Well-Posedness in Critical Spaces for the Full Compressible Mhd Equations

Acta Mathematica Scientia 2013,33B(4):1153–1176 http://actams.wipm.ac.cn

WELL-POSEDNESS IN CRITICAL SPACES FOR THE FULL COMPRESSIBLE MHD EQUATIONS∗


)†

Dongfen BIAN (

The Graduate School of China Academy of Engineering Physics, Beijing 100088, China E-mail: [email protected]; bian [email protected]

 )

Boling GUO (

Institute of Applied Physics and Computational Mathematics, Beijing 100088, China E-mail: [email protected]

Abstract In this paper we prove local well-posedness in critical Besov spaces for the full compressible MHD equations in RN , N ≥ 2, under the assumptions that the initial density is bounded away from zero. The proof relies on uniform estimates for a mixed hyperbolic/parabolic linear system with a convection term. Key words full compressible MHD equations; Besov spaces; critical spaces; LittlewoodPaley theory; local well-posedness 2010 MR Subject Classification

1

76W05; 76N10

Introduction

In this paper, we consider the following full compressible magnetohydrodynamics (MHD) equations on RN , N ≥ 2,   ∂t ρ + div(ρu) = 0,       ∂t H − curl(u × H) − σ∆H = 0,       1   ∂t (ρu) + div(ρu ⊗ u − H ⊗ H) + ∇(p + |H|2 ) − µ∆u − (µ + λ)∇divu = ρf, 2 (1.1) µ  ⊤ 2 2 2   ∂ (ρθ) + div(ρuθ) − k∆θ + pdivu = |∇u + ∇u | − µ(divu) + σ(curlH) , t   2      divH = 0,      (ρ, H, u) = (ρ , H , u ), |t=0

0

0

0

which describes the motion of electrically conducting media in the presence of a magnetic field. Here ρ, u, H, p and θ denote the density, velocity, magnetic field, pressure and temperature, respectively. The pressure p is a suitably smooth function of ρ. We denote by σ, λ and µ the ∗ Received

October 13, 2011; revised January 6, 2012. author: Dongfen BIAN.

† Corresponding

1154

ACTA MATHEMATICA SCIENTIA

Vol.33 Ser.B

viscosity coefficients of the fluid, which are assumed to satisfy σ > 0, µ > 0 and λ+2µ > 0. Such a condition ensures ellipticity for the operator µ∆ + (λ + µ)∇ div and is satisfied in the physical cases (where λ + 2µ/N ≥ 0). If H = 0, then equations (1.1) are compressible Navier-Stokes equations, moreover, if divu = 0, then equaitons (1.1) are classical incompressible Navier-Stokes equations. For smooth initial data such that the density ρ0 is bounded and bounded away from zero (i.e., 0 < ρ ≤ ρ0 (x) ≤ M ), existence and uniqueness of local classical solutions to the compressible Navier-Stokes equations have been known for a long time (see the pioneering work of Nash [1] or the paper of Itaya [2]). Matsumura and Nishida [3] proved the global wellposedness for compressible Navier-Stokes equations for smooth data close to equilibrium. The reader may refer to [4–12] for more recent advances on the subject. Concerning the global existence of weak solutions to compressible Navier-Stokes equations for the large initial data, readers refer to [13–16], and refer to [4, 17–19] and references therein for the viscous shallow water equations. Recently, Fan-Zhou [20] obtained a blow-up criterion of the local strong solutions in terms of the temperature and positive density for the compressible Navier-Stokes equations of viscous heat-conductive fluids in a periodic domain T3 with zero heat conductivity k = 0, similar to the Beale-Kato-Majda criterion for ideal incompressible flows. Due to the physical importance and mathematical challenges, the study on (1.1) attracted many physicists and mathematicians [21–26]. Existence and uniqueness of (weak, strong or smooth) solutions in one dimension can be found in [27–31] and the references cited therein. This paper is devoted to the study of the well-posedness of system (1.1) for N ≥ 2 in the critical spaces. Recently, Danchin obtained several important well-posedness for compressible Navier1 0 2 Stokes equations [5–9]. Chen-Miao-Zhang [4] obtained the local well-posedness in B˙ 2,1 × (B˙ 2,1 ) for the viscous shallow water equations and for compressible Navier-Stokes equations with N density dependent viscosities in the Besov spaces B˙ p [32]. Zhou-Xin-Fan [33] proved wellp,1

posedness for the density-dependent incompressible Euler equations in the critical Besov spaces. Bian and Yuan obtained local well-posedness in the critical Besov spaces [34] and super critical Besov spaces [35] for the compressible MHD equations. In the absence of heat conduction, it was proved by Xin that any non-zero smooth solution with initially compact supported density would blow up in finite time (see [36]). This result was generalized to the cases for the nonbarotropic compressible Navier-Stokes system with heat conduction [37] and for non-compact but rapidly decreasing at far field initial densities [38]. As a reasonable starting point, we will therefore restrict our work to solutions such that ρ remains positive. We will also suppose that ρ tends to some constant ρ¯ > 0 at infinity. To explain the precise meaning of critical spaces, let us consider the incompressible Navier-Stokes equations   ∂ u − ∆u + u · ∇u + ∇p = 0, t (1.2)  divu = 0.

It is easy to find that if (u, p) is a solution of (1.2), then

uλ (t, x) = λu(λ2 t, λx), pλ (t, x) = λ2 p(λ2 t, λx)

(1.3)

is also a solution of (1.2). For equations (1.2), we say a functional space X is critical if the N corresponding norm is invariant under the scaling of (1.3). Obviously, H˙ 2 −1 is a critical space.

No.4

1155

D.F. Bian & B.L. Guo: WELL-POSEDNESS IN CRITICAL SPACES

N Fujita and Kato [39] proved the well-posedness of (1.2) in H˙ 2 −1 , see also [40–42] and references therein for the well-posedness in other critical spaces. For the full compressible MHD equations, let us introduce the following transformation

ρλ (t, x) := ρ(λ2 t, λx), Hλ (t, x) := λH(λ2 t, λx), uλ (t, x) := λu(λ2 t, λx), θλ (t, x) := λ2 θ(λ2 t, λx).

(1.4)

If (ρ, H, u, θ) solves (1.1), so does (ρλ , Hλ , uλ , θλ ) provided the pressure law has been changed into λ2 p. This motivates the following definition. Definition 1.1 A functional space B1 × B2 × B3 × B4 is called a critical space for (1.1) if the associated norm is invariant under the transformation (ρ, H, u, θ) −→ (ρλ , Hλ , uλ , θλ ) (up to a constant independent of λ). In addition to have a norm invariant by (1.4), appropriate function spaces for solving the system (1.1) must provide a control on L∞ norm of the density (in order to avoid vacuum and loss of ellipticity). For that reason, we restricted our study to the case where the initial data (ρ0 , H0 , u0 , θ0 ) and external force f are such that, for some positive constant ρ¯, N

N

N

−1

−1

N

−2

p p p p (ρ0 − ρ¯) ∈ B˙ p,1 , u0 ∈ B˙ p,1 , H0 ∈ B˙ p,1 , θ0 ∈ B˙ p,1

N

−1

p and f ∈ L1loc (R+ ; B˙ p,1 ).

To simplify the notation, we assume from now on that ρ¯ = 1. Hence, as long as ρ does not vanish, the equations for (a := ρ−1 − 1, H, u, θ) read:   ∂t a + u · ∇a = (1 + a)divu,         ∂t H + u · ∇H − σ∆H = H · ∇u − (∇ · u)H, (1.5) ∂t u + u · ∇u − (1 + a)Au + ∇(g(a)) = F,     ∂t θ + u · ∇θ − k(1 + a)∆θ = G,      divH = 0

with A := µ∆ + (λ + µ)∇div, F = (1 + a)(H · ∇H − ∇H · H) + f , G = (1 + a)(−pdivu + µ2 |∇u + ∇u⊤ |2 + µ(divu)2 + σ(curlH)2 ) and where g is a smooth function which may be computed from the pressure function p. Our main result is as follows. N N N p −1 N p −1 N p −2 Theorem 1.1 Assume that (u0 , H0 , θ0 ) ∈ (B˙ p,1 ) × (B˙ p,1 ) × B˙ p,1 , f ∈ L1loc (R+ ; N

N

p −1 p with 1 + a0 bounded away from zero. Then there exists a positive time B˙ p,1 ) and a0 ∈ B˙ p,1 T such that (a) Existence: If p ∈ [2, N ], system (1.5) has a solution (a, H, u, θ) ∈ ETp with N

N

N

N

p p −1 p +1 p −1 ETp := C([0, T ]; B˙ p,1 ) × (C([0, T ]; B˙ p,1 ) ∩ L1 (0, T ; B˙ p,1 ))N × (C([0, T ]; B˙ p,1 ) N

N

N

p +1 p −2 p ∩L1 (0, T ; B˙ p,1 ))N × (C([0, T ]; B˙ p,1 ) ∩ L1 (0, T ; B˙ p,1 ));

(b) Uniqueness: If p ∈ [2, N ], then the uniqueness holds in ETp . Remark 1.1 If we choose p = 2, then we obtain the results as similarly as that in [9]. So we only need to prove Theorem 1.1 in the case of p > 2. For completeness, we also prove the case p = 2. We follow the idea as in [9] to prove the well-posedness in critical Besov

1156

ACTA MATHEMATICA SCIENTIA

Vol.33 Ser.B

spaces for the full compressible MHD equations. But in our case for the general p the estimates of the linearized momentum equation are very difficult. We use Proposition 3.2, Proposition 3.4, Proposition 3.5 and Remark 3.2 which were proved by [34], and apply the communicator estimate to overcome the difficulty by the couple of the velocity field and the temperature. We also use the Littlewood-Paley theory and Bony’s paraproduct decomposition to prepare some new a priori estimates for the product of two functions and a communicator. Thus we obtain the well-posedness in general critical Besov spaces (for 2 ≤ p ≤ N ) for the full compressible MHD equations. Notation Throughout the paper, C stands for a “harmless ” constant, and we sometimes use the notation A . B as an equivalent of A ≤ CB. The notation A ≈ B means that A . B and B . A.

