕ ∗ w)(τ )| + |V · ∇ w, e2(t−τ ) (ϕ ∗ ϕ ∗ w)(τ )| + |h · ∇ h, e2(t−τ ) (ϕ ∗ ϕ ∗ w)(τ )| + |h · ∇ B, e2(t−τ ) (ϕ ∗ ϕ ∗ w)(τ )|
+ |B · ∇ h, e
2(t−τ )
(ϕ ∗ ϕ ∗ w)(τ )| dτ .
(4.10)
Using the same process that used in deriving (4.10) we can obtain (4.9). The proof is thus complete. Corollary 4.2. Let E(t) ∈ C1 [0, ∞) and E(t) ≥ 0. Let (w, h) be a weak solution to (3.1) satisfying the generalized energy inequality (4.3) and (4.4). Then for every ϕ ∈ S(R3 ), we have (w, h) satisfies
E (t )w(t ) − ϕ ∗ w(t )22 + 2 +
t
E
s
+2
t
s
s
+2 +2
t
t s
t
E s
t s
E (τ )∇ w(τ ) − ϕ ∗ ∇ w(τ )22 dτ ≤ E (s)w(s) − ϕ ∗ w(s)22
(τ )w(τ ) − ϕ ∗ w(τ )22 dτ + 2
s
t
E (τ )|w · ∇ w, ϕ ∗ ϕ ∗ w − 2ϕ ∗ w(τ )|dτ
E (τ )|h · ∇ h, w − 2ϕ ∗ w + ϕ ∗ ϕ ∗ w(τ )|dτ + 2 E (τ )|V · ∇ w, ϕ ∗ ϕ ∗ w − 2ϕ ∗ w(τ )|dτ + 2
s
t
t s
E (τ )|w · ∇ V, w − 2ϕ ∗ w + ϕ ∗ ϕ ∗ w(τ )|dτ
E (τ )|h · ∇ B, w − 2ϕ ∗ w + ϕ ∗ ϕ ∗ w(τ )|dτ
E (τ )|B · ∇ h, w − 2ϕ ∗ w + ϕ ∗ ϕ ∗ w(τ )|dτ .
E (t )h(t ) − ϕ ∗ h(t )22 + 2 +
t s
E (τ )∇ h(τ ) − ϕ ∗ ∇ h(τ )22 dτ ≤ E (s)h(s) − ϕ ∗ h(s)22
(τ )h(τ ) − ϕ ∗ h(τ )22 dτ + 2
t s
E (τ )|w · ∇ h, ϕ ∗ ϕ ∗ h − 2ϕ ∗ h(τ )|dτ
(4.11)
B. Yuan, L. Bai / Applied Mathematics and Computation 269 (2015) 443–455
+2
s
s
+2 +2
t
t
t s
E (τ )|h · ∇ w, h − 2ϕ ∗ h + ϕ ∗ ϕ ∗ h(τ )|dτ + 2 E (τ )|V · ∇ h, ϕ ∗ ϕ ∗ h − 2ϕ ∗ h(τ )|dτ + 2
t s
t s
453
E (τ )|h · ∇ V, h − 2ϕ ∗ h + ϕ ∗ ϕ ∗ h(τ )|dτ
E (τ )|w · ∇ B, h − 2ϕ ∗ h + ϕ ∗ ϕ ∗ h(τ )|dτ
E (τ )|B · ∇ w, h − 2ϕ ∗ h + ϕ ∗ ϕ ∗ h(τ )|dτ .
(4.12)
Proof. Let ψ(x, t ) = ζn (x) − ϕ(x) in (4.3), where ζn (x) = n−3 ζn ( nx ) is a smooth and compactly supported approximation of the t Dirac measure. Then s E (τ )|ψ (τ ) ∗ w(τ ), ψ ( τ ) ∗ w(τ )|dτ = 0. Combining the fact of ϕ ∗ ϕ ∗ w = ζn ∗ ζn ∗ w − 2ζn ∗ ϕ ∗ w + ϕ ∗ ϕ ∗ w and the divergence free for w, and passing to the limit as n → ∞ it yields the inequality (4.11). Arguing similarly to above we obtain (4.12). This completes the proof. 4.2. The proof of Theorem 1.2 We use the Fourier splitting method [13] to prove the decay of the weak solution (w, h). The proof is spilt into two steps, that is, the estimates for the low frequency part and the high frequency part of the energy. For each step, we use the Generalized energy inequality and Corollaries 4.1 and 4.2 with suitable functions E(t) and ψ . The L2 −norm of the Fourier transform of the weak solution (w, h) is divided into the following two parts.
ˆ (t )L2 + (1 − ϕ) ˆ (t )L2 , ˆ L2 ≤ ϕˆ w ˆ w w(t )L2 = wˆ (t )L2 = wˆ (t ) − wˆ (t )ϕˆ + wˆ (t )ϕ
(4.13)
h(t )L2 = hˆ (t )L2 = hˆ (t ) − hˆ (t )ϕˆ + hˆ (t )ϕ ˆ L2 ≤ ϕˆ hˆ (t )L2 + (1 − ϕ) ˆ hˆ (t )L2 , where we choose ϕ(ξ ˆ ) = exp ( − |ξ |2 ). Step 1: Estimates for the low frequencies of w. By Corollary 4.1 and Plancherel identity, we have
ϕˆ wˆ (t )2L2 ≤ e(t−s) ϕ ∗ w(s)2L2 + 2
t
s
|w · ∇ w, e2(t−τ ) (ϕ ∗ ϕ ∗ w)(τ )|
+ |w · ∇ V, e2(t−τ ) (ϕ ∗ ϕ ∗ w)(τ )| + |V · ∇ w, e2(t−τ ) (ϕ ∗ ϕ ∗ w)(τ )| + |h · ∇ h, e2(t−τ ) (ϕ ∗ ϕ ∗ w)(τ )| + |h · ∇ B, e2(t−τ ) (ϕ ∗ ϕ ∗ w)(τ )|
+ |B · ∇ h, e2(t−τ ) (ϕ ∗ ϕ ∗ w)(τ )| dτ K1 (t, s) + 2
s
t
(|K2 | + |K3 | + |K4 | + |K5 | + |K6 | + |K7 |)dτ ,
(4.14)
It is not difficult to see that the estimates of K5 is the same as K2 while K4 , K6 and K7 are the same as K3 . Therefore, it suffices to estimate K1 , K2 , K3 . For the fixed s ≥ 0, by the Plancherel identity and the Lebesgue dominated convergence theorem it follows that
|K1 | = e(t−s) ϕ ∗ w(s)2L2 = (2π )3 e−(t−s)|ξ | ϕˆ wˆ (s)2L2 → 0 as t → ∞. 2
(4.15)
Hölder, Young inequalities and Sobolev embedding theorem together give
|K2 | ≤ ϕ ∗ ϕ ∗ (w · ∇ w)L2 e2(t−τ ) wL2 ≤ C ϕ ∗ ϕ 6 wL6 ∇ wL2 wL2 ≤ C ϕ ∗ ϕ 6 w0 L2 ∇ w2L2 . L5 L5 By the L2 − estimate for the heat semigroup, Hölder inequality and Sobolev embedding, one can deduce that
|K3 | ≤ C V L3 wL6 ∇ e2(t−τ ) (ϕ ∗ ϕ ∗ w)L2 ≤ C V ˙ 1 ∇ w2L2 ϕ ∗ ϕL1 . H2 Combining the above estimates into (4.14), and taking t → ∞ we get
ϕˆ wˆ (t )2L2 ≤ K1 (t, x) + C (w0 L2 + V H˙ 12 + BH˙ 12 )
t s
(∇ w(τ )2L2 + ∇ h(τ )2L2 )dτ .
(4.16)
Let s > 0 be fixed and large on the right-hand side of (4.16). The term K1 (t, x) tends to zero as t → ∞ by (4.15). From (3.9), it can t ˆ (t )22 → 0 as t → ∞. be deduced that the quantity s (∇ w(τ )22 + ∇ h(τ )22 )dτ converges 0 as s → ∞. Therefore, ϕˆ w L
L
L
Step 2: Estimate the high frequencies for w. Here we use Corollary 4.2 with the test function ϕ satisfying ϕ(ξ ˆ ) = exp{−|ξ |2 }
ˆ (t )L2 = w(t ) − ϕ ∗ w(t )L2 , it is easy to see and a function E(t) > 0 to be determined below. In this step, we estimate (1 − ϕ) ˆ w that we only need to estimate each term on the right-hand of (4.11) successively. To simplify the presentation, we denote
S1
t
E s
(τ )w(τ ) − ϕ ∗ w(τ )22 dτ − 2
t s
E (τ )∇ w(τ ) − ϕ ∗ ∇ w(τ )22 dτ ,
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B. Yuan, L. Bai / Applied Mathematics and Computation 269 (2015) 443–455
S2
t
E (τ )|w · ∇ w, ϕ ∗ ϕ ∗ w − 2ϕ ∗ w(τ )|dτ ,
s
S3
t
E (τ )|h · ∇ h, w − 2ϕ ∗ w + ϕ ∗ ϕ ∗ w(τ )|dτ ,
s
S4
t
E (τ )|w · ∇ V, w − 2ϕ ∗ w + ϕ ∗ ϕ ∗ w(τ )|dτ ,
s
S5
t
E (τ )|V · ∇ w, ϕ ∗ ϕ ∗ w − 2ϕ ∗ w(τ )|dτ ,
s
S6
t
E (τ )|h · ∇ B, w − 2ϕ ∗ w + ϕ ∗ ϕ ∗ w(τ )|dτ ,
s
S7
t
E (τ )|B · ∇ h, w − 2ϕ ∗ w + ϕ ∗ ϕ ∗ w(τ )|dτ .
s
Now we estimate S1 to S7 .
S1 =
t
E
(τ )
s
+
t
E
s
|ξ |>G(t )
(τ )
2
t
dξ dτ − 2 E (τ )
(1 − ϕ(ξ ˆ ˆ )) (ξ , τ ) w
s
|ξ |≤G(t )
|ξ |>G(t )
2
t
dξ dτ − 2 E (τ )
(1 − ϕ(ξ ˆ ˆ )) (ξ , τ ) w
s
2
|ξ |(1 − ϕ(ξ ˆ (ξ , τ )
dξ dτ ˆ ))w
|ξ |≤G(t )
2
dξ dτ .
|ξ |(1 − ϕ(ξ ˆ ˆ )) (ξ , τ ) w
α 2 We choose E (t ) = (1 + t )α and G2 (t ) = 2(1+t ) with fixed α > 0, thus E (t ) − 2E (t )G (t ) = 0. So
t
E
s
(τ )
≤
t
E s
=
|ξ |>G(t )
(τ )
2
t
dξ dτ − 2 E (τ )
(1 − ϕ(ξ ˆ ˆ )) (ξ , τ ) w
s
|ξ |>G(t )
2
t
dξ dτ − 2 E (τ )G2 (τ )
(1 − ϕ(ξ ˆ ˆ )) (ξ , τ ) w
2 E (τ ) − 2E (τ )G (τ )
t
s
|ξ |>G(t )
2
|ξ |(1 − ϕ(ξ ˆ (ξ , τ )
dξ dτ ˆ ))w
s
|ξ |>G(t )
|ξ |>G(t )
2
(1 − ϕ(ξ ˆ (ξ , τ )
dξ dτ = 0. ˆ ))w
For |ξ | ≤ G(t), it follows that 1 − ϕ(ξ ˆ ) = 1 − e−|ξ | , thus, we obtain
2
dξ dτ
(1 − ϕ(ξ ˆ ˆ )) (ξ , τ ) w
2
t
E s
(τ )
|ξ |≤G(t )
2
t
dξ dτ ≤ E (τ )
(1 − ϕ(ξ ˆ ˆ )) (ξ , τ ) w
s
|ξ |≤G(t )
2
2
|ξ wˆ (ξ , τ ) dξ dτ
2
t t
4
dξ dτ ≤ C w0 2L2 E (τ )G4 (τ )dτ
ˆ ≤ E (τ )G (τ ) (ξ , τ ) w
|ξ |≤G(t ) s s
≤C
s
t
(1 + τ )α−3 dτ .
t ˆ (ξ , τ )|2 dξ dτ ≥ 0. Since E(t) ≥ 0, we get s E (τ ) |ξ |≤G(t ) |(1 − ϕ(ξ ˆ ))w t α −3 Therefore, S1 ≤ C s (1 + τ ) dτ . Set η = ϕ ∗ ϕ − 2ϕ , and applying the Hölder and Young inequalities, one has
S2 ≤
s
t
E (τ )w · ∇ w
3 L2
η ∗ wL3 dτ ≤ C ηL 65 w0 L2
t
s
E (τ )∇ w2L2 dτ .
Similar arguments as in deriving S2 we derive that
S3 ≤ C η
6 L5
w0 L2
t s t
E (τ )∇ h2L2 dτ , S4 ≤ C V ˙
H
S5 ≤ C V ˙
1 2
ηL1
S6 ≤ C B ˙
1 2
(1 + ηL1 )
H
H
s
E (τ )∇ w2L2 , S7 ≤ C B ˙
H
t s
(1 + ηL1 )
ηL1
s
t
t s
E (τ )∇ w2L2 dτ ,
E (τ )∇ wL2 ∇ hL2 dτ ,
E (τ )∇ hL2 ∇ wL2 dτ .
Thus,
ˆ (t )2L2 ≤ (1 − ϕ) ˆ w
1 2
1 2
E (s) C ˆ (s)2L2 + ˆ w (1 − ϕ) E (t ) E (t )
t s
(1 − τ )α−3 dτ +
C E (t )
t s
E (τ )(∇ w2L2 + ∇ h2L2 )dτ .
(4.17)
B. Yuan, L. Bai / Applied Mathematics and Computation 269 (2015) 443–455
455
For fixed s > 0, we compute the lim sup of both sides of (4.17), as t → ∞. Since E (t ) = (1 + t )α with α > 0, the first term on the right-hand side of (4.17) tends to zero.
limsup t→∞
Since
E (τ ) E (t )
=
1 (1 + t )α
τ α ( 1+ 1+t )
s
t
(1 − τ )α−3 dτ = limsup
≤ 1 for τ ∈ [0, t], for any small ε > 0, choosing s large enough, one has
ˆ (t )2L2 ≤ C limsup (1 − ϕ) ˆ w t→∞
t→∞
(1 + s)α−2 = 0. (1 + t )2 (α − 2)
s
∞
(∇ w2L2 + ∇ h2L2 )dτ < ε .
Combining the above estimates of the high and low frequencies for w, this completes the proof of wL2 → 0 as t → ∞. Obviously, we can take a similarly argument to h, and then can complete the proof of Theorem 1.2. Acknowledgments The research of B Yuan was partially supported by the National Natural Science Foundation of China (no. 11471103). References [1] W. Borchers, T. Miyakawa, L2 decay for Navier–Stokes flows in unbounded domains, with application to exterior stationary flows, Arch. Ration. Mech. Anal. 118 (1992) 273–295. [2] P. Constantin, C. Foias, Navier–Stokes Equations, Chicago University Press, Chicago, 1988. [3] L. Caffarelli, R. Kohn, L. Nirenberg, Partial regularity of suitable weak solutions of the Navier–Stokes equations, Commun. Pure Appl. Math. 35 (1982) 771–831. [4] M. Hess, M. Hieber, A. Mahalov, J. Saal, Nonlinear stability of Ekman boundary layers, Bull. London Math. Soc. 42 (2010) 691–706. [5] H. Kozono, Asymptotic stability of large solutions with large perturbation to the Navier–Stokes equations, J. Funct. Anal. 176 (2000) 153–197. [6] R. Kajikiya, T. Miyakawa, On L2 decay of weak solutions of the Navier–Stokes equations in Rn , Math. Z. 192 (1986) 135–148. [7] G. Karch, D. Pilarczyk, Asymptotic stability of landau solutions to Navier–Stokes system, Arch. Ration. Mech. Anal. 202 (2011) 115–131. [8] G. Karch, D. Pilarczyk, M. Schonbek, L2 − asymptotic stability of mild solutions to Navier–Stokes system in R3 , [math.AP] 30 Aug2013, arXiv:1308.6667v1. [9] J. Leray, étude de diverses équations integrales non lineaires et de quelques problèmes que pose l’hydrodynamique, J. Math. Pure Appl. 9 (1933) 1–82. [10] J. Leray, Sur le mouvement d’un liquide visqeux emplissant l’espace, Acta Math. 63 (1934) 193–248. [11] C.X. Miao, B.Q. Yuan, On the well-posedness of the Cauchy problem for an MHD system in Besov spaces, Math. Methods Appl. Sci. 32 (2009) 53–76. [12] T. Ogawa, S. Rajopadhye, M. Schonbek, Energy decay for a weak solution of the Navier–Stokes equation with slowly varying external forces, J. Funct. Anal. 144 (1997) 325–358. [13] M.E. Schonbek, l2 decay for weak solutions of the Navier–Stokes equations, Arch. Ration. Mech. Anal. 88 (1985) 209–222. [14] M. Wiegner, Decay results for weak solutions of the Navier–Stokes equations in Rn , J. London Math. Soc. 35 (1987) 303–313.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
L2 − asymptotic stability of the mild solution to the 3D MHD equation Baoquan Yuan∗, Linna Bai School of Mathematics and Information Science, Henan Polytechnic University, Henan, 454000, China
a r t i c l e
i n f o
a b s t r a c t
MSC classiffication: 35B40 35D30 35Q35
In this paper, we investigate the L2 − asymptotic stability of a global-in-time mild solution of the small initial value problem to the 3D magnetohydrodynamic (MHD) equations. We construct a weak solution to the MHD equation, and then we show this weak solution converges to the mild solution in the sense of L2 energy norm.
Keywords: Mild solution 3D MHD equation Weak solution
© 2015 Elsevier Inc. All rights reserved.
1. Introduction In this paper, we consider the following 3D magnetohydrodynamic (MHD) equations
ut + u · ∇ u − b · ∇ b − u + ∇ p = 0, bt + u · ∇ b − b · ∇ u − b = 0, ∇ · u = ∇ · b = 0, u(x, 0) = u0 (x), b(x, 0) = b0 (x),
(1.1)
where u = (u1 (x, t ), u2 (x, t ), u3 (x, t )) is the velocity of the flow, b = (b1 (x, t ), b2 (x, t ), b3 (x, t )) is the magnetic field, p is the scalar 1
pressure, and (u0 , b0 ) ∈ H 2 (R3 ) is the given initial data. It is well-known that the global-in-time solution with the small initial data to the Eq. (1.1) was obtained in reference [11]. 1 They showed that for (u0 , b0 ) ∈ H˙ 2 (R3 ), there exists a positive time T such that the Eq. (1.1) has a unique solution (u, b) in L4 ([0, T ]; H˙ 1 (R3 )) which also belongs to
C ([0, T ]; H˙ 2 (R3 )) ∩ L2 ([0, T ]; H˙ 2 (R3 )). 1
3
(1.2)
Let T(u0 ,b0 ) denote the maximal time of existence of such a solution, if there exists ε 0 > 0 such that
(u0 , b0 )H˙ 12 ≤ ε0 ,
(1.3)
then T(u0 ,b0 ) = ∞. Moreover, (u, b) satisfies the energy equality
u(t )2L2 + b(t )2L2 + 2
0
t
(∇ u(t )2L2 + ∇ b(t )2L2 )dτ = u0 2L2 + b0 2L2 .