2

Littlewood-Paley Theory and the Functional Spaces

Let us introduce the homogeneous Littlewood-Paley decomposition which relies upon a dyadic partition of unity. We can choose a radial function ϕ ∈ S(RN ) supported in C = {ξ ∈ RN , 43 ≤ |ξ| ≤ 38 } such that X ϕ(2−q ξ) = 1 if ξ 6= 0. q∈Z

The frequency localization operator ∆q and Sq are defined by X ∆q f = ϕ(2−q D)f, Sq f = ∆q f for q ∈ Z. k≤q−1

Furthermore, the above dyadic decomposition has nice properties of quasi-orthogonality: ∆q ∆k f ≡ 0 if |q − k| ≥ 2 and ∆q (Sk−1 f ∆k f ) ≡ 0 if |q − k| ≥ 5.

(2.1)

We denote the space Sh′ (RN ) by the dual space of Sh (RN ) = {f ∈ S(RN ); Dα fˆ(0) = 0; ∀α ∈ Nd multi-index}. It also can be identified by the quotient space S ′ (RN )/P with the polynomials space P. The formal equality X f= ∆k f k∈Z

holds true for f ∈ Sh′ (RN ) and is called the homogeneous Littlewood-Paley decomposition. Let us turn to the definition of the Besov spaces that we need. Definition 2.1 For s ∈ R, p, r ∈ [1, +∞]. The homogeneous Besov space is defined by s B˙ p,r = {f ∈ Sh′ (RN ) : kf kB˙ s < +∞}, p,r

where kf kB˙ s := p,r

  1r X   ks r   (2 k∆k f kp ) , k∈Z ks

    sup 2 k∆k f kp , k∈Z

if r < +∞; (2.2) if r = +∞.

1157

No.4

D.F. Bian & B.L. Guo: WELL-POSEDNESS IN CRITICAL SPACES

and

˜ q (B˙ s ) which is initiated in [43]. We next introduce the Besov-Chemin-Lerner space L p,r T Definition 2.2 Let (p, r) ∈ [1, +∞]2 , T ∈ [0, +∞] and s ∈ R. We set

Z T 1/q

ks q

kf kL˜ q (B˙ p,r k∆ f (t)k dt := 2 s ) k p

T

0

lr

s s ˜ q (B˙ p,r L ) := {u ∈ LqT (B˙ p,r ), kf kL˜ q (B˙ p,r s ) < +∞}. T T

By the Minkowski inequality, we have ˜ q (B˙ s ) ֒→ Lq (B˙ s ) if q > r, L p,r p,r T T q ˙s q ˙s ˜ LT (Bp,r ) ֒→ LT (Bp,r ) if q < r and ˜ q (B˙ s ) = Lq (B˙ s ) if q = r. L p,r p,r T T Let us conclude this section by collecting some useful lemmas. Lemma 2.1 [32, 44] Let 1 ≤ p ≤ q ≤ +∞. Assume that f ∈ Lp (RN ), then for any γ ∈ (N ∪ 0)N , there exist two constants C1 , C2 independent of f, j such that 1

1

suppfˆ ⊆ {|ξ| ≤ A0 2j } ⇒ k∂ γ f kq ≤ C1 2j|γ|+jN ( p − q ) kf kp , suppfˆ ⊆ {A1 2j ≤ |ξ| ≤ A2 2j } ⇒ kf kp ≤ C2 2−j|γ| sup k∂ β f kp . |γ|=|β|

Lemma 2.2 [5, 32] Let 1 < p < ∞, and a ≥ a ¯ > 0 be a bounded continuous function. p N Assume that u ∈ L (R ) and suppˆ u ⊆ {R1 ≤ |ξ| ≤ R2 }. Then there exists a constant c depending only on N and R2 /R1 such that Z Z p−1 c¯ aR12 2 |u|p dx ≤ − div(a∇u)|u|p−2 udx. p N N R R Lemma 2.3 [34] Let s1 ≤ Np , s2 < Np , s1 + s2 ≥ N max(0, 2p − 1), and 1 ≤ p ≤ ∞. s1 s2 Assume that f ∈ B˙ p,1 and g ∈ B˙ p,∞ . Then there holds kf gk

s +s2 − N p

1 ˙ p,∞ B

s2 . ≤ Ckf kB˙ s1 kgkB˙ p,∞ p,1

s Lemma 2.4 Let s > 0, and 1 ≤ p ≤ ∞, Assume that f, g ∈ B˙ p,1 ∩ L∞ . Then there holds kf gkB˙ s ≤ C(kf kB˙ s kgkL∞ + kf kL∞ kgkB˙ s ). p,1

p,1

p,1

2 p

s1 + s2 > N max(0, − 1) (s1 , s2 ≤ Np , s1 + s2 ≥ Lemma 2.5 Let s1 , s2 < s1 s2 N max(0, p2 − 1), if r = 1), and 1 ≤ p ≤ ∞. Assume that f ∈ B˙ p,r and g ∈ B˙ p,r . Then there holds s1 kgk ˙ s2 . kf gk s1+s2 − Np ≤ Ckf kB˙ p,r Bp,r N p,

˙ p,r B

N

p +1 Lemma 2.6 Let s ∈ (−N min( p1 , p1′ ), Np + 1], and 1 ≤ p ≤ ∞. Assume that f ∈ B˙ p,1 s and g ∈ B˙ p,1 . Then there holds X 2js k[f · ∇, ∆j ]gkLp ≤ Ckf k Np +1 kgkB˙ s .

j

˙ B p,1

p,1

1158

ACTA MATHEMATICA SCIENTIA

Vol.33 Ser.B N

N

−p p +1 . Then and g ∈ B˙ p,∞ Lemma 2.7 If s = − Np , and 2 ≤ p ≤ ∞. Assume that f ∈ B˙ p,1 there holds N sup 2j(− p ) k[f · ∇, ∆j ]gkLp ≤ Ckf k Np +1 kgk − Np . ˙ p,∞ B

B˙ p,1

j∈Z

[s]+3

Lemma 2.8 Let s > 0, and 1 ≤ p ≤ ∞. Assume that F ∈ Wloc s Then for any f ∈ L∞ ∩ B˙ p,1 , we have

(R) with F (0) = 0.

kF (f )kB˙ s ≤ C(1 + kf kL∞ )[s]+2 kf kB˙ s . p,1

p,1

[s]+3

Lemma 2.9 Let s ∈ (− Np , Np ), and 1 ≤ p ≤ ∞. Assume that F ∈ Wloc

(R) with

N p

s , we have F ′ (0) = 0. Then for any f, g ∈ B˙ p,∞ ∩ L∞ , and that f − g belongs to B˙ p,∞

kF (f ) − F (g)kB˙ s

p,∞

≤ C(kf k

N

˙ p B p,1

+ kgk

N

˙ p B p,1

)kf − gkB˙ s . p,∞

Lemma 2.10 If Rj = ∆j (a∂k w) − ∂k (a∆j w). There exists C = C(N, σ) such that X 2jσ kRj kLp ≤ Ckak Np +1 kwkB˙ σ , B˙ p,1

j

p,1

whenever −N/p < σ ≤ 1 + N/p. Lemma 2.11 If Rj = ∆j (a∂k w) − ∂k (a∆j w) and σ = − N p . There exists C = C(N, σ) such that N sup 2j(− p ) kRj kL1T (Lp ) ≤ Ckak N +1 kwk −N . p p ˜ ∞ (B˙ L p,1 T

j

˜ 1 (B˙ p,∞ ) L T

)

s ˜ q (B˙ p,r Remark 2.1 Lemmas 2.3–2.7 and Lemma 2.10 still remain true for the spaces L ) T q ˙s or LT (Bp,r ). The indices s, p, r behave just as in the stationary case whereas the time exponent q behaves according to H¨ older inequality. For example, the estimate in Lemma 2.5 becomes

kf gk

s +s2 − N p

1 ˜ q (B˙ p,r L T

)

s1 kgk ˜ q2 ˙ s2 , ≤ Ckf kL˜ q1 (B˙ p,r ) L (Bp,r ) T

T

where q11 + q12 = 1q and 1 ≤ q, q1 , q2 ≤ ∞. Readers can prove Lemmas 2.4–2.11 by using Bony’s decomposition and Lemma 2.1, see [5, 8, 32, 34, 45] for more details.

3

Estimates of the Linear Transport and Momentum Equations

Let us first recall standard estimates in Besov spaces for the following linear transport equation:   ∂ f + u · ∇f = F, t (3.1)  f|t=0 = f0 . Proposition 3.1 Let s ∈ (−N min( p1 , p1′ ), Np + 1), 1 ≤ p, r ≤ +∞, and s = 1 + Np , if r = N

p s s 1. Let u be a vector field such that ∇u ∈ L1T (B˙ p,r ∩L∞ ). Assume that f0 ∈ B˙ p,r , F ∈ L1T (B˙ p,r ) and f is the solution of (3.1). Then there holds for t ∈ [0, T ],   Z t kf kL˜ ∞(B˙ s ) ≤ eCU (t) kf0 kB˙ s + e−CU (τ ) kF kB˙ s dτ , t

p,r

p,r

0

p,r

No.4

1159

D.F. Bian & B.L. Guo: WELL-POSEDNESS IN CRITICAL SPACES

where U (t) :=

Rt 0

k∇uk

N

p ˙ p,r B ∩L∞

s dτ . If r < +∞, then f belongs to C([0, T ]; B˙ p,r ).

The proof is similar as that of the case r = 1 which can be found in [8, 9]. The difference P is that we use the estimate of commutator ( k[u · ∇, ∆q ]akrLp 2rqs )1/r ≤ k∇uk Np kakB˙ s ˙ p,r ∩L∞ B

q∈Z

instead of k[u · ∇, ∆q ]akLp ≤ Ccq 2

−qs

k∇uk

N

˙ p B p,1

p,r

kakB˙ s , here we omit the proof. We now focus p,1

on the mass equation associated to (1.5)   ∂ a + u · ∇a = (1 + a)divu, t  a|t=0 = a0 .

(3.2) N

p Proposition 3.2 [34] Let u be a vector field such that ∇u ∈ L1T (B˙ p,1 ), and s ∈ N 1 1 s . (−N min( p , p′ ), p ], 1 ≤ p ≤ +∞. Assume that a is the solution of (3.2) and a0 ∈ B˙ p,1 Then there holds for t ∈ [0, T ],

kakL˜ ∞ (B˙ s t

X

p ≤ 2qs k∆q akL∞ t (L )

q≥m

X

p,1 )

≤ eCU (t) ka0 kB˙ s + eCU (t) − 1,

X

2qs k∆q a0 kLp + (1 + ka0 kB˙ s )(eCU (t) − 1), p,1

p,1

q≤m

Rt 0

(3.4)

q≥m

m p ≤ C2 2qs k∆q (a − a0 )kL∞ ka0 kB˙ s t (L )

where U (t) :=

(3.3)

p,1

k∇uk

N

˙ p B p,1

Z

0

t

kuk

N

˙ p B p,1

dτ + (1 + ka0 kB˙ s )(eCU (t) − 1), (3.5) p,1

dτ .