(1.4)
The purpose of this paper can be described as follows. Assume that (V(x, t), B(x, t)) is a global-in-time mild solution of the small initial value problem (1.1) with (V0 , B0 ) 1 ≤ min{ε0 , 1}, then we show that the system (1.1) has a weak solution (u(x, t), H2
∗
Corresponding author. Tel.: +863913987808. E-mail addresses: [email protected], [email protected] (B. Yuan), [email protected] (L. Bai).
http://dx.doi.org/10.1016/j.amc.2015.07.006 0096-3003/© 2015 Elsevier Inc. All rights reserved.
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B. Yuan, L. Bai / Applied Mathematics and Computation 269 (2015) 443–455
b(x, t)) in the sense of Leray corresponding to the initial datum (V0 , B0 ) perturbed by an arbitrarily large divergence free L2 −vector and magnetic fields, and this weak solution converges to the mild solution (V(x, t), B(x, t)) in the sense of L2 −norm as t → ∞. More specifically, we denote (u(x, t), b(x, t)) a weak solution to the Eq. (1.1) with the initial data u0 = V0 + w0 , b0 = B0 + h0 , where 1
(w0 , h0 ) ∈ H 2 (R3 ), then the functions w(x, t ) = u(x, t ) − V (x, t ), h(x, t ) = b(x, t ) − B(x, t ) satisfy the following perturbed initial value problem
⎧ ⎪ ⎨wt + w · ∇ w − h · ∇ h − w = −w · ∇V − V · ∇ w + h · ∇ B + B · ∇ h − ∇π , ht + w · ∇ h − h · ∇ w − h = −w · ∇ B + B · ∇ w + h · ∇ V − V · ∇ h, ∇ ⎪ ⎩ · w = ∇ · h = 0, w(x, 0) = w0 (x), h(x, 0) = h0 (x).
(1.5)
Therefore, we first construct a weak solution (w, h) of the initial problem (1.5), then we show the L2 −norms of w and h tend to 0 as t → ∞. On the topic of asymptotic stability for the solutions, there have been many classical results to the Navier–Stokes equation [1,4–8,10,12–14]. The energy decay problem of weak solutions to the Navier–Stokes equation was originally suggested by Leray in his pioneering papers [9,10]. Later, Schonbek [13] proved the algebraic decay for the Cauchy problem in R3 for large data in L1 ∩ L2 by using the approximate solutions of Caffarelli et al. [3]. Kajikiya and Miyakawa [6] improved the result of [13] by a systematic use of a modified version of the method in [13]. Wiegner [14] also showed the decay results for weak solutions to the Navier–Stokes equation. In 1997, Ogawa et al. investigated the energy decay problem of the weak solution to the Navier–Stokes equation with slowly varying external forces [12]. Later, Kozono studied the weak solutions u of the Navier–Stokes equations in Serrin’s class u ∈ Lα (0, ∞; Lq ()) for α2 + 3q = 1 with 3 < q ≤ ∞. They show that although the initial and external disturbances from u are large, the weak solution of the perturbed problem can also decay to zero as t → ∞ [5]. Recently, in [8], Karch et al. established an asymptotic stability for a global-in-time small mild solution to the Navier–Stokes equation in R3 under arbitrarily large initial L2 − perturbations. Motivated by Ogawa et al. [12] and Karch et al. [8], in our paper, we discuss the L2 − asymptotic stability of the mild solution to the MHD system in R3 . The key idea is to construct a weak solution (w(x, t), h(x, t)) of the perturbed Eq. (1.5) in the sense of Leray–Hopf by the Fridrich method, and then we show them L2 − decay to zero as t → ∞. To this end, we state our results. Theorem 1.1 (Existence of the weak solutions). Let (V = V (x, t ), B = B(x, t )) be a global-in-time solution to the small initial value problem(1.1) with V0 1 + B0 1 ≤ min{ε0 , 1}. Denote V0 = V (·, 0), B0 = B(·, 0) and let (w0 , h0 ) ∈ L2 (R3 ) be arbitrary. Then, the H2
H2
Cauchy problem (1.1) with the initial condition u0 = V0 + w0 , b0 = B0 + h0 has a global weak solution (u, b) of the form
u(x, t ) = V (x, t ) + w(x, t ), b(x, t ) = B(x, t ) + h(x, t ),
where (w(x, t), h(x, t)) is a weak solution of the corresponding perturbed problem (1.5) satisfying
(w, h) ∈ Cw ([0, T ]; L2 (R3 )) ∩ L2 ([0, T ]; H˙ 1 (R3 )) f or each T > 0. Theorem 1.2 (Asymptotic behavior of mild solutions). As t → ∞, the weak solution (u(x, t), b(x, t)) of the system (1.1) tend to the mild solution (V = V (x, t ), B = B(x, t )) in the sense of L2 −norm. That is to say
w(t )2 = u(t ) − V (t )2 → 0, h(t )2 = b(t ) − B(t )2 → 0,
where (w(x, t), h(x, t)) is constructed in Theorem 1.1. The rest of this paper is organized as follows. In Section 2, we give some definitions and Lemmas which will be used in the following proofs. Sections 3 and 4 devoted to prove Theorems 1.1 and 1.2, respectively. 2. Preliminaries In this section we introduce the definitions of Leray–Hopf weak solution and the cut-off operator, and then recall some lemmas and notations which will be used in the following parts. Definition 2.1. For (u0 (x), b0 (x)) ∈ L2 (R3 ) and T > 0. We call (u, b) the weak solution of Leray–Hopf type (Leray–Hopf weak solution) if and only if 1. (u, b) ∈ L∞ (0, T ; L2 (R3 )) ∩ L2 (0, T ; H˙ 1 (R3 )); t t t 2. Rd u(x, t ) · ϕ(x)dx − Rd u(x, 0) · ϕ(x)dx + 0 Rd u · ∇ u · ϕ(x)dxdτ − 0 Rd b · ∇ b · ϕ(x)dxdτ = 0 Rd u(x, τ ) · ϕ(x)dxdτ ; t t t Rd b(x, t ) · ϕ(x)dx − Rd b(x, 0) · ϕ(x)dx + 0 Rd u · ∇ b · ϕ(x)dxdτ − 0 Rd b · ∇ u · ϕ(x)dxdτ = 0 Rd b(x, τ ) · ϕ(x)dxdτ for any test functions ϕ ∈ C0∞ (R3 ) satisfying ∇ · ϕ = 0.
B. Yuan, L. Bai / Applied Mathematics and Computation 269 (2015) 443–455
445
Definition 2.2. Let N > 0 be an integer, χ (ξ ) be a characteristic function satisfying
χ (ξ ) =
1, 0,
if if
ξ ∈ B(0; N), ξ ∈ B(0; N),
where B(0; N) denotes the ball of radius N centered at the origin. Then the cut-off operator JN is defined as
ˆ J
N f (ξ ) = χB(0;N) (ξ ) f (ξ ). Remark 2.1. We denote L2N = { f ∈ L2 (Rd )| supp fˆ ⊂ B(0; N)}. If f ∈ L2N , then f ∈ Hs for any s > 0, and JN f L2 ≤ f L2 . Lemma 2.1. (Hodge decomposition) For any vector field u ∈ L2 (Rd ) ∩ C ∞ (Rd ), there exist w and p such that u = w + ∇ p with w and p satisfying 1. w ∈ L2 (Rd ) ∩ C ∞ (Rd ), ∇ p ∈ L2 (Rd ) ∩ C ∞ (Rd ) and ∇ · w = 0; 2. Rd w · ∇ pdx = 0; 3. Dα u22 = Dα w22 + Dα ∇ p22 for any multi-index α ≥ 0. L
L
L
Now let’s introduce the operator P, an orthogonal projection operator, which maps u onto its divergence-free part w and we denote Pu = w, moreover PuL2 = wL2 ≤ uL2 . Lemma 2.2. (Picard Lemma) Let B be a Banach space and O ⊂ B be an open set. Assume F: O → B satisfies the local Lipschitz condition:
∀ X ∈ O, there exists a neighborhood of X denoted by U = U (X ) and a Lipschitz constant L = L(X ) such that F (X ) − F (X )B ≤ L(X )X − X B for any X , X ∈ U (X ). Then, for X0 ∈ U, the ODE
dX = F (X ), dt X (0) = X0 ,
(2.1)
has a unique local solution, that is , ∃ T > 0 and X ∈ C1 ([0, T]; O) such that X solves (2.1). In addition, either X = X (t ) exists for all time or ∃ T∗ > 0 such that X(t) leaves O as t → T∗ . Lemma 2.3. (Lions–Aubin Lemma) Let q1 , q2 ∈ (1, ∞) and δ 2 < δ 1 be real numbers. Assume {fm } satisfies 1. {fm } is uniformly bounded in Lq1 ([0, T ]; H δ1 (Rd )); 2. For any ρ ∈ C0∞ (Rd ), ρ ∂∂ftm in Lq2 ([0, T ]; H δ2 (Rd )) is uniformly bounded, or ρ fm (x, t1 ) − ρ fm (x, t2 )H δ2 ≤ C |t1 − t2 |. Then, there exists a subsequence of fm (still denoted by fm ) and f ∈ Lq1 ([0, T ]; H δ1 (Rd )) such that, for any ρ ∈ C0∞ (Rd ), ρ fm − ρ f → 0 in Lq1 ([0, T ]; H δ (Rd )), for any δ ∈ (δ 2 , δ 1 ). Remark 2.2. For details of the proofs of Lemma 2.3, please refer to reference [2]. Now we introduce some notations. 1. S(R3 ) is the Schwartz class of rapidly decreasing functions; 2. f, g stands for the inner-product which is defined as f, g = f (x)g(x)dx; 3. The Fourier transform of a function f(x) is defined by 3 fˆ(ξ ) = (2π )− 2
R3
e−ix·ξ f (x)dx.
Throughout this paper, C denotes a generic positive constant which may change from one line to the other line. 3. Existence of weak solution to the perturbed equations In this section, we shall employ Friedrichs’ methods to prove Theorem 1.1, the proof is divided into three steps. Step 1: Local existence of an approximate solution. Fix an integer N > 0, and consider
⎧ N N N N N N ⎪ ⎪∂t w − (PJN w ) + PJN (PJN w · ∇ PJN w ) − PJN (PJN h · ∇ PJN h ) ⎪ ⎪ N N N N ⎪ ⎪ ⎨ = −PJN (PJN w · ∇V ) − PJN (V · ∇ PJN w ) + PJN (PJN h · ∇ B) + PJN (B · ∇ PJN h ), N N N N N N ∂t h − ( JN h ) + JN ( JN w · ∇ JN h ) − JN ( JN h · ∇ JN w ) ⎪ ⎪ ⎪ = −JN ( JN wN · ∇ B) − JN (V · ∇ JN hN ) + JN ( JN hN · ∇ V ) + JN (B · ∇ JN wN ), ⎪ ⎪ ⎪ ⎩ N w (x, 0) = JN w0 (x), hN (x, 0) = JN h0 (x).
(3.1)
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B. Yuan, L. Bai / Applied Mathematics and Computation 269 (2015) 443–455
Now we show the existence and uniqueness of an approximate solution. We apply the Picard Lemma with B = O = L2N .
⎧ N ∂t w = (PJN wN ) − PJN (PJN wN · ∇ PJN wN ) + PJN (PJN hN · ∇ PJN hN ) ⎪ ⎪ ⎪ ⎪ ⎨ −PJN (PJN wN · ∇V ) − PJN (V · ∇ PJN wN ) + PJN (PJN hN · ∇ B) + PJN (B · ∇ PJN hN ) F1 (wN , hN ), ⎪ ∂t hN = ( JN hN ) − JN ( JN wN · ∇ JN hN ) + JN ( JN hN · ∇ JN wN ) − JN ( JN wN · ∇ B) ⎪ ⎪ ⎪ ⎩ −JN (V · ∇ JN hN ) + JN ( JN hN · ∇ V ) + JN (B · ∇ JN wN ) F2 (wN , hN ).
(3.2)
First, we verify that if (wN , hN ) ∈ L2N , then F1 (wN , hN ), F2 (wN , hN ) ∈ L2N .
F1 (wN , hN )L2 ≤ PJN wN · ∇ PJN wN L2 + PJN hN · ∇ PJN hN L2 + (PJN wN )L2 + PJN wN · ∇ V L2 + V · ∇ PJN wN L2 + PJN hN · ∇ BL2 + B · ∇ PJN hN L2 I1 + I2 + I3 + I4 + I5 + I6 + I7 .
(3.3)
¨ We successively estimate I1 to I7 . Combining Holder inequality, Plancherel identity and properties of the Fourier transform, we obtain
I1 ≤ PJN wN L∞ ∇ PJN wN L2 ≤ CN 2 wN 2L2 . 5
Similarly, it can be shown that
I2 ≤ CN 2 hN 2L2 . 5
By the Plancherel identity we have
I3 ≤ ( JN wN )L2 = From (1.4), we know
t 0
|ξ |≤N
2 N ||ξ |2 J N w (ξ )| dξ
12
≤ N2 wN L2 .
∇V 2L2 dτ < ∞, it can be deduced that for almost every t, ∇V L2 < ∞, hence
I4 ≤ PJN wN L∞ ∇ V L2 ≤ CN 2 wN L2 ∇ V L2 3
for a.e. t.
Arguing similarly to I4 it can be derived, for a.e. t, that
I6 ≤ CN 2 hN L2 ∇ BL2 , 3
and
I5 ≤ V L2 ∇ PJN wN L∞ ≤ CN 2 V L2 wN L2 , 5
I7 ≤ CN 2 BL2 hN L2 . 5
Therefore, inserting the above estimates into (3.3), one has
F1 (wN , hN )L2 ≤ CN 2 (wN 2L2 + hN 2L2 + wN L2 + hN L2 ) + N 2 (wN L2 + hN L2 )(∇V L2 + ∇ BL2 ) < ∞. 5
3
Use an argument similar to that used in evaluating F1 (wN , hN ) to obtain that
F2 (wN , hN )L2 ≤ CN 2 (wN 2L2 + hN 2L2 + wN L2 + hN L2 ) + N 2 (wN L2 + hN L2 )(∇V L2 + ∇ BL2 ) < ∞. 5
3
N N N 2 Next we verify F1 , F2 satisfy the Lipschitz condition. Let (wN 1 , h1 ) and (w2 , h2 ) ∈ LN ,
F1 (wN1 , hN1 ) − F1 (wN2 , hN2 )L2 ≤ PJN wN1 · ∇ PJN wN1 − PJN wN2 · ∇ PJN wN2 L2 + PJN hN1 · ∇ PJN hN1 − PJN hN2 · ∇ PJN hN2 L2 +PJN (wN1 − wN2 )L2 + PJN wN1 · ∇ V − PJN wN2 · ∇ V L2 +V · ∇ PJN wN1 − V · ∇ PJN wN2 L2 + PJN hN1 · ∇ B − PJN hN2 · ∇ BL2 +B · ∇ PJN hN1 − B · ∇ PJN hN2 L2 Q1 + Q2 + Q3 + Q4 + Q5 + Q6 + Q7 . ¨ Holder inequality, Plancherel identity and properties of the Fourier transform together give
Q1 ≤ PJN (wN1 − wN2 ) · ∇ PJN wN1 L2 + PJN wN2 · ∇ PJN (wN1 − wN2 )L2 ≤ N 2 (wN1 L2 + wN2 L2 )wN1 − wN2 L2 . 5
By an analogous argument with Q1 , we have
Q2 ≤ N 2 (hN1 L2 + hN2 L2 )hN1 − hN2 L2 , 5
Q3 ≤ PJN (wN1 − wN2 )L2 ≤ N2 wN1 − wN2 L2 ,
Q4 ≤ N 2 wN1 − wN2 L2 ∇ V L2 , and Q6 ≤ N 2 hN1 − hN2 L2 ∇ BL2 for a.e. t, 3
Q5 ≤ N 2 V L2 wN1 − wN2 L2 , 5
3
Q7 ≤ N 2 BL2 hN1 − hN2 L2 . 5
(3.4)
B. Yuan, L. Bai / Applied Mathematics and Computation 269 (2015) 443–455
447
Combining the above estimates, we obtain, for almost every t, that
F1 (wN1 , hN1 ) − F1 (wN2 , hN2 )L2 ≤ CN 2 (V H1 + wN1 L2 + wN2 L2 )wN1 − wN2 L2 5
+ CN 2 (BH 1 + hN1 L2 + hN2 L2 )hN1 − hN2 L2 . 5
Using a similar argument of the estimate to
F1 (wN1 , hN1 ) − F1 (wN2 , hN2 )L2
(3.5)
yields
F2 (wN1 , hN1 ) − F2 (wN2 , hN2 )L2 ≤ CN (BH1 + hN1 L2 + hN2 L2 )wN1 − wN2 L2 5 2
+ CN 2 (V H 1 + wN1 L2 + wN2 L2 )hN1 − hN2 L2 . 5
(wN , hN )
C 1 ([0, T ]; L2N )
∈ such By Picard’s Lemma, ∃ T > 0 and Note that Eq. (3.1) is equivalent to the following system
that(wN ,
hN )
(3.6)
solves (3.1).
⎧ N ∂t w − wN + PJN (wN · ∇ wN ) − PJN (hN · ∇ hN ) = −PJN (wN · ∇V ) ⎪ ⎪ ⎪ ⎪ ⎪ −PJN (V · ∇ wN ) + PJN (hN · ∇ B) + PJN (B · ∇ hN ), ⎪ ⎨ ∂t hN − hN + JN (wN · ∇ hN ) − JN (hN · ∇ wN ) = −JN (wN · ∇ B) ⎪ ⎪ ⎪ −JN (V · ∇ hN ) + JN (hN · ∇V ) + JN (B · ∇ wN ), ⎪ ⎪ ⎪ ⎩ ∇ · wN = 0, ∇ · hN = 0.
(3.7)
In fact, since (wN , hN ) is a solution of (3.1) and JN2 = JN , P2 = P, taking P and JN on both sides of (3.1) respectively, and by the uniqueness of solutions, we have wN = PwN = JN wN , hN = JN hN . Multiplying the two equations of (3.7) by wN and hN , respectively, integrating and adding the resulting equations together it follows that
1 d (wN 2L2 + hN 2L2 ) + ∇ wN 2L2 + ∇ hN 2L2 = − PJN (wN · ∇V ) · wN dx 2 dt R3 N N N N + PJN (h · ∇ B) · w dx − JN (w · ∇ B) · h dx + JN (hN · ∇ V ) · hN dx R3
R3
R3
J1 + J2 + J3 + J4 .
(3.8)
¨ Using the Holder inequality and Sobolev embedding one can deduce
N N
|J1 | = (w ⊗ V ) : PJN ∇ w dx
≤ C wN L6 V L3 ∇ wN L2 3 R ≤ C wN H˙ 1 V ˙
1
H2
∇ wN L2 = C V H˙ 12 ∇ wN 2L2 .
Similarly, it can be shown that
|J2 | ≤ BH˙ 12 hN H˙ 1 ∇ wN L2 , |J3 | ≤ BH˙ 12 wN H˙ 1 ∇ hN L2 , |J4 | ≤ V H˙ 12 ∇ hN 2L2 . Inserting the above estimates into (3.8), and taking the initial value small enough so that V inequality implies
wN 2L2 + hN 2L2 +
t 0
1
H2
+ B
1
H2
< min{ε0 , 1}, Gronwall
(∇ wN 2L2 + ∇ hN 2L2 )dτ ≤ w0 2L2 + h0 2L2 .