Focusing on the magnetic equation associated to (1.5):     ∂t H + u · ∇H − σ∆H = H · ∇u − (∇ · u)H, divH = 0,    Ht=0 = H0 ,

(3.6)

we get the following proposition.

N

p Proposition 3.3 Let u be a vector field such that ∇u ∈ L1T (B˙ p,1 ), and s ∈ (−N min( p1 , 1 N ˙s p′ ), p ], 1 ≤ p < +∞. Assume that H is the solution of (3.6) and H0 ∈ Bp,1 . Then there holds for t ∈ [0, T ], kHkL˜ ∞ (B˙ s ) + σ e kHkL1(B˙ s+2 ) ≤ eCU (t) kH0 kB˙ s , (3.7) t p,1 p,1 t p,1 Rt where U (t) := 0 k∇uk Np dτ .

˙ B p,1

Proof Using ∆q to (3.6) yields

∂t ∆q H + u · ∇∆q H − σ∆∆q H = [u · ∇, ∆q ]H + ∆q (H · ∇u − (∇ · u)H).

(3.8)

Multiplying both sides of (3.8) by |∆q H|p−2 ∆q H, we get by integrating by parts over RN for the resulting equation that 1 d p−1 k∆q Hkpp + c22q 2 k∆q Hkpp p dt p ≤ C(k[u · ∇, ∆q ]Hkp + k∆q (H · ∇u − (∇ · u)H)kp )k∆q Hkp−1 + p

1 p

Z

RN

|∆q H|p divudx,

1160

ACTA MATHEMATICA SCIENTIA

then choosing σ e ≤ c p−1 p2 , we have Z t Z 2q k∆q Hkp dτ ≤ k∆q H0 kp + C k∆q Hkp + σ e2

Vol.33 Ser.B

t

k∆q Hkp kdivukL∞ + k[u · ∇, ∆q ]Hkp
+k∆q (H · ∇u − (∇ · u)H)kp dτ, (3.9)

0

0

from which and applying Lemma 2.5 and Lemma 2.6, it follows that Z t kHkL˜ ∞ (B˙ s ) + σ e kHkL˜ 1(B˙ s+2 ) ≤ kH0 kB˙ s + C kHkB˙ s kuk t

t

p,1

p,1

p,1

0

p,1

N +1

˙ p B p,1

dτ.

(3.10)

By Gronwall inequality, we infer that

+σ e kHkL1(B˙ s+2 ) ≤ eCU (t) kH0 kB˙ s .

kHkL˜ ∞ (B˙ s t

p,1 )

t

p,1

p,1

(3.11)

2 Let us state estimates in Besov spaces for the following constant coefficient parabolic equations:   ∂ u − µ∆u − (λ + µ)∇divu = g, t (3.12)  u|t=0 = u0 .

Proposition 3.4 [34] Assume that µ ≥ 0, and that λ + 2µ ≥ 0, p ∈ [1, +∞]. Then there exists a universal constant κ such that for all s ∈ R and T ∈ R+ , kukL˜ ∞ (B˙ s

p,1 )

T

≤ ku0 kB˙ s + kgkL1 (B˙ s ) ,

κ¯ ν kukL1 (B˙ s+2 ) ≤ T

p,1

p,1

X

T

p,1 2q

2qs (1 − e−κ¯ν 2

T

)(k∆q u0 kLp + k∆q gkL1T (Lp ) )

q∈Z

with ν¯ := λ + 2µ. This is an easy adaption of the result of [5]. Remark 3.1 When p = 2, Proposition 3.4 is as same as Proposition 5 in [9]. We now consider the following parabolic system which is obtained by linearizing the momentum equation in (1.5)   ∂ u + v · ∇u + u · ∇w − b(µ∆u + (λ + µ)∇divu) = g, t (3.13)  u|t=0 = u0 .

s s Above u is the unknown function. We assume that u0 ∈ B˙ p,1 and g ∈ L1 (0, T ; B˙ p,1 ) that v N

p +1 and w are time dependent vector-fields with coefficients in L1 (0, T ; B˙ p,1 ) that b is bounded s ˜ ∞ (B˙ p,1 by below a positive constant b and that a := b − 1 belongs to L ). T ¯ Proposition 3.5 [34] Let ν := bµ, and ν¯ := λ + 2µ = µ + |λ + µ|, p ∈ [2, ∞). Let m ∈ Z be such that bm := 1 + Sm a satisfies

inf

(t,x)∈[0,T )×RN

bm (t, x) ≥ b/4.

(3.14)

Then there exist three constants c, C, κ (with c, C depending on N and on s, and κ universal) such that if in addition we have ka − Sm ak

N

˙ p L∞ T (Bp,1 )

≤ cν/¯ ν.

(3.15)

No.4

1161

D.F. Bian & B.L. Guo: WELL-POSEDNESS IN CRITICAL SPACES

Then setting V (t) :=

Z

t

kvk

0

N +1 ˙ p B p,1

dτ, W (t) :=

Z

t

kwk

0

and Zm (t) := 22m ν¯2 ν −1

Z

N +1

p B˙ p,1

dτ

t

kak2

N

˙ p B p,1

0

dτ,

for s ∈ (− Np , Np ], we have for all t ∈ [0, T ], kukL˜ ∞ (B˙ s t

p,1 )

+ κνkukL1 (B˙ s+2 ) ≤ e t

C(V +W +Zm )(t)

  Z t −C(V +W +Zm )(τ ) ku0 kB˙ s + e kgkB˙ s dτ . p,1

p,1

p,1

0

Remark 3.2 [34] For s = − Np , p ∈ [2, ∞), under condition (3.14), there exist three constants c, C, κ (with c, C depending on N and on s, and κ universal) such that if ka − Sm ak

N

˜ ∞ (B˙ p ) L t p,1

≤ cν/¯ ν

(3.16)

≤ c.

(3.17)

and 22m ν¯2 ν −1 tkak2

N

˜ ∞ (B˙ p ) L t p,1

Then we have for all t ∈ [0, T ], kuk

4

−N

˙ p L∞ t (Bp,∞ )

+ κνkuk

− N +2

p ˜ 1(B˙ p,∞ L t

)

≤ 2eC(V +W )(t) (ku0 k

−N

p B˙ p,∞

+ kgk

−N

p ˜ 1 (B˙ p,∞ L ) t

).

Estimates of the Energy Equation Let us finally recall standard estimates in Besov spaces for the following equation:   ∂ θ + u · ∇θ − kb∆θ = g, t  θ|t=0 = θ0 .

(4.1)

s s Above θ is the unknown function. We assume that θ0 ∈ B˙ p,1 and g ∈ L1 (0, T ; B˙ p,1 ) that u is N

p +1 time dependent vector-fields with coefficients in L1 (0, T ; B˙ p,1 ) that b is denoted as same as that in (3.13). Proposition 4.1 Let p ∈ (1, ∞) and m ∈ Z be such that bm := 1 + Sm a satisfies

inf

(t,x)∈[0,T )×RN

bm (t, x) ≥ b/4.

(4.2)

Then there exist three constants c, C, ηe (with c, C depending on N and on s, and ηe universal) such that if in addition ka − Sm ak

and setting U (t) :=

Z

0

N

˙ p L∞ T (Bp,1 )

≤ ck −1 ,

t

kuk

N +1 ˙ p B p,1

2 2m

dτ, Zm (t) := k 2

(4.3)

Z

0

t

kak2

N

˙ p B p,1

dτ

1162

ACTA MATHEMATICA SCIENTIA

Vol.33 Ser.B

for s ∈ (−N min( 1p , p1′ ), Np ], we have for all t ∈ [0, T ], kθkL˜ ∞ (B˙ s t

p,1

  Z t −C(U+Zm )(τ ) C(U+Zm )(t) s+2 + η e kθk kθ k + e kgk dτ . ≤ e s s 1 ˙ 0 B˙ ) B L (B˙ ) t

p,1

p,1

p,1

0

Remark 4.1 For s = − Np , p ∈ [2, ∞), under condition (4.2), there exist three constants c, C, ηe (with c, C depending on N and on s, and ηe universal) such that if ka − Sm ak

N

˜ ∞ (B˙ p ) L t p,1

≤ ck −1

(4.4)

≤ c.

(4.5)

and k 2 22m tkak2

N

˜ ∞ (B˙ p ) L t p,1

Then we have for all t ∈ [0, T ], kθk

−N

˙ p L∞ t (Bp,∞ )

+ ηekθk

− N +2

p ˜ 1 (B˙ p,∞ L t

)

≤ 2eCU (t) (kθ0 k

−N

p B˙ p,∞

+ kgk

−N

p ˜ 1 (B˙ p,∞ L ) t

).

N

˜ ∞ (B˙ p ) then assumptions (4.2)–(4.3) are satisfied for m large Remark 4.2 If a ∈ L p,1 T enough. Remark 4.3 Proposition 4.1 is a particular case of Proposition 3.5 obtained in [34], for completeness, we give its proof in the following. Proof of Proposition 4.1 Let us rewrite (4.1) as follows: ∂t θ − bm k∆θ + u · ∇θ = g + Em

(4.6)

with Em = k∆θ(Id − Sm )a. Now applying ∆q to both sides of equation (4.6) yields that ∂t ∆q θ − kdiv(bm ∇∆q θ) + u · ∇∆q θ = [u · ∇, ∆q ]θ + ∆q g + ∆q Em + Rq

(4.7)

with Rq = k(∆q (bm ∆θ) − div(bm ∇∆q θ)). Multiplying both sides of (4.7) by |∆q θ|p−2 ∆q θ, integrating by parts over RN , by Lemma 2.2 and H¨older’s inequality and condition (4.2), we get for 1 < p < ∞, d b (p − 1) k∆q θkp + kc22q k∆q θkp dt 4 p2 1 ≤ kdivukL∞ k∆q θkp + k[u · ∇, ∆q ]θkp + k∆q gkp + k∆q Em kp + kRq kp . p Integrating with respect to t yields that Z t Z t 1 2q kdivukL∞ k∆q θkp + k[u · ∇, ∆q ]θkp k∆q θkp + 3e η2 k∆q θkLp dτ ≤ k∆q θ0 kp + p 0 0  +k∆q gkp + k∆q Em kp + kRq kp dτ, (4.8) where we choose ηe satisfies ηe ≤

cbk(p−1) 12p2 .