(3.9)
By Picard Lemma, (wN , hN ) exists for all time. There exists a subsequence of (wN , hN ) (still denoted by (wN , hN )) and (w, h) in L∞ ([0, ∞); L2 ) ∩ L2 ([0, ∞); H 1 ) such that wN w, hN h in the weak sense of L∞ ([0, T ); L2 ) ∩ L2 ([0, T ); H 1 ) for any T > 0. Step 2: Strong convergence. 4
For any ρ ∈ C0∞ (Rd ), we show (ρ∂t wN , ρ∂t hN ) ∈ L 3 ([0, T ]; H −1 (R3 )). By the definition of H −1
ρ∂t wN
= H −1
sup
g∈H 1 ,gH 1
< ρ∂ wN , g > .
=1
Multiply both sides of the first equation of (3.7) by ρ g and integrate it over R3 to get
R3
ρ∂t wN · g(x)dx = − + +
R3
R
3
R3
ρ PJN (wN · ∇ wN ) · g(x)dx + ρ wN · g(x)dx −
R3
R3
ρ PJN (hN · ∇ hN ) · g(x)dx
ρ PJN (V · ∇ wN ) · g(x)dx
ρ PJN (hN · ∇ B) · g(x)dx −
L1 + L2 + L3 + L4 + L5 + L6 + L7 .
R3
ρ PJN (wN · ∇V ) · g(x)dx +
R3
ρ PJN (B · ∇ hN ) · g(x)dx (3.10)
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B. Yuan, L. Bai / Applied Mathematics and Computation 269 (2015) 443–455
¨ Applying the Holder and Gagliardo–Nirenberg inequalities, one has
1 3 |L1 | =
− wN ⊗ wN : PJN ∇(ρ g)
≤ wN 2L4 ∇(ρ g)L2 ≤ C wN L22 wN 2˙ 1 gH 1 . H R3
Similarly, it can be shown that 1
3
|L2 | ≤ C hN L22 hN H2˙ 1 . ¨ The estimate of L3 can also be obtained by Holder inequality
|L3 | =
− ∇ρ · ∇ wN · gdx − ρ · ∇ wN · ∇ gdx
≤ C wN H˙ 1 . R3
R3
Arguing similarly as the estimate of L1 we derive that 1
3
1
3
|L4 | ≤ V L4 wN L4 ∇(ρ g)L2 ≤ C V L42 V H4˙ 1 wN L42 wN H4˙ 1 , and 1
3
1
3
1
3
1
3
1
3
1
3
|L5 | ≤ C BL42 BH4˙ 1 hN L42 hN H4˙ 1 , |L6 | ≤ C V L42 V H4˙ 1 wN L42 wN H4˙ 1 , |L7 | ≤ C BL42 BH4˙ 1 hN L42 hN H4˙ 1 . Thus it follows that
ρ∂t wN
3
3
3
3
H
H
H
H
≤ C 1 + wN 2˙ 1 + hN 2˙ 1 + V 2˙ 1 + B 2˙ 1 . H −1
(3.11)
Consequently, one has
ρ∂t wN
≤ C (T, V0 L2 , B0 L2 , w0 L2 , h0 L2 ).
4
(3.12)
L 3 ([0,T ];H −1 )
Arguing similarly to that used in deriving (3.12) it gives
ρ∂t hN
≤ C (T, V0 L2 , B0 L2 , w0 L2 , h0 L2 ).
4
(3.13)
L 3 ([0,T ];H −1 )
By the Lions–Aubin Lemma, there exists (wN , hN ) such that, for any ρ ∈ C0∞ (R3 ),
(ρ wN , ρ hN ) → (ρ w, ρ h) in L2 ([0, T ]; L2 (R3 )).
(3.14)
Step 3: Passing to the limit for the weak solution (w, h) of (1.5).
Since (wN , hN ) is the smooth solution of the Eq. (3.7), then for any test function ϕ ∈ C0∞ (R3 ) and ∇ · ϕ = 0, multiplying the first equation of (3.7) by ϕ , and integrating it over R3 × [0, t ), we have the identity (3.15). Let (w, h) be the limit of the approximate solution (wN , hN ) as N → ∞. Our aim is to prove that the limit function (w, h) is a weak solution to (1.5). First, we pass the limit for the first equation
R3
wN (x, t )ϕ(x)dx − − + −
t 0
t 0
t 0
R3
R3
R3
R3
wN0 (x, t )ϕ(x)dx =
hN ⊗ hN : PJN ∇ϕ(x)dxdτ + wN ⊗ V : PJN ∇ϕ(x)dxdτ + hN ⊗ B : PJN ∇ϕ(x)dxdτ −
As N → ∞, from Step 2, we obtain
R3
and
t 0 t
R3
R3 0 t 0
R3
R3
(wN − w)ϕ dxdτ ≤
Obviously,
(
wN0
− w0 )∇ϕ dx ≤ C
wN ⊗ wN : PJN ∇ϕ(x)dxdτ
wN · ϕ(x)dxdτ V ⊗ wN : PJN ∇ϕ(x)dxdτ
B ⊗ hN : PJN ∇ϕ(x)dxdτ .
(3.15)
ρ wN − ρ w2L2 dτ → 0, so ρ wN − ρ wL2 → 0 for a.e. t. Hence, we can easily get
0
1
0
R3
R3
0
(wN − w)ϕ dx ≤ (wN − w)ϕ 2 L2 ϕ 2 L2 → 0
t
T
t
0
t
1
as N → ∞,
(3.16)
(wN − w)(ϕ) 2 L2 (ϕ) 2 L2 dτ → 0 as N → ∞.
(3.17)
1
|w0 | dx 2
|ξ |≥N
We now estimate the nonlinear terms of (3.15).
1
12 →0
as N → ∞.
(3.18)
B. Yuan, L. Bai / Applied Mathematics and Computation 269 (2015) 443–455
t
[wN ⊗ wN : PJN ∇ϕ(x) − w ⊗ w : ∇ϕ(x)]dxdτ
R3
0
t
≤
449
R3
0
(wN ⊗ wN ) : ( JN ∇ϕ − ∇ϕ)dxdτ +
t R3
0
(wN ⊗ wN : ∇ϕ − w ⊗ w : ∇ϕ)dxdτ
M1 + M2 , ¨ By Holder and Gagliardo–Nirenberg inequalities M1 can be estimated as
M1 ≤
t
0
wN 2L4 dτ · JN ∇ϕ − ∇ϕL2 ≤ C (T, w0 L2 )JN ∇ϕ − ∇ϕL2 → 0
as N → ∞,
¨ Holder inequality and (3.14) together give
t
M2 =
R3
0
(wNj wNk − w j wk )∂ j ϕk dxdτ
√ T w0 L2 ∂ j ϕk (wN − w)L2 → 0
≤
as N → ∞.
Hence,
t R3
0
[wN ⊗ wN : PJN ∇ϕ(x) − w ⊗ w : ∇ϕ(x)]dxdτ → 0
as N → ∞.
(3.19)
Using an argument similar to that used in deriving (3.19) it yields
t
R3
0
[hN ⊗ hN : PJN ∇ϕ(x) − h ⊗ h : ∇ϕ(x)]dxdτ → 0 as N → ∞.
(3.20)
The estimate of the fourth to seventh terms on the right hand-side of (3.15) can be handled by precisely the same process as (3.19).
t
R3
0
t R3
0
t R3
0
and
t R3
0
[wN ⊗ V : PJN ∇ϕ(x) − w ⊗ V : ∇ϕ(x)]dxdτ → 0 as N → ∞,
[V ⊗ wN : PJN ∇ϕ(x) − V ⊗ w : ∇ϕ(x)]dxdτ → 0
R3
as N → ∞,
(3.23)
[B ⊗ hN : PJN ∇ϕ(x) − B ⊗ h : ∇ϕ(x)]dxdτ → 0
as N → ∞.
(3.24)
w(x, t )∇ϕ(x)dx − − + −
t 0
R3
0
R3
0
R3
t t
R3
w0 (x, t )∇ϕ(x)dx =
h ⊗ h : ∇ϕ(x)dxdτ +
t
w ⊗ V : ∇ϕ(x)dxdτ + h ⊗ B : ∇ϕ(x)dxdτ −
R3
0
R3
0
t
R3
0
R3
h(x, t )∇ϕ(x)dx −
− + −
t 0
R3
0
R3
0
R3
t t
R3
V ⊗ h : ∇ϕ(x)dxdτ + h ⊗ V : ∇ϕ(x)dxdτ −
t 0
R3
0
R3
0
R3
t t
R3
0
w ⊗ w : ∇ϕ(x)dxdτ
V ⊗ w : ∇ϕ(x)dxdτ
B ⊗ h : ∇ϕ(x)dxdτ .
h0 (x, t )∇ϕ(x)dx =
h ⊗ w : ∇ϕ(x)dxdτ +
t
w · ϕ(x)dxdτ
t
Using an identical process we can deduce that
(3.22)
[hN ⊗ B : PJN ∇ϕ(x) − h ⊗ B : ∇ϕ(x)]dxdτ → 0
Combing (3.16)–(3.24), one has
as N → ∞,
(3.21)
t 0
R3
(3.25)
w ⊗ h : ∇ϕ(x)dxdτ
h · ϕ(x)dxdτ w ⊗ B : ∇ϕ(x)dxdτ B ⊗ w : ∇ϕ(x)dxdτ .
Therefore, (w(x, t), h(x, t)) is a weak solution to the Eq. (1.5).
(3.26)
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B. Yuan, L. Bai / Applied Mathematics and Computation 269 (2015) 443–455
4. Decay for the weak solution This section is devoted to the proof of decay for the weak solution of Theorem 1.2. We need a generalized energy inequality and some corollaries. 4.1. The generalized energy inequality For the purpose of proving the generalized energy inequality, we give the following Lemma. Lemma 4.1. For every T > 0, the approximate solution (wN , hN ) constructed in Theorem 1.1 strongly converges to (w, h) in L2 ([0, T ]; L2 (R3 )). Proof. For every R > 0, define the cut-off function ϕR (x) = ϕ( Rx ), where
ϕ(x) ∈
t
0
C0∞
(R ) and ϕ(x) = 3
wN − w2L2 dt =
0
0
≤
wN − wN ϕR 2L2 =
t
wϕR − w2L2 ≤ 5
|wN − wN ϕR |2 dx ≤
R3
|wN |2 dx → 0
|x|≥R
|x|≥R
|x| ≥ 2, |x| ≤ 1.
wN − wN ϕR 2L2 dt +
Similarly, we have
for for
wN − wN ϕR + wN ϕR − wϕR + wϕR − w2L2 dt
t
≤5
0, 1,
|w|2 dx → 0
By the Lions–Aubin Lemma, we have proof is thus complete.
t 0
t 0
wN ϕR − wϕR 2L2 dt +
|x|≥2R
|wN |2 dx + 4
R≤|x|≤2R
t
0
wϕR − w2L2 dt.
|wN |2 dx
as R → ∞.
(4.1)
as R → ∞.
(4.2)
wN − w2L2 ds → 0 as N → ∞. Similarly, one has
t 0
hN − h2L2 ds → 0 as N → ∞. The
Lemma 4.2. (Generalized energy inequality) Let E(t) ∈ C1 [0, ∞) with E(t) ≥ 0 and ψ(t, x) ∈ C 1 ([0, ∞); S(R3 )). Then the weak solution (w, h) in L∞ ([0, ∞); L2 ) ∩ L2 ([0, ∞); H1 ) of the problem (1.5) satisfies the following generalized energy inequality
E (t )ψ(t ) ∗ w(t )22 + 2
E (τ )ψ(τ ) ∗ ∇ w(τ )22 ≤ E (s)ψ(s) ∗ w(s)22
s
+
t
t
E s
(τ )ψ(τ ) ∗ w(τ )22 dτ + 2
s
t
E (τ )
ψ (τ ) ∗ w(τ ), ψ(τ ) ∗ w(τ )
|w · ∇ w(τ ), ψ ∗ ψ ∗ w(τ )| + |h · ∇ h(τ ), ψ ∗ ψ ∗ w(τ )| |w · ∇V, ψ ∗ ψ ∗ w(τ )| + |V · ∇ w, ψ ∗ ψ ∗ w(τ )| + |h · ∇ B, ψ ∗ ψ ∗ w(τ )| + |B · ∇ h, ψ ∗ ψ ∗ w(τ )| dτ , + +
E (t )ψ(t ) ∗ h(t )22 + 2
+
t
s t
E s
(4.3)
E (τ )ψ(τ ) ∗ ∇ h(τ )22 ≤ E (s)ψ(s) ∗ h(s)22
(τ )ψ(τ ) ∗ h(τ ) τ + 2 2 2d
t s
E (τ ) ψ (τ ) ∗ h(τ ), ψ(τ ) ∗ h(τ )
|w · ∇ h(τ ), ψ ∗ ψ ∗ h(τ )| + |h · ∇ w(τ ), ψ ∗ ψ ∗ h(τ )| + |h · ∇ V, ψ ∗ ψ ∗ h(τ )| + |V · ∇ h, ψ ∗ ψ ∗ h(τ )| + |w · ∇ B, ψ ∗ ψ ∗ h(τ )| + |B · ∇ w, ψ ∗ ψ ∗ h(τ )| dτ . +
(4.4)
In reference [12], Ogawa et al. proved the Generalized Energy inequality for Navier–Stokes equation. For the sake of completeness, we give the detailed proof. Proof. We first recall the equation
∂t wN = −PJN (wN · ∇ wN ) + PJN (hN · ∇ hN ) + wN − PJN (wN · ∇V ) − PJN (V · ∇ wN ) + PJN (hN · ∇ B) + PJN (B · ∇ hN ).
(4.5)
B. Yuan, L. Bai / Applied Mathematics and Computation 269 (2015) 443–455
451
To deduce the term E (t )ψ ∗ wN (t )22 , we multiply (4.5) by E(t)ψ ∗ ψ ∗ wN and integrate it over R3 × [s, t], by the Leibniz d (E (t )ψ dt
formula
∂t wN (τ ), one has
L
∗ wN (t )2L2 ) = E (t )ψ ∗ wN (t )2L2 + 2E (t )ψ (τ ) ∗ wN (τ ), ψ(τ ) ∗ wN (τ ) + 2E (t )ψ(τ ) ∗ wN (τ ), ψ(τ ) ∗
E (t )ψ ∗ wN (t )2L2 + 2
+2
t s t s
E (τ )∇ψ ∗ wN 2L2 dτ ≤ E (s)ψ ∗ wN (τ )2L2 +
E (τ )
t s
E (τ )ψ ∗ wN 2L2 dτ
|ψ (τ ) ∗ wN (τ ), ψ(τ ) ∗ wN (τ )| + |PJN (wN · ∇ wN ), ψ ∗ ψ ∗ wN |
|PJN (hN · ∇ hN ), ψ ∗ ψ ∗ wN | + |PJN (wN · ∇V ), ψ ∗ ψ ∗ wN | + |PJN (V · ∇ wN ), ψ ∗ ψ ∗ wN | + |PJN (hN · ∇ B), ψ ∗ ψ ∗ wN | + |PJN (B · ∇ hN ), ψ ∗ ψ ∗ wN | dτ . +
(4.6)
For the purpose of proving the generalized energy inequality (4.3), we aim to take the limit as N → ∞ in (4.6), so we estimate each term of them. According to Lemma 4.1, we have wN → w in L2 (0, T ; L2 (R3 )). Therefore, one can easily get
ψ ∗ wN (t )L2 → ψ ∗ w(t )L2 , ∇ψ ∗ wN (t )L2 → ∇ψ ∗ w(t )L2 , for a.e. t. Now we estimate the nonlinear terms in (4.6).
PJN (wN · ∇ wN ), ψ ∗ ψ ∗ wN − w · ∇ w, ψ ∗ ψ ∗ w dτ s t = E (τ ) JN (wN · ∇ wN ), ψ ∗ ψ ∗ wN − (wN · ∇ wN ), ψ ∗ ψ ∗ wN dτ s t + E (τ ) wN · ∇ wN , ψ ∗ ψ ∗ wN − w · ∇ w, ψ ∗ ψ ∗ wN dτ s t N + E (τ ) w · ∇ w, ψ ∗ ψ ∗ w − w · ∇ w, ψ ∗ ψ ∗ w dτ t
E (τ )
s
R11 + R12 + R13 . Using the facts that wN ⊗ wN ∈ L2 (0, T ; L2 (R3 )) and ∇ψ ∗ ψ ∗ wN ∈ L2 (R3 ), we obtain
R11 ≤ C (T, w0 L2 )JN ∇(ψ ∗ ψ ∗ wN ) − ∇(ψ ∗ ψ ∗ wN )L2 → 0 as N → ∞. To estimate the term R12 , we use the Hölder and Young’s inequalities and Lemma 4.1 to obtain
R12 =
t
s
E (τ )
≤C
t s
R3
N N N N (w · ∇ w )ψ ∗ ψ ∗ w − (w · ∇ w)ψ ∗ ψ ∗ w dxdτ
E (τ )wN − wL2 dτ → 0 as N → ∞.
Similar to R12 one has
R13 ≤
t s
E (τ )w2L2 ∇(ψ ∗ ψ)L2 wN − wL2 dτ → 0 as N → ∞.
Hence, one has
t s
E (τ )PJN (wN · ∇ wN ), ψ ∗ ψ ∗ wN − w · ∇ w, ψ ∗ ψ ∗ w dτ → 0 as N → ∞.
Using a same process as the above it gives that
t s
E (τ )PJN (hN · ∇ hN ), ψ ∗ ψ ∗ wN − h · ∇ h, ψ ∗ ψ ∗ w dτ → 0 as N → ∞,
E (τ )PJN (w · ∇ V ), ψ ∗ ψ ∗ w − w · ∇ V, ψ ∗ ψ ∗ w dτ → 0 as N → ∞, s t N N E (τ )PJN (V · ∇ w ), ψ ∗ ψ ∗ w − V · ∇ w, ψ ∗ ψ ∗ w dτ → 0 as N → ∞, s t N N E (τ )PJN (h · ∇ B), ψ ∗ ψ ∗ w − h · ∇ B, ψ ∗ ψ ∗ w dτ → 0 as N → ∞,
t
s
N
N
(4.7)
452
B. Yuan, L. Bai / Applied Mathematics and Computation 269 (2015) 443–455
t
E (τ )PJN (B · ∇ hN ), ψ ∗ ψ ∗ wN − B · ∇ h, ψ ∗ ψ ∗ w dτ → 0 as N → ∞.