By Lemmas 2.1 and 2.5 we obtain

kdivukL∞ k∆q θkp . 2−qs cq kuk

N +1

˙ p B p,1

kθkB˙ s , p,1

(4.9)

No.4

D.F. Bian & B.L. Guo: WELL-POSEDNESS IN CRITICAL SPACES

k∆q gkp . 2−qs cq kgkB˙ s ,

(4.10)

p,1

kEm kp . 2

−qs

cq ka − Sm ak

N

˙ p B p,1

1163

kθkB˙ s+2 ,

(4.11)

p,1

N

p where the embedding relation B˙ p,1 ֒→ L∞ has been used. By Lemmas 2.1, 2.6 and 2.10 we have k[u · ∇, ∆q ]θkp . cq 2−qs kuk Np +1 kθkB˙ s (4.12)

˙ B p,1

p,1

and kRq kp . cq 2−qs 2m kak

N

p B˙ p,1

kθkB˙ s+1 .

(4.13)

p,1

Inserting estimates (4.9)–(4.13) into (4.8), and multiplying the both sides by 2qs and summing up for q ∈ Z, we obtain that, for all t ∈ [0, T ], kθkL˜ ∞ (B˙ s ≤ kθ0 kB˙ s

p,1

+Ck

+ 3e η kθkL˜ 1 (B˙ s+2 ) t p,1 Z t + kgkL˜ 1 (B˙ s ) + C kuk

p,1 )

t

Z

t

p,1

0

N +1

˙ p B p,1

kθkB˙ s dτ p,1

t

(ka − Sm ak

0

N p B˙ p,1

kθkB˙ s+2 + 2m kak p,1

N p B˙ p,1

kθkB˙ s+1 )dτ.

(4.14)

p,1

By choosing m large enough one has that Ck(ka − Sm ak

N

˙ p L∞ T (Bp,1 )

By the interpolation inequality 1

) ≤ ηe. 1

kθkB˙ s+1 ≤ kθkB2˙ s kθkB2˙ s+2 p,1

p,1

p,1

and Young’s inequality, it follows that Ck2m kak

N ˙ p B p,1

C 2 k 2 22m kak2 N kθkB˙ s . p,1 4e η ˙ p B p,1

kθkB˙ s+1 ≤ ηekθkB˙ s+2 + p,1

p,1

From estimate (4.14) we arrive at

+ ηekθkL˜ 1 (B˙ s+2 ) t p,1 Z t (kuk + kgkL˜ 1 (B˙ s ) + C

kθkL˜ ∞ (B˙ s

p,1 )

t

≤ kθ0 kB˙ s

p,1

t

p,1

0

N +1

˙ p B p,1

+ k 2 22m kak2

N

˙ p B p,1

)kθkL˜ ∞ B˙ s ) dτ, τ

p,1

and the Gronwall inequality implies the desired estimate. 2 Proof of Remark 4.1 The proof is similar to that of Proposition 4.1. The changes are that we use Lemmas 2.3, 2.7 and 2.11 instead of Lemmas 2.5, 2.6 and 2.10. Estimates (4.9)–(4.13) are changed by N

sup 2q(− p ) kdivukL∞ k∆q θkp . kuk q∈Z N

sup 2q(− p ) k∆q gkp . kgk q∈Z

−N

p ˙ p,∞ B

N +1

˙ p B p,1

kθk

−N

p B˙ p,∞

,

N

sup 2q(− p ) k∆q Em kL1t (Lp ) . ka − Sm ak q∈Z

N

sup 2q(− p ) k[u · ∇, ∆q ]θkp . kuk q∈Z

,

N +1

˙ p B p,1

N

˜ ∞ (B˙ p ) L t p,1

kθk

−N

p B˙ p,∞

kθk

, for p ≥ 2,

− N +2

p ˜ 1 (B˙ p,∞ L t

)

1164

ACTA MATHEMATICA SCIENTIA

Vol.33 Ser.B

and N

sup 2q(− p ) kRq kL1t (Lp ) . kbm k q∈Z

kθk

N +1

˜ ∞ (B˙ p L t p,1

. 2m kak

)

N ˜ ∞ (B˙ p ) L t p,1

kθk

− N +1

p ˜ ∞ (B˙ p,∞ L t

)

− N +1 p ˜ ∞ (B˙ p,∞ L ) t

.

Taking supremum over q ∈ Z in (4.8) and by the above estimates, we have kθk

−N

p ˜ ∞ (B˙ p,∞ ) L t

≤ kθ0 k

−N

p B˙ p,∞

+ 3e ηkθk

+ kgk

− N +2

p ˜ 1 (B˙ p,∞ L t

+C

−N

p ˜ 1 (B˙ p,∞ L ) t

 +Ck ka − Sm ak

Z

) t

kuk

0

N ˜ ∞ (B˙ p ) L t p,1

kθk

N +1

˙ p B p,1

− N +2 p ˜ 1 (B˙ p,∞ L ) t

kθk

−N

p ˙ p,∞ B

+ 2m kak

dτ kθk

N ˜ ∞ (B˙ p ) L t p,1

− N +1 p ˜ 1 (B˙ p,∞ L ) t

 .

The interpolation kθk

Z

N

− N +1 p ˜ 1 (B˙ p,∞ L ) t

= sup 2q(− p +1) q

t

k∆q θkLp dτ

0

= sup 2

N q(− 2p +1)

q N

≤ sup 2q(− p +2) q

Z

t

k∆q θkLp dτ

0 Z t

k∆q θkLp dτ

0

≤ sup 2

q(− N p

+2)

q

≤ kθk

Z

t

0

− N +2

p ˜ 1 (B˙ p,∞ L t


12

k∆q θkLp dτ

√ 1 tkθk 2

1 2


21


21

2

N q(− 2p )

Z

t

k∆q θkLp dτ

0

N

sup 2q(− p ) q

Z

t

sup 2 q

)

,

k∆q θkLp dτ

0

q(− N p


12

Z

t


12

k∆q θkLp dτ

0


12

−N

p ˜ ∞ (B˙ p,∞ L ) t

)

holds. Use an argument similar to that used in proving Proposition 4.1, it is easy to prove Remark 4.1. 2

5

Proof of the Existence Throughout the proof, we denote ν = bµ and ν¯ = λ + 2µ = µ + |λ + µ|. Step 1 The Approximate Solution Sequence We smooth out the data as follows: an0 = Sn a0 , H0n = Sn H0 , un0 = Sn u0 and θ0n = Sn θ0 .

A standard linearized argument (as in the proof of Theorem 4.2 in [5]) will ensure system (1.5) with the smooth data (an0 , H0n , un0 , θ0n ) has a solution (an , H n , un , θn ) on a time interval [0, Tn ] for some Tn > 0 such that N

N

+1

p p an ∈ C([0, Tn ]; B˙ p,1 ∩ B˙ p,1 ), N

N

N

N

p −1 p p +1 p +2 H n ∈ C([0, Tn ]; B˙ p,1 ∩ B˙ p,1 ) ∩ L1 (0, Tn ; B˙ p,1 ∩ B˙ p,1 ), N

N

N

N

p −1 p p +1 p +2 ∩ B˙ p,1 ) ∩ L1 (0, Tn ; B˙ p,1 ∩ B˙ p,1 ) un ∈ C([0, Tn ]; B˙ p,1

No.4

1165

D.F. Bian & B.L. Guo: WELL-POSEDNESS IN CRITICAL SPACES

and

N

N

−2

p θn ∈ C([0, Tn ]; B˙ p,1

N

−1

N

+1

p p p ∩ B˙ p,1 ) ∩ L1 (0, Tn ; B˙ p,1 ∩ B˙ p,1 ).

(5.1)

In what follows, we also denote by Tn the maximal lifespan of the solution (an , H n , un , θn ). Step 2 Uniform Estimates We aim at getting uniform estimates in ETp for T small enough. For that, we need to introduce the solution unL to the linear system ∂t unL − AunL = f n , unL (0) = un0 .

(5.2)

Now, the vectorfield u ˜n := un − unL satisfies the parabolic system   ˜n + unL · ∇˜ un + u ˜n · ∇un − (1 + an )A˜ un = an AunL − unL · ∇unL − ∇g(an )   ∂t u +(1 + an )(H n · ∇H n − ∇H n · H n ),    n u ˜ (0) = 0.

Let ¯b := 1 + sup a0 (x), A0 := 1 + 2ka0 k x∈RN

kf k

N −1

p L1 (R+ ;B˙ p,1

)

, D0 := kθ0 k

N −2

p B˙ p,1

N p B˙ p,1

, B0 := 2kH0 k

N −1

p B˙ p,1

, U0 := 2ku0 k

N −1

p B˙ p,1 n n

+

, T ∈ (0, Tn ). We assume that the solutions (an , H , u , θn )

˜0 , V0 , m, c (to be determined later): satisfy the following inequalities for some positive η, U (H1) kan − Sm an k

N

˙ p L∞ T (Bp,1 )

(H2) CT kan k2

N ˙ p L∞ T (Bp,1 )

≤ min(cν ν¯−1 , ck −1 ),

≤ min(¯ ν −2 2−2m ν, k −2 2−2m ),

1 b ≤ 1 + an (t, x) ≤ 2¯b for all (t, x) ∈ [0, T ] × RN , 2 N ≤ A0 , (H4) kan k p

(H3)

˜ ∞ (B˙ L p,1 ) T

(H5) kH n k (H6) kunL k (H7) k˜ un k (H8) kθk

N −1

˜ ∞ (B˙ p L p,1 T

)

)

≤ B0 ,

)

+ νk˜ un k

N −1

˜ ∞ (B˙ p L p,1 T

)

N −2

˜ ∞ (B˙ p L p,1 T

N +1

p L1T (B˙ p,1

≤ η,

N +1

p L1T (B˙ p,1

+σ ekH n k

)

+ ηekθk

N +1

p L1T (B˙ p,1 N

p ) L1T (B˙ p,1

)

˜0 η, ≤U

≤ V0 .