s
Combining the above estimates, we obtain the inequality (4.3). By an argument similar to that used in deriving the inequality (4.3) we can produce (4.4). The proof is thus complete. From the generalized energy inequality, we have the following corollaries. Note that the detailed proof is stated in references [8] and [12] and we sketch it for the completeness. Corollary 4.1. Let (w,h) be a weak solution to (3.1) satisfying the generalized energy inequality (4.3) and (4.4). Then for every ϕ ∈ S(R3 ), we have
ϕ ∗ w
2 L2
≤ e(t−s) ϕ ∗ w(s)2L2 + 2
t
|w · ∇ w, e2(t−τ ) (ϕ ∗ ϕ ∗ w)(τ )|
s
+ |w · ∇ V, e2(t−τ ) (ϕ ∗ ϕ ∗ w)(τ )| + |V · ∇ w, e2(t−τ ) (ϕ ∗ ϕ ∗ w)(τ )| + |h · ∇ h, e2(t−τ ) (ϕ ∗ ϕ ∗ w)(τ )| + |h · ∇ B, e2(t−τ ) (ϕ ∗ ϕ ∗ w)(τ )|
+ |B · ∇ h, e (ϕ ∗ ϕ ∗ w)(τ )| dτ , t ϕ ∗ h2L2 ≤ e(t−s) ϕ ∗ h(s)2L2 + 2 |w · ∇ h, e2(t−τ ) (ϕ ∗ ϕ ∗ h)(τ )| 2(t−τ )
(4.8)
s
2(t−τ )
(ϕ ∗ ϕ ∗ h)(τ )| + |V · ∇ h, e2(t−τ ) (ϕ ∗ ϕ ∗ h)(τ )| + |w · ∇ B, e (ϕ ∗ ϕ ∗ h)(τ )| + |h · ∇V, e2(t−τ ) (ϕ ∗ ϕ ∗ h)(τ )| + |B · ∇ w, e2(t−τ ) (ϕ ∗ ϕ ∗ h)(τ )| dτ . + |h · ∇ w, e
2(t−τ )
(4.9)
Proof. Take E(t) ≡ 1 and ψ(τ ) = e(t−τ ) ϕ in (4.3). Then ψ(τ ) ∗ w(τ ) = e(t−τ ) ϕ ∗ w(τ ) and
ψ (τ ) ∗ w(τ ), ψ(τ ) ∗ w(τ ) − ∇ψ(τ ) ∗ w(τ )2L2 = −e(t−τ ) ϕ ∗ w(τ ), e(t−τ ) ϕ ∗ w(τ ) − ∇ e(t−τ ) ϕ ∗ w(τ )2L2 = 0. Hence, we have
ϕ ∗ w(t )
2 L2
≤ e(t−s) ϕ ∗ w(s)2L2 + 2
t
|w · ∇ w, e2(t−τ ) (ϕ ∗ ϕ ∗ w)(τ )|
s
+ |w · ∇ V, e2(t−τ ) (ϕ ∗ ϕ ∗ w)(τ )| + |V · ∇ w, e2(t−τ ) (ϕ ∗ ϕ ∗ w)(τ )| + |h · ∇ h, e2(t−τ ) (ϕ ∗ ϕ ∗ w)(τ )| + |h · ∇ B, e2(t−τ ) (ϕ ∗ ϕ ∗ w)(τ )|
+ |B · ∇ h, e
2(t−τ )
(ϕ ∗ ϕ ∗ w)(τ )| dτ .
(4.10)
Using the same process that used in deriving (4.10) we can obtain (4.9). The proof is thus complete. Corollary 4.2. Let E(t) ∈ C1 [0, ∞) and E(t) ≥ 0. Let (w, h) be a weak solution to (3.1) satisfying the generalized energy inequality (4.3) and (4.4). Then for every ϕ ∈ S(R3 ), we have (w, h) satisfies
E (t )w(t ) − ϕ ∗ w(t )22 + 2 +
t
E
s
+2
t
s
s
+2 +2
t
t s
t
E s
t s
E (τ )∇ w(τ ) − ϕ ∗ ∇ w(τ )22 dτ ≤ E (s)w(s) − ϕ ∗ w(s)22
(τ )w(τ ) − ϕ ∗ w(τ )22 dτ + 2
s
t
E (τ )|w · ∇ w, ϕ ∗ ϕ ∗ w − 2ϕ ∗ w(τ )|dτ
E (τ )|h · ∇ h, w − 2ϕ ∗ w + ϕ ∗ ϕ ∗ w(τ )|dτ + 2 E (τ )|V · ∇ w, ϕ ∗ ϕ ∗ w − 2ϕ ∗ w(τ )|dτ + 2
s
t
t s
E (τ )|w · ∇ V, w − 2ϕ ∗ w + ϕ ∗ ϕ ∗ w(τ )|dτ
E (τ )|h · ∇ B, w − 2ϕ ∗ w + ϕ ∗ ϕ ∗ w(τ )|dτ
E (τ )|B · ∇ h, w − 2ϕ ∗ w + ϕ ∗ ϕ ∗ w(τ )|dτ .
E (t )h(t ) − ϕ ∗ h(t )22 + 2 +
t s
E (τ )∇ h(τ ) − ϕ ∗ ∇ h(τ )22 dτ ≤ E (s)h(s) − ϕ ∗ h(s)22
(τ )h(τ ) − ϕ ∗ h(τ )22 dτ + 2
t s
E (τ )|w · ∇ h, ϕ ∗ ϕ ∗ h − 2ϕ ∗ h(τ )|dτ
(4.11)
B. Yuan, L. Bai / Applied Mathematics and Computation 269 (2015) 443–455
+2
s
s
+2 +2
t
t
t s
E (τ )|h · ∇ w, h − 2ϕ ∗ h + ϕ ∗ ϕ ∗ h(τ )|dτ + 2 E (τ )|V · ∇ h, ϕ ∗ ϕ ∗ h − 2ϕ ∗ h(τ )|dτ + 2
t s
t s
453
E (τ )|h · ∇ V, h − 2ϕ ∗ h + ϕ ∗ ϕ ∗ h(τ )|dτ
E (τ )|w · ∇ B, h − 2ϕ ∗ h + ϕ ∗ ϕ ∗ h(τ )|dτ
E (τ )|B · ∇ w, h − 2ϕ ∗ h + ϕ ∗ ϕ ∗ h(τ )|dτ .
(4.12)
Proof. Let ψ(x, t ) = ζn (x) − ϕ(x) in (4.3), where ζn (x) = n−3 ζn ( nx ) is a smooth and compactly supported approximation of the t Dirac measure. Then s E (τ )|ψ (τ ) ∗ w(τ ), ψ ( τ ) ∗ w(τ )|dτ = 0. Combining the fact of ϕ ∗ ϕ ∗ w = ζn ∗ ζn ∗ w − 2ζn ∗ ϕ ∗ w + ϕ ∗ ϕ ∗ w and the divergence free for w, and passing to the limit as n → ∞ it yields the inequality (4.11). Arguing similarly to above we obtain (4.12). This completes the proof. 4.2. The proof of Theorem 1.2 We use the Fourier splitting method [13] to prove the decay of the weak solution (w, h). The proof is spilt into two steps, that is, the estimates for the low frequency part and the high frequency part of the energy. For each step, we use the Generalized energy inequality and Corollaries 4.1 and 4.2 with suitable functions E(t) and ψ . The L2 −norm of the Fourier transform of the weak solution (w, h) is divided into the following two parts.
ˆ (t )L2 + (1 − ϕ) ˆ (t )L2 , ˆ L2 ≤ ϕˆ w ˆ w w(t )L2 = wˆ (t )L2 = wˆ (t ) − wˆ (t )ϕˆ + wˆ (t )ϕ
(4.13)
h(t )L2 = hˆ (t )L2 = hˆ (t ) − hˆ (t )ϕˆ + hˆ (t )ϕ ˆ L2 ≤ ϕˆ hˆ (t )L2 + (1 − ϕ) ˆ hˆ (t )L2 , where we choose ϕ(ξ ˆ ) = exp ( − |ξ |2 ). Step 1: Estimates for the low frequencies of w. By Corollary 4.1 and Plancherel identity, we have
ϕˆ wˆ (t )2L2 ≤ e(t−s) ϕ ∗ w(s)2L2 + 2
t
s
|w · ∇ w, e2(t−τ ) (ϕ ∗ ϕ ∗ w)(τ )|
+ |w · ∇ V, e2(t−τ ) (ϕ ∗ ϕ ∗ w)(τ )| + |V · ∇ w, e2(t−τ ) (ϕ ∗ ϕ ∗ w)(τ )| + |h · ∇ h, e2(t−τ ) (ϕ ∗ ϕ ∗ w)(τ )| + |h · ∇ B, e2(t−τ ) (ϕ ∗ ϕ ∗ w)(τ )|
+ |B · ∇ h, e2(t−τ ) (ϕ ∗ ϕ ∗ w)(τ )| dτ K1 (t, s) + 2
s
t
(|K2 | + |K3 | + |K4 | + |K5 | + |K6 | + |K7 |)dτ ,
(4.14)
It is not difficult to see that the estimates of K5 is the same as K2 while K4 , K6 and K7 are the same as K3 . Therefore, it suffices to estimate K1 , K2 , K3 . For the fixed s ≥ 0, by the Plancherel identity and the Lebesgue dominated convergence theorem it follows that
|K1 | = e(t−s) ϕ ∗ w(s)2L2 = (2π )3 e−(t−s)|ξ | ϕˆ wˆ (s)2L2 → 0 as t → ∞. 2
(4.15)
Hölder, Young inequalities and Sobolev embedding theorem together give
|K2 | ≤ ϕ ∗ ϕ ∗ (w · ∇ w)L2 e2(t−τ ) wL2 ≤ C ϕ ∗ ϕ 6 wL6 ∇ wL2 wL2 ≤ C ϕ ∗ ϕ 6 w0 L2 ∇ w2L2 . L5 L5 By the L2 − estimate for the heat semigroup, Hölder inequality and Sobolev embedding, one can deduce that
|K3 | ≤ C V L3 wL6 ∇ e2(t−τ ) (ϕ ∗ ϕ ∗ w)L2 ≤ C V ˙ 1 ∇ w2L2 ϕ ∗ ϕL1 . H2 Combining the above estimates into (4.14), and taking t → ∞ we get
ϕˆ wˆ (t )2L2 ≤ K1 (t, x) + C (w0 L2 + V H˙ 12 + BH˙ 12 )
t s
(∇ w(τ )2L2 + ∇ h(τ )2L2 )dτ .