In what follows, we will show that if conditions (H1) to (H8) are satisfied for some T > 0, then they are actually satisfied with strict inequalities. Since all those conditions depend continuously on the time variable and are satisfied initially, a standard bootstrap argument will ensure that (H1) to (H8) are indeed satisfied for T . First of all, we get by Proposition 3.2 that kan k

N

˜ ∞ (B˙ p ) L p,1 T

≤ eCU

n

(t)

kan0 k

N p B˙ p,1

+ eCU

n

(t)

− 1 ≤ eCU

n

(t)

ka0 k

N p B˙ p,1

+ eCU

n

(t)

− 1,

(5.3)

.

(5.4)

and by Proposition 3.3, we have kH n k

N −1

˜ ∞ (B˙ p L p,1 T

)

+σ ekH n k

N +1 p L1T (B˙ p,1 )

≤ eCU

n

(t)

kH0n k

N −1

p B˙ p,1

≤ eCU

n

(t)

kH0 k

N −1

p B˙ p,1

1166

ACTA MATHEMATICA SCIENTIA

We denote U n (t) :=

Rt 0

kun k

˜ n (T ) + we have U n (T ) ≤ U

N +1

˜ n (t) := dτ , U

p B˙ p,1 n UL (T ).

Rt 0

k˜ un k

N +1

p B˙ p,1

Vol.33 Ser.B

dτ , ULn (t) :=

We take η small enough such that

Rt 0

kunL k

N +1

p B˙ p,1

˜0 )η < log 2, C(1 + ν −1 U

dτ , then

(5.5)

according to (H6) and (H7), eCU

n

(T )

˜ n +U n )(T )

≤ eC(U

L

< 2.

(5.6)

Plugging (5.6) in (5.3) and (5.4), we thus get kan k

N

˜ ∞ (B˙ p ) L p,1 T

< 1 + 2ka0 k

N p B˙ p,1

= A0

and kH n k

+σ e kH n k

N −1

˜ ∞ (B˙ p L p,1 T

)

N +1

˜ 1 (B˙ p L p,1 T

)

< 2kH0 k

N −1

p B˙ p,1

= B0 ,

which ensure that (H4) and (H5) are satisfied with strict inequalities. Next, by Proposition 3.4, it yields kunL k κ¯ ν kunLk

N +1 p ) L1T (B˙ p,1

≤ ku0 k

N −1

˜ ∞ (B˙ p L p,1 T

≤

X

)

N −1

p B˙ p,1

+ kf k

N

2q

2q( p −1) (1 − e−κ¯ν 2

T

N −1

p L1T (B˙ p,1

)

= U0 ,

(5.7)

)(k∆q u0 kLp + k∆q f kL1T (Lp ) ).

q∈Z

Hence, taking T small enough such that X N 2q 2q( p −1) (1 − e−κ¯ν 2 T )(k∆q u0 kLp + k∆q f kL1T (Lp ) ) < κη¯ ν,

(5.8)

q∈Z

then (H6) is satisfied with a strict inequality. Since 1 + S m an = 1 + an + S m an − an , N

p ֒→ L∞ insure that assumptions (H1) and (H3) combined with the embedding B˙ p,1

inf

(t,x)∈[0,T ]×RN

(1 + Sm an )(t, x) ≥

1 b, 4

(5.9)

provided c can be chosen small enough and m is large enough such that min(cν ν¯−1 , ck −1 ) ≤

1 b. 4

Applying Proposition 3.5 and (H2), we get k˜ un k

un k + κνk˜

N −1

˜ ∞ (B˙ p L t p,1 n

≤ CeC(UL +U

n

)

)(T )

Z

N −1

p B˙ p,1

)

t

0

+k∇g(an )k

N +1

p L1t (B˙ p,1

(kan AunL k

N −1

p B˙ p,1 n n

+ kunL · ∇unL k

N −1

p B˙ p,1

+ k(1 + a )(H · ∇H n − ∇H n · H n )k

N −1

p B˙ p,1

)dτ.

(5.10)

No.4

1167

D.F. Bian & B.L. Guo: WELL-POSEDNESS IN CRITICAL SPACES

We use Lemmas 2.5 and 2.8 to get for p < 2N , kan AunL k

N −1

p B˙ p,1

kunL · ∇unL k k∇g(an )k

≤ C ν¯kan k

N

p B˙ p,1

≤ CkunL k

N −1

p B˙ p,1

≤ kg(an )k

N −1

p B˙ p,1

kunLk

N −1

˙ p B p,1

N +1

p B˙ p,1

kunLk

,

N +1

p B˙ p,1

, N

N

p B˙ p,1

≤ C(1 + kan kL∞ )[ p ]+2 kan k

k(1 + an )(H n · ∇H n − ∇H n · H n )k

N −1

p B˙ p,1

≤ Ckan k

N

p B˙ p,1

N

p B˙ p,1

kH n k

, N −1

p B˙ p,1

kH n k

N +1

p B˙ p,1

.

Inserting these estimates into (5.10) gives k˜ un k

+ νk˜ un k

N −1

˜ ∞ (B˙ p L t p,1

)

 n C(UL +U n )(T ) kunL k ≤ Ce n

+(1 + ka k

L∞

[N p ]+2

)

N +1

p L1t (B˙ p,1

N +1 p L1T (B˙ p,1 )

n

T ka k

)

ν¯kan k N

˙ p L∞ T (Bp,1 )

+

N ˙ p L∞ T (Bp,1 )

Z

+ kunL k

N −1 ˙ p L∞ ) T (Bp,1

t n

ka k 0

N

n

p B˙ p,1

kH k

N −1

p B˙ p,1


n

kH k

N +1

p B˙ p,1

dτ



N

≤ 2Cη(¯ ν A0 + U0 ) + 2C(1 + kan kL∞ )[ p ]+2 T A0 + 2CA0 B02 η. We chose T small enough such that N

C(1 + kan kL∞ )[ p ]+2 T A0 < C ν¯A0 η.

(5.11)

Let U˜0 = 4C ν¯A0 + 2CU0 + 2CA0 B02 , then (H7) is satisfied with a strict inequality. We apply the second inequality of Proposition 3.2 which gives for all m ∈ Z, X N X N n ˜n p ≤ 2q p k∆q an kL∞ 2q p k∆q a0 kLp + (1 + ka0 k Np )(eC(UL +U )(T ) − 1). T (L ) q≥m

B˙ p,1

q≥m

Plugging (5.5), (H6) and (H7) in (5.12), we get X N ≤ 2q p k∆q a0 kLp + C(1 + ka0 k kan − Sm an k N p ˙ L∞ T (Bp,1 )

q≥m

N

p B˙ p,1

˜0 )η. )(1 + ν −1 U

(5.12)

(5.13)

(H1) is strictly satisfied provided that m large enough satisfies X qN 2 p k∆q a0 kLp < min(cν ν¯−1 , ck −1 ) q≥m

and η further satisfies C(1 + ka0 k

N p B˙ p,1

˜0 )η < min(cν ν¯−1 , ck −1 ). )(1 + ν −1 U

Next, condition (H2) is strictly satisfied provided  −2m  2 ν 2−2m T < min , , C ν¯2 A20 Ck 2 A20 because of (H4).

(5.14)

1168

ACTA MATHEMATICA SCIENTIA

Vol.33 Ser.B

Now we have to check whether (H3) is satisfied with a strict inequality. For that, we apply the fact X an − a0 = Sm (an − an0 ) + (Id − Sm )(an − an0 ) − ∆q a0 . q≥n

N p

Using B˙ p,1 ֒→ L∞ and assuming that n ≥ m, kan − a0 kL∞ (0,T ×RN ) ≤ Ckan − a0 k

N

˙ p L∞ T (Bp,1 )

 ≤ C kSm (an − an0 )k

N ˙ p L∞ T (Bp,1 )

+ kan − Sm an k

N ˙ p L∞ T (Bp,1 )

+2

X

q≥m

 N 2q p k∆q a0 kLp .

Taking c small enough and m large enough such that X qN kan − Sm an k N +2 2 p k∆q a0 kLp < b/4. p ˙ L∞ T (Bp,1 )

For kSm (an − an0 )k

N

˙ p L∞ T (Bp,1 )

, by the third inequality of the Proposition 3.2,

kSm (an − an0 )k ≤

X

q≥m

N

˙ p L∞ T (Bp,1 )

N

p 2q p k∆q (an − an0 )kL∞ t (L )

q≤m

≤ C2m ka0 k

N p B˙ p,1

√ T kun k

N p L2T (B˙ p,1 )

+ (1 + ka0 k

n

N p B˙ p,1

˜ n )(T )

)(eC(UL +U

− 1).

(5.15)

If we choose η small enough, then the second term in the right hand of the above inequality may be bounded by b/8. By interpolation, (5.7), (H6) and (H7), we get q 1 1 n 2 n 2 ˜0 η)(1 + ν −1 U ˜0 ). N ≤ ku k kun k η(U0 + U N +1 ku k N −1 ≤ p L2T (B˙ p,1 )

p L1T (B˙ p,1

)

˙ p L∞ T (Bp,1

)

We can rewrite (5.15) as that n

kSm (a −

an0 )k

N ˙ p L∞ T (Bp,1 )

m

≤ C2 ka0 k

N p B˙ p,1

√ q ˜0 η)(1 + ν −1 U ˜0 ) + b/8. T η(U0 + U

Assuming in addition that T satisfies C2m ka0 k

N p B˙ p,1

√ q ˜0 η)(1 + ν −1 U ˜0 ) < b/8. T η(U0 + U

and applying b ≤ 1 + a0 ≤ ¯b yields that (H3) is satisfied with a strict inequality. By Proposition 4.1, we get kθn k ≤ Ce

N −2

˜ ∞ (B˙ p L t p,1

CU (t)

)

 kθ0n k

+ ηekθn k

N −2 p B˙ p,1

+

N

p L1t (B˙ p,1 )

Z

t

k(1 + an )(−p(an )divun + 0

µ |∇un + ∇(un )⊤ |2 2

(5.16)

No.4

1169

D.F. Bian & B.L. Guo: WELL-POSEDNESS IN CRITICAL SPACES

n 2

n 2

+µ(divu ) + σ(curlH ) )k

N −2

p B˙ p,1

dτ



 N ≤ CeCU (t) D0 + (1 + kan kL∞ )[ p ]+2 kan k +kan k

N ˙ p L∞ T (Bp,1 )

Z

N ˙ p L∞ T (Bp,1 )

t

kun k2

N p B˙ p,1

0

dτ + σkan k

 Z t [N ]+3 p kun k ≤ 2C D0 + (1 + A0 ) 0

N −1

p B˙ p,1

N ˙ p L∞ T (Bp,1 )

dτ + A0

Z

t

kun k

0 Z t

Z

kH n k2

N p B˙ p,1

0

B˙ p,1

dτ

dτ



t n 2

ku k 0

We chose T small enough such that Z t Z t [N ]+3 n p (1 + A0 ) ku k Np −1 dτ + A0 kun k2 0

N −1

p B˙ p,1

N p B˙ p,1

0

N

p B˙ p,1

dτ + σA0

Z

n 2

kH k

0

dτ + σA0

Z



t N

p B˙ p,1

dτ .

t

kH n k2

N

p B˙ p,1

0

dτ

N 2 < (1 + A0 )[ p ]+3 U˜0 η + A0 U˜0 η 2 + σA0 B02 .

(5.17)
N 2 Let V0 = 2C D0 + (1 + A0 )[ p ]+3 U˜0 η + A0 U˜0 η 2 + σA0 B02 , then (H8) is satisfied with a strict inequality. Let T ∗ be the supremum of all time T such that (5.8), (5.11), (5.14), (5.16) and (5.17) are satisfied. We need to prove that Tn ≥ T ∗ . If Tn < T ∗ , then we can prove that N

N

+1

˙ p ˜∞ ˙ p an ∈ L Tn (Bp,1 ∩ Bp,1 ), N

−1

˜∞ ˙ p Hn ∈ L Tn (Bp,1

N

N

+1

p p ) ∩ L1Tn (B˙ p,1 ∩ B˙ p,1

N

N

N

N

N

+2

p ∩ B˙ p,1 ),

N

N

˜∞ ˙ p −1 ∩ B˙ p ) ∩ L1T (B˙ p +1 ∩ B˙ p +2 ) un ∈ L Tn (Bp,1 p,1 p,1 p,1 n and

N

N

˜ ∞ (B˙ p −2 ∩ B˙ p −1 ) ∩ L1 (B˙ p ∩ B˙ p +1 ), θn ∈ L Tn Tn p,1 p,1 p,1 p,1

thus the solution (an , H n , un , θn ) can be continued beyond T ∗ . We thus have Tn ≥ T ∗ . Step 3 Existence of a Solution We will use a compact argument to prove that the approximate sequence (an , H n , un , θn )n∈N tends (up to a subsequence) to some function (a, H, u, θ) which belongs to ETp and satisfies (1.5) in the sense of distribution. N N p +1 ˙ p −1 Since {un } is uniformly bounded in L1T (B˙ p,1 ) ∩ L∞ T (Bp,1 ), we get by the interpolation N

2

p −1+ r that {un }n∈N is also uniformly bounded in LrT (B˙ p,1 ) for any r ∈ [1, +∞]. Applying Lemma 2.5, we obtain for p < 2N ,

kun · ∇an k

N −1

p B˙ p,1

k(1 + an )divun k

≤ Ckun k N −1

˙ p B p,1

N

p B˙ p,1

kan k

≤ Ckun k

N

p B˙ p,1

N

p B˙ p,1

,

+ Ckun k

N

p B˙ p,1

kan k

N

p B˙ p,1

.

By interpolation, kun k

1

N p L2T (B˙ p,1 )

≤ kun k 2

N +1 p L1T (B˙ p,1 )

n

≤ (k˜ u k

1

kun k 2

N +1 p L1T (B˙ p,1 )

N −1

˙ p L∞ T (Bp,1

+

kunL k

) 1

N +1 p L1T (B˙ p,1 )

) 2 (k˜ un k

N −1 ˙ p L∞ ) T (Bp,1

+ kunL k

1

N −1 ˙ p L∞ ) T (Bp,1

)2 .

1170

ACTA MATHEMATICA SCIENTIA

Vol.33 Ser.B

From ∂t an = −un · ∇an + (1 + an )divun , we easily gather that {∂t an }n∈N is uniformly bounded N N N p −1 ˜ ∞ (B˙ p −1 ∩ B˙ p ) and equicontinuous on [0, T ] in L2T (B˙ p,1 ). Hence (an )n∈N is bounded in L T p,1 p,1 N

N

N

N

p −1 p −1 p p −1 with values in B˙ p,1 . Since the embedding B˙ p,1 ∩ B˙ p,1 ֒→ B˙ p,1 is locally compact, and N

N

N

p ˜ ∞ (B˙ p ∩ B˙ p −1 ). , we conclude (an )n∈N tends to some function a in L (an0 )n∈N → a0 in B˙ p,1 p,1 p,1 T N

˜ ∞ (B˙ p ). Therefore, we have a ∈ L p,1 T On the other hand, by Lemma 2.5, we conclude for p < 2N , kun · ∇H n k kH n · ∇un k

N −1

≤ Ckun k

N −1

≤ Ckun k

p B˙ p,1

p B˙ p,1

k(∇ · un )H n k k∆H n k

kH n k

N +1

kH n k

p B˙ p,1

≤ Ckun k

N −1

p B˙ p,1

≤ CkH n k

N −1

p B˙ p,1

N −1

p B˙ p,1

,

N −1

,

p B˙ p,1

kH n k

N +1

p B˙ p,1

N +1

p B˙ p,1

N +1

p B˙ p,1

N −1

p B˙ p,1

,

.

Using the fact ∂t H n = −un · ∇H n + H n · ∇un − (∇ · un )H n + σ∆H, similarly, we can prove N N −1 +1 that {∂t H n }n∈N is uniformly bounded in L1 (B˙ p ). Combing with H n ∈ L1 (B˙ p ), we can p,1

T

T

N

p,1

N

˜ ∞ (B˙ p −1 ) ∩ L1 (B˙ p +1 ). verify that (H )n∈N tends to some function H in L p,1 p,1 T T Similarly, in virtue of Lemma 2.5 and Lemma 2.8, we can show that for p ≤ N , n

kun · ∇θn k

≤ Ckun k

N −2

p B˙ p,1 n

k(1 + an )∆θ k

N −2

p B˙ p,1

N −1

p B˙ p,1 n

≤ Cka k

k(1 + an )p(an )divun k

kθn k N

˙ p B p,1

,

N

p B˙ p,1 kθn k Np B˙

,

p,1

≤ C(1 + kan k

N −2

p B˙ p,1

N

N p B˙ p,1

)[ p ]+3 kan k

N p B˙ p,1

kun k

N −1

p B˙ p,1

,

µ k(1 + an )( |∇un + ∇un⊤ |2 + µ(divun )2 + σ(curlH n )2 )k Np −2 ≤ Ckan k Np k(un , H n )k2 N . p 2 B˙ p,1 B˙ p,1 B˙ p,1 Applying the quality ∂t θn = −un · ∇θn + k(1 + an )∆θn + (1 + an )(−p(an )divun + µ2 |∇un + N ∇un⊤ |2 + µ(divun )2 + σ(curlH n )2 ), together with θn ∈ L1 (B˙ p ), we can prove that (θn )n∈N p,1

T

N ˜ ∞ (B˙ p −2 ) L T p,1

N

p ). L1T (B˙ p,1

tends to some function θ in ∩ Moreover, from Lemma 2.5 and Lemma 2.8, by choosing ς = min( 2N p − 1, 1), we have kunL · ∇˜ un k k˜ un · ∇un k

N −1−ς

≤ CkunL k

N −1−ς

≤ Ck˜ un k

p B˙ p,1

p B˙ p,1 n

k(1 + an )A˜ u k kan AunL k kunL ·

N −1−ς

p B˙ p,1

N −1−ς

≤

p,1

n

n

N −1

p B˙ p,1

k˜ un k

kun k

≤ C ν¯(1 + kan k

≤ C ν¯kan k

p B˙ p,1 ∇unL k Np −1−ς B˙

N −1

p B˙ p,1

N p B˙ p,1

kunL k

N −1

p B˙ p,1

N +1−ς

,

N

p B˙ p,1

)k˜ un k

N +1−ς

N +1−ς

p B˙ p,1

,

,

p B˙ p,1 CkunL k Np −1 kunL k Np +1−ς , B˙ B˙

n

p,1

n

k(1 + a )(∇H · H − H · ∇H )k k∇g(an )k

,

p B˙ p,1

p,1

n

N +1−ς

p B˙ p,1

≤ kg(an )k

N −1−ς

p B˙ p,1

≤ C(1 + kan k

N

p B˙ p,1

)kH n k

N

N

p B˙ p,1

≤ C(1 + kan kL∞ )[ p ]+2 kan k

N

p B˙ p,1

.

N −1

p B˙ p,1

kH n k

N +1−ς

p B˙ p,1

,

No.4

1171

D.F. Bian & B.L. Guo: WELL-POSEDNESS IN CRITICAL SPACES

Then, by interpolation and the equation ∂t u˜n = −unL · ∇˜ un − u ˜n · ∇un + (1 + an )A˜ un + an AunL − unL · ∇unL − ∇g(an ) +(1 + an )(∇H n · H n − H n · ∇H n ). ς 1+ 2−ς

We get {∂t u˜n }n∈N is uniformly bounded in LT N

N

N

N

N

−1−ς

p (B˙ p,1

N

N

−1

p + B˙ p,1 ). Since the embedding

N

p −1−ς p −1 p −1−ς p −1 p p −1 B˙ p,1 ∩ B˙ p,1 ֒→ B˙ p,1 and B˙ p,1 ∩ B˙ p,1 ֒→ B˙ p,1 are locally compact, this ensures N

N

N

N

p +1 ˜ ∞ (B˙ p −1−ς ∩ B˙ p −1 ). Hence u ˜ ∞ (B˙ p −1 ). Combining u u ˜n ∈ L ˜n → u ˜ in L ˜n ∈ L1T (B˙ p,1 ), we p,1 p,1 p,1 T T N

N

N

p +1 ˜ ∞ (B˙ p −1 ) ∩ L1 (B˙ p +1 ). conclude u˜n → u ˜ in L1T (B˙ p,1 ) and we prove that u˜ ∈ L p,1 p,1 T T In order to obtain the limit of un , we now focus on equation (5.2). Let uL satisfy

∂t uL − AuL = f, uL (0) = u0 .