(4.16)
Let s > 0 be fixed and large on the right-hand side of (4.16). The term K1 (t, x) tends to zero as t → ∞ by (4.15). From (3.9), it can t ˆ (t )22 → 0 as t → ∞. be deduced that the quantity s (∇ w(τ )22 + ∇ h(τ )22 )dτ converges 0 as s → ∞. Therefore, ϕˆ w L
L
L
Step 2: Estimate the high frequencies for w. Here we use Corollary 4.2 with the test function ϕ satisfying ϕ(ξ ˆ ) = exp{−|ξ |2 }
ˆ (t )L2 = w(t ) − ϕ ∗ w(t )L2 , it is easy to see and a function E(t) > 0 to be determined below. In this step, we estimate (1 − ϕ) ˆ w that we only need to estimate each term on the right-hand of (4.11) successively. To simplify the presentation, we denote
S1
t
E s
(τ )w(τ ) − ϕ ∗ w(τ )22 dτ − 2
t s
E (τ )∇ w(τ ) − ϕ ∗ ∇ w(τ )22 dτ ,
454
B. Yuan, L. Bai / Applied Mathematics and Computation 269 (2015) 443–455
S2
t
E (τ )|w · ∇ w, ϕ ∗ ϕ ∗ w − 2ϕ ∗ w(τ )|dτ ,
s
S3
t
E (τ )|h · ∇ h, w − 2ϕ ∗ w + ϕ ∗ ϕ ∗ w(τ )|dτ ,
s
S4
t
E (τ )|w · ∇ V, w − 2ϕ ∗ w + ϕ ∗ ϕ ∗ w(τ )|dτ ,
s
S5
t
E (τ )|V · ∇ w, ϕ ∗ ϕ ∗ w − 2ϕ ∗ w(τ )|dτ ,
s
S6
t
E (τ )|h · ∇ B, w − 2ϕ ∗ w + ϕ ∗ ϕ ∗ w(τ )|dτ ,
s
S7
t
E (τ )|B · ∇ h, w − 2ϕ ∗ w + ϕ ∗ ϕ ∗ w(τ )|dτ .
s
Now we estimate S1 to S7 .
S1 =
t
E
(τ )
s
+
t
E
s
|ξ |>G(t )
(τ )
2
t
dξ dτ − 2 E (τ )
(1 − ϕ(ξ ˆ ˆ )) (ξ , τ ) w
s
|ξ |≤G(t )
|ξ |>G(t )
2
t
dξ dτ − 2 E (τ )
(1 − ϕ(ξ ˆ ˆ )) (ξ , τ ) w
s
2
|ξ |(1 − ϕ(ξ ˆ (ξ , τ )
dξ dτ ˆ ))w
|ξ |≤G(t )
2
dξ dτ .
|ξ |(1 − ϕ(ξ ˆ ˆ )) (ξ , τ ) w
α 2 We choose E (t ) = (1 + t )α and G2 (t ) = 2(1+t ) with fixed α > 0, thus E (t ) − 2E (t )G (t ) = 0. So
t
E
s
(τ )
≤
t
E s
=
|ξ |>G(t )
(τ )
2
t
dξ dτ − 2 E (τ )
(1 − ϕ(ξ ˆ ˆ )) (ξ , τ ) w
s
|ξ |>G(t )
2
t
dξ dτ − 2 E (τ )G2 (τ )
(1 − ϕ(ξ ˆ ˆ )) (ξ , τ ) w
2 E (τ ) − 2E (τ )G (τ )
t
s
|ξ |>G(t )
2
|ξ |(1 − ϕ(ξ ˆ (ξ , τ )
dξ dτ ˆ ))w
s
|ξ |>G(t )
|ξ |>G(t )
2
(1 − ϕ(ξ ˆ (ξ , τ )
dξ dτ = 0. ˆ ))w
For |ξ | ≤ G(t), it follows that 1 − ϕ(ξ ˆ ) = 1 − e−|ξ | , thus, we obtain
2
dξ dτ
(1 − ϕ(ξ ˆ ˆ )) (ξ , τ ) w
2
t
E s
(τ )
|ξ |≤G(t )
2
t
dξ dτ ≤ E (τ )
(1 − ϕ(ξ ˆ ˆ )) (ξ , τ ) w
s
|ξ |≤G(t )
2
2
|ξ wˆ (ξ , τ ) dξ dτ
2
t t
4
dξ dτ ≤ C w0 2L2 E (τ )G4 (τ )dτ
ˆ ≤ E (τ )G (τ ) (ξ , τ ) w
|ξ |≤G(t ) s s
≤C
s
t
(1 + τ )α−3 dτ .
t ˆ (ξ , τ )|2 dξ dτ ≥ 0. Since E(t) ≥ 0, we get s E (τ ) |ξ |≤G(t ) |(1 − ϕ(ξ ˆ ))w t α −3 Therefore, S1 ≤ C s (1 + τ ) dτ . Set η = ϕ ∗ ϕ − 2ϕ , and applying the Hölder and Young inequalities, one has
S2 ≤
s
t
E (τ )w · ∇ w
3 L2
η ∗ wL3 dτ ≤ C ηL 65 w0 L2
t
s
E (τ )∇ w2L2 dτ .
Similar arguments as in deriving S2 we derive that
S3 ≤ C η
6 L5
w0 L2
t s t
E (τ )∇ h2L2 dτ , S4 ≤ C V ˙
H
S5 ≤ C V ˙
1 2
ηL1
S6 ≤ C B ˙
1 2
(1 + ηL1 )
H
H
s
E (τ )∇ w2L2 , S7 ≤ C B ˙
H
t s
(1 + ηL1 )
ηL1
s
t
t s
E (τ )∇ w2L2 dτ ,
E (τ )∇ wL2 ∇ hL2 dτ ,
E (τ )∇ hL2 ∇ wL2 dτ .
Thus,
ˆ (t )2L2 ≤ (1 − ϕ) ˆ w
1 2
1 2
E (s) C ˆ (s)2L2 + ˆ w (1 − ϕ) E (t ) E (t )
t s
(1 − τ )α−3 dτ +
C E (t )
t s
E (τ )(∇ w2L2 + ∇ h2L2 )dτ .
(4.17)
B. Yuan, L. Bai / Applied Mathematics and Computation 269 (2015) 443–455
455
For fixed s > 0, we compute the lim sup of both sides of (4.17), as t → ∞. Since E (t ) = (1 + t )α with α > 0, the first term on the right-hand side of (4.17) tends to zero.
limsup t→∞
Since
E (τ ) E (t )
=
1 (1 + t )α
τ α ( 1+ 1+t )
s
t
(1 − τ )α−3 dτ = limsup
≤ 1 for τ ∈ [0, t], for any small ε > 0, choosing s large enough, one has
ˆ (t )2L2 ≤ C limsup (1 − ϕ) ˆ w t→∞
t→∞
(1 + s)α−2 = 0. (1 + t )2 (α − 2)
s
∞
(∇ w2L2 + ∇ h2L2 )dτ < ε .
Combining the above estimates of the high and low frequencies for w, this completes the proof of wL2 → 0 as t → ∞. Obviously, we can take a similarly argument to h, and then can complete the proof of Theorem 1.2. Acknowledgments The research of B Yuan was partially supported by the National Natural Science Foundation of China (no. 11471103). References [1] W. Borchers, T. Miyakawa, L2 decay for Navier–Stokes flows in unbounded domains, with application to exterior stationary flows, Arch. Ration. Mech. Anal. 118 (1992) 273–295. [2] P. Constantin, C. Foias, Navier–Stokes Equations, Chicago University Press, Chicago, 1988. [3] L. Caffarelli, R. Kohn, L. Nirenberg, Partial regularity of suitable weak solutions of the Navier–Stokes equations, Commun. Pure Appl. Math. 35 (1982) 771–831. [4] M. Hess, M. Hieber, A. Mahalov, J. Saal, Nonlinear stability of Ekman boundary layers, Bull. London Math. Soc. 42 (2010) 691–706. [5] H. Kozono, Asymptotic stability of large solutions with large perturbation to the Navier–Stokes equations, J. Funct. Anal. 176 (2000) 153–197. [6] R. Kajikiya, T. Miyakawa, On L2 decay of weak solutions of the Navier–Stokes equations in Rn , Math. Z. 192 (1986) 135–148. [7] G. Karch, D. Pilarczyk, Asymptotic stability of landau solutions to Navier–Stokes system, Arch. Ration. Mech. Anal. 202 (2011) 115–131. [8] G. Karch, D. Pilarczyk, M. Schonbek, L2 − asymptotic stability of mild solutions to Navier–Stokes system in R3 , [math.AP] 30 Aug2013, arXiv:1308.6667v1. [9] J. Leray, étude de diverses équations integrales non lineaires et de quelques problèmes que pose l’hydrodynamique, J. Math. Pure Appl. 9 (1933) 1–82. [10] J. Leray, Sur le mouvement d’un liquide visqeux emplissant l’espace, Acta Math. 63 (1934) 193–248. [11] C.X. Miao, B.Q. Yuan, On the well-posedness of the Cauchy problem for an MHD system in Besov spaces, Math. Methods Appl. Sci. 32 (2009) 53–76. [12] T. Ogawa, S. Rajopadhye, M. Schonbek, Energy decay for a weak solution of the Navier–Stokes equation with slowly varying external forces, J. Funct. Anal. 144 (1997) 325–358. [13] M.E. Schonbek, l2 decay for weak solutions of the Navier–Stokes equations, Arch. Ration. Mech. Anal. 88 (1985) 209–222. [14] M. Wiegner, Decay results for weak solutions of the Navier–Stokes equations in Rn , J. London Math. Soc. 35 (1987) 303–313.