(5.18)

Let equations (5.2) substract equations (5.18), we get ∂t (unL − uL ) − A(unL − uL ) = f n − f, ( unL − uL )(0) = un0 − u0 . By Proposition 3.4, one can give kunL − uL k

N −1

˜ ∞ (B˙ p L p,1 T

)

κ¯ ν kunL − uL k ≤ ≤

X

2

q∈Z kun0

q( N p −1)

− u0 k

≤ kun0 − u0 k

N −1

p B˙ p,1

+ kf n − f k

→ 0, n → ∞,

N −1

p L1T (B˙ p,1

)

N +1

p L1T (B˙ p,1

(1 − e N −1

p B˙ p,1

)

−κ¯ ν 22q T

)(k∆q (un0 − u0 )kLp + k∆q f n − ∆q f kL1T (Lp ) )

+ kf n − f k

N

→ 0, n → ∞.

N −1

p L1T (B˙ p,1

)

N

˜ ∞ (B˙ p −1 ) ∩ L1 (B˙ p +1 ). Finally setting u = u˜ + uL , we conclude that Hence, unL → uL in L p,1 p,1 T T (a, H, u, θ) is a solution of (1.5). Finally, in order to prove continuity in time for a, it suffices to make use of Proposition N N +1 3.1. In fact, a ∈ L∞ (B˙ p ) and u ∈ L1 (B˙ p ) insure that T

p,1

T

k(1 + a)divuk

N

p ) L1T (B˙ p,1

p,1

≤ Ckuk

(1 + kak

N +1

p L1T (B˙ p,1

)

N

N

˙ p L∞ T (Bp,1 )

) < ∞,

N

N

p p p so we get that ∂t a ∈ L1T (B˙ p,1 ) and prove that a ∈ C([0, T ]; B˙ p,1 ) making use of a0 ∈ B˙ p,1 . SimN

−1

p ilarly, continuity for H, u, θ may be proved by applying that H0 belongs to B˙ p,1 N p

−1 B˙ p,1

N p

−2 B˙ p,1

× and that (∂t u, ∂t H, ∂t θ) ∈ belongs to Furthermore, having N

N

N p

−1 L1T (B˙ p,1 )

N

×

N p

−1 L1T (B˙ p,1 )

N

and (u0 , θ0 ) N

p −2 × L1T (B˙ p,1 ).

N

1 ˙ p +1 N 1 ˙ p +1 N ˜∞ ˙ p ˜ ∞ ˙ p −1 ˜ ∞ ˙ p −1 (a, H, u, θ) ∈ L T (Bp,1 ) × (LT (Bp,1 ) ∩ LT (Bp,1 )) × (LT (Bp,1 ) ∩ LT (Bp,1 )) N

N

1 ˙ p ˜∞ ˙ p −2 ×(L T (Bp,1 ) ∩ LT (Bp,1 ))

and following the argument in [7], we can show that N

N

N

p p −1 p +1 (a, H, u, θ) ∈ C([0, T ]; B˙ p,1 ) × (C([0, T ]; B˙ p,1 ) ∩ L1 (0, T ; B˙ p,1 ))N N

N

N

N

p −1 p +1 p −2 p ×(C([0, T ]; B˙ p,1 ) ∩ L1 (0, T ; B˙ p,1 ))N × (C([0, T ]; B˙ p,1 ) ∩ L1 (0, T ; B˙ p,1 )).

2

1172

6

ACTA MATHEMATICA SCIENTIA

Vol.33 Ser.B

Proof of the Uniqueness

In this section, we prove the uniqueness of the solution. Assume that (a1 , u1 , H 1 , θ1 ) ∈ ETp and (a2 , u2 , H 2 , θ2 ) ∈ ETp are two solutions of (1.1) with the same initial data. Let δa = a2 − a1 , δH = H 2 − H 1 , δu = u2 − u1 and δθ = θ2 − θ1 . Then the system for (δa, δH, δu, δθ) reads  2 2 1 1   ∂t δa + u · ∇δa = δadivu − δu · ∇a + (1 + a )divδu,     ∂t δH + u2 · ∇δH − σ∆δH = −δu · ∇H 1 + H 2 · ∇δu + δH · ∇u1 − H 2 ∇ · δu − δH∇ · u1 ,       ∂t δu + u2 · ∇δu + δu · ∇u1 − (1 + a1 )Aδu = δaAu2 − ∇(g(a2 ) − g(a1 )) + H 2 · ∇δH       +δH · ∇H 1 − ∇δH · H 2 − ∇H 1 · δH + a2 δH · ∇H 2 + a2 H 1 · ∇δH + δaH 1 · ∇H 1       −a2 ∇δH · H 2 − δa∇H 1 · H 2 − a1 ∇H 1 · δH,       ∂ δθ + u2 · ∇δθ − k(1 + a1 )∆δθ = −δu · ∇θ1 + kδa∆θ2 − δap(a2 )divu2   t µ (6.1) −(1 + a1 )p(a1 )divδu − (1 + a1 )(p(a2 ) − p(a1 ))divu2 + δa|∇u2 + ∇(u2 )⊤ |2  2     µ   + (1 + a1 )|∇u1 + ∇(u1 )⊤ |(|∇u2 + ∇(u2 )⊤ | − |∇u1 + ∇(u1 )⊤ |)   2     µ    + (1 + a1 )|∇u2 + ∇(u2 )⊤ |(|∇u2 + ∇(u2 )⊤ | − |∇u1 + ∇(u1 )⊤ |)   2    2 2  +µδa(divu ) + µ(1 + a1 )divδudiv(u1 + u2 ) + σδa(curlH 2 )2       +σ(1 + a1 )curlδHcurl(H 1 + H 2 ),     (δa, δH, δu, δθ)t=0 = (0, 0, 0, 0). Rt In what follows, we set U i (t) = 0 kui (τ )k Np +1 dτ for i = 1, 2. Due to the inclusion relation ETp

B˙ p,1

ETN ,

⊆ it suffices to prove the uniqueness of the solution in ETN . So we take p = N in the sequel. We aplly Proposition 3.1 to get for any t ∈ [0, T ], Z t 2 2 (6.2) e−CU (τ ) kδadivu2 − δu · ∇a1 + (1 + a1 )divδukB˙ 0 dτ. kδakB˙ 0 ≤ CeCU (t) p,∞

p,∞

0

By Lemma 2.3, we have kδadivu2 kB˙ 0

≤ CkδakB˙ 0 ku2 kB˙ 2 ,

kδu · ∇a1 kB˙ 0

≤ CkδukB˙ 1 ka1 kB˙ 1

p,∞

p,∞

p,∞

p,1

p,∞

p,1

1

≤ CkδukB˙ 1 ka1 kB˙ 1 , p,1

p,1

1

k(1 + a )divδukB˙ 0

p,∞

≤ CkδukB˙ 1 (1 + ka kB˙ 1 ) ≤ CkδukB˙ 1 (1 + ka1 kB˙ 1 ). p,∞

p,1

p,1

p,1

Plugging these estimates into (6.2), we get by Gronwall inequality that Z t 2 CU 2 (t) kδakB˙ p,∞ ≤ Ce e−CU (τ ) kδukB˙ 1 (1 + ka1 kB˙ 1 )dτ. 0 p,1

0

p,1

(6.3)

On the other hand, one can use Proposition 3.1 to deal with the second equation of (6.1) as follows: Z t 2 CU 2 (t) kδHkB˙ 0 + σ ekδHkL˜ 1 (B˙ 2 ) ≤ Ce e−CU (τ ) k − δu · ∇H 1 + H 2 · ∇δu p,∞

t

p,∞

0

+δH · ∇u1 − H 2 ∇ · δu − δH∇ · u1 kB˙ 0 dτ. p,∞

(6.4)

No.4

1173

D.F. Bian & B.L. Guo: WELL-POSEDNESS IN CRITICAL SPACES

1 1 Similarly, applying Lemma 2.3, Lemma 2.5, the embedding B˙ p,1 ֒→ B˙ p,∞ and Gronwall inequality gives Z t 2 1 2 kδHkB˙ 0 ≤ CeC(U +U )(t) e−CU (τ ) kδukB˙ 1 (kH 1 kB˙ 1 + kH 2 kB˙ 1 )dτ. (6.5) p,∞

p,1

0

p,1

p,1

By Remark 4.1, one has −1 + η kδθkB˙ p,∞ ekδθkL˜ 1 (B˙ p,∞ 1 ) t Z t 2 2 −1 e−CU (τ ) k − δu · ∇θ1 + kδa∆θ2 − δap(a2 )divu2 kB˙ p,∞ ≤ CeCU (t)

0

+k − (1 + a1 )(p(a2 ) − p(a1 ))divu2 +

µ −1 δa|∇u2 + ∇(u2 )⊤ |2 kB˙ p,∞ 2

µ −1 +k (1 + a1 )|∇u1 + ∇(u1 )⊤ |(|∇u2 + ∇(u2 )⊤ | − |∇u1 + ∇(u1 )⊤ |)kB˙ p,∞ 2 µ −1 +k (1 + a1 )|∇u2 + ∇(u2 )⊤ |(|∇u2 + ∇(u2 )⊤ | − |∇u1 + ∇(u1 )⊤ |)kB˙ p,∞ 2 −1 +kµδa(divu2 )2 + µ(1 + a1 )divδudiv(u1 + u2 ) + σδa(curlH 2 )2 kB˙ p,∞ −1 dτ. +kσ(1 + a1 )curlδHcurl(H 1 + H 2 ) − (1 + a1 )p(a1 )divδukB˙ p,∞

Making use of Lemma 2.3, Lemma 2.5, Lemma 2.8 and Lemma 2.9, we can prove −1 + η kδθkB˙ p,∞ ekδθkL˜ 1 (B˙ 1 ) p,∞ t Z t 2 ≤ CeCU (t) kδHkB˙ 0 k(H 1 , H 2 )kB˙ 2 ka1 kB˙ 1 p,∞

0

+kδakB˙ 0

1

p,∞

k(a , a

2

)k2B˙ 1

p,1

p,1

p,1

2

2

ku kB˙ 1 + (1 + ka kB˙ 1 )4 ku2 kB˙ 1 p,1

p,1

p,1




kθ1 kB˙ 1 + (1 + ka1 kB˙ 1 )5 + k(u1 , u2 )kB˙ 2 ka1 kB˙ 1 p,∞ p,1 p,1 p,1 p,1 
+kδakB˙ 1 kθ2 kB˙ 1 + ku2 kB˙ 2 ku2 kB˙ 0 + k(u2 , H 2 )k2B˙ 1 dτ. +kδukB˙ 0

p,1

p,1

p,1

p,1


(6.6)

p,1

Now, dealing with the third equation of (6.1), we get by taking advantage of Remark 3.2 for any t ∈ [0, T ], −1 kδukL˜ ∞(B˙ p,∞ ˜ 1(B˙ 1 ) ) + κνkδukL p,∞ t t Z t 1 2 −1 ≤ CeC(U +U )(t) (kδaAu2 − ∇(g(a2 ) − g(a1 ))kB˙ p,∞

0

−1 +kH 2 · ∇δH + δH · ∇H 1 − ∇δH · H 2 − ∇H 1 · δHkB˙ p,∞ −1 +ka2 δH · ∇H 2 + a2 H 1 · ∇δH + δaH 1 · ∇H 1 kB˙ p,∞ −1 )dτ. +ka2 ∇δH · H 2 + δa∇H 1 · H 2 + a1 ∇H 1 · δHkB˙ p,∞

0 0 Making use of Lemma 2.3, Lemma 2.5, Lemma 2.9 and B˙ p,1 ֒→ B˙ p,∞ yields −1 kδukL˜ ∞ (B˙ p,∞ ˜ 1 (B˙ 1 ) ) + κνkδukL t p,∞ t Z t 1 2 ≤ CeC(U +U )(t) (kH 1 k2B˙ 1 + kH 1 kB˙ 1 kH 2 kB˙ 1 p,1

p,1

0 2

1

p,1

2

+ka kB˙ 1 + ka kB˙ 1 + ku kB˙ 2 )kδakB˙ 0 p,1

p,1

p,1

p,∞

+(k(H 1 , H 2 )kB˙ 1 + ka2 kB˙ 1 kH 2 kB˙ 1 + ka1 kB˙ 1 kH 1 kB˙ 1 p,1 p,1 p,1 p,1 p,1
+ka2 kB˙ 1 kH 1 kB˙ 1 )kδHkB˙ p,∞ dτ. 0 p,1

p,1

(6.7)

1174

ACTA MATHEMATICA SCIENTIA

Vol.33 Ser.B

Inserting (6.3) and (6.5) into (6.7), we get −1 kδukL˜ ∞ (B˙ p,∞ ˜ 1(B˙ 1 ) + κνkδukL

≤ Ce Z ×

p,∞ )

t

t

C(U 1 +U 2 )(t)

1

(1 + ka kL˜ ∞ (B˙ 1

p,1 )

T

+ kH 1 kL˜ ∞ (B˙ 1

+ kH 2 kL˜ ∞ (B˙ 1 ) )

p,1 )

T

p,1

T

t

kH 1 k2B˙ 1 + kH 1 kB˙ 1 kH 2 kB˙ 1 + ka1 kB˙ 1 + ka2 kB˙ 1 p,1

p,1

0 2

1

p,1

2

p,1

2

p,1

2

+ku kB˙ 2 + k(H , H )kB˙ 1 + ka kB˙ 1 kH kB˙ 1 p,1 p,1 p,1 p,1
+ka1 kB˙ 1 kH 1 kB˙ 1 + ka2 kB˙ 1 kH 1 kB˙ 1 kδukL1τ (B˙ 1 ) dτ. p,1

p,1

p,1

p,1

(6.8)

p,1

We introduce the following logarithmic interpolation inequality which helps to deal with the inequality (6.8). Lemma 6.1 [10, 32] Let s ∈ R. Then for any 1 ≤ p, ρ ≤ +∞ and 0 < ǫ ≤ 1, we have kf kL˜ ρ (B˙ s

p,1 )

T

≤C

  s−ǫ + kf k ˜ ρ ˙ s+ǫ kf kL˜ ρ (B˙ p,∞ ) LT (Bp,∞ ) T log e + . kf kL˜ ρ (B˙ s )

kf kL˜ ρ (B˙ s

p,∞ )

T

ǫ

p,∞

T

From Lemma 6.1, it follows that kδukL1(B˙ 1

p,1 )

t

 kδukL˜ 1(B˙ 0 ) + kδukL˜ 1 (B˙ 2 )  p,∞ p,∞ t t , log e + ≤ CkδukL˜ 1 (B˙ p,∞ 1 ) t kδukL˜ 1 (B˙ p,∞ 1 ) t

which combines with (6.8) gives that for any t ∈ [0, Te], −1 kδukL˜ ∞ (B˙ p,∞ ˜ 1 (B˙ 1 ) + κνkδukL

≤ Ce Z ×

p,∞ )

t

t

C(U 1 +U 2 )(t)

1

(1 + ka kL˜ ∞ (B˙ 1

p,1 )

T

+ kH 1 kL˜ ∞ (B˙ 1

p,1 )

T

+ kH 2 kL˜ ∞ (B˙ 1 ) ) p,1

T

t

kH 1 k2B˙ 1 + kH 1 kB˙ 1 kH 2 kB˙ 1 + ka1 kB˙ 1 + ka2 kB˙ 1 + ku2 kB˙ 2 p,1

p,1

0

1

2

p,1

2

p,1

2

p,1

1

p,1

1

+k(H , H )kB˙ 1 + ka kB˙ 1 kH kB˙ 1 + ka kB˙ 1 kH kB˙ 1 p,1 p,1 p,1 p,1 p,1  
C T dτ, +ka2 kB˙ 1 kH 1 kB˙ 1 kδukL˜ 1 (B˙ 1 ) log e + p,∞ τ p,1 p,1 kδukL˜ 1 (B˙ 1 ) τ

p,∞

where CT = kδukL˜ 1 (B˙ 0 ) + kδukL˜ 1 (B˙ 2 ) . p,∞ τ p,∞ τ Since kH 1 k2B˙ 1 + kH 1 kB˙ 1 kH 2 kB˙ 1 + ka1 kB˙ 1 + ka2 kB˙ 1 + ku2 kB˙ 2 + k(H 1 , H 2 )kB˙ 1 + p,1

p,1

2

2

1

p,1

1

p,1

2

p,1

1

p,1

ka kB˙ 1 kH kB˙ 1 +ka kB˙ 1 kH kB˙ 1 +ka kB˙ 1 kH kB˙ 1 is integrable on [0, T ], e p,1 p,1 p,1 p,1 p,1 p,1 ka1 kL˜ ∞ (B˙ 1 ) + kH 1 kL˜ ∞ (B˙ 1 ) + kH 2 kL˜ ∞ (B˙ 1 ) ) and CT are bounded, and T

p,1

T

p,1

0

1

(1+

p,1

T

Z

p,1

C(U 1 +U 2 )(t)

1 r log(e +

CT r

)

dr = +∞.

e −1 = kδukL∞ (B˙ p,∞ ) = 0 for any t ∈ [0, T ], which t together with (6.3), (6.5) yields that for any t ∈ [0, Te], kδakB˙ 0 = 0, kδHkB˙ 0 = 0. Hence, p,∞ p,∞ (δa, δu, δH) = (0, 0, 0) on [0, Te]. In virtue of (6.6), Littlewood-Paley theory and the definition of Besov spaces, we can easily show that kδθk ˙ −1 = kδθk ˙ 1 = 0. Thus, δθ = 0 on [0, Te].

Osgood lemma apllied insures that kδukL˜ 1 (B˙ 1 t

p,∞ )

Bp,∞

Bp,∞

A continuity argument concludes that (a1 , H 1 , u1 , θ1 ) = (a2 , H 2 , u2 , θ2 ) on [0, T ]. We complete the proof of uniqueness of the solution to (1.5) in ETp . 2

No.4

D.F. Bian & B.L. Guo: WELL-POSEDNESS IN CRITICAL SPACES

1175

References [1] Nash J. Le probl` eme de Cauchy pour les ´ equations diff´ erentielles d’un fluide g´ en´ eral. Bull Soc Math France, 1962, 90: 487–497 [2] Itaya N. On initial value problem of the motion of compressible viscous fluid, especially on the problem of uniqueness. J Math Kyoto Univ, 1976, 16: 413–427 [3] Matsumura A, Nishida T. The initial value problem for the equations of motion of viscous and heatconductive gases. J Math Kyoto Univ, 1980, 20: 67–104 [4] Chen Q L, Miao C X, Zhang Z F. On the well-posedness for the viscous shallow water equations. SIAM J Math Anal, 2008, 40: 443–474 [5] Danchin R. Local theory in critical spaces for compressible viscous and heat-conductive gases. Comm Partial Differ Equ, 2001, 26: 1183–1233 [6] Danchin R. Global existence in critical spaces for flows of compressible viscous and heat-conductive gases. Arch Rational Mech Anal, 2001, 160: 1–39 [7] Danchin R. Global existence in critical spaces for compressible Navier-Stokes equations. Invent Math Anal, 2000, 141: 579–614 [8] Danchin R. On the uniqueness in critical spaces for compressible Navier-Stokes equations. Nonlinear Differ Equ Appl, 2005, 12: 111–128 [9] Danchin R. Well-posedness in critical spaces for barotropic viscous fluids with truly not constant density. Comm Partial Differ Equ, 2007, 32: 1373–1397 [10] Danchin R. Density-dependent incompressible viscous fluids in critical Spaces. Proc Roy Soc Edinburgh Sect A, 2003, 133: 1311–1334 [11] Kobayashi T, Shibata Y. Dacay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in R3 . Comm Math Phys, 1999, 200: 621–660 [12] Hoff D, Zumbrun K. Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow. Indiana Univ Math J, 1995, 44: 603–676 [13] Bresch D, Desjardins B. Existence of global weak solutiona for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model. Commun Math Sci, 2003, 238: 211–223 [14] Bresch D, Desjardins B, Lin C K. On some compressible fluid models: Korteweg, lubrication, and shallow water systems. Comm Partial Differ Equ, 2003, 28: 843–868 [15] Lion P L. Mathematical Topics in Fluid Mechanics, Vol 2. Oxford Lecture Series in Mathematics and its Applications. Oxford: Clarendon Press, 1998 [16] Mellet A, Vasseur A. On the barotropic compressible Navier-Stokes equations. Comm Partial Differ Equ, 2007, 32: 431–452 [17] Bresch D, Desjardins B. M´ etivier G. Recent mathematical results and open problems about shallow water systems//Analysis and Simulation of Fluid Dynamics. Adv Math Fluid Mech. Basel: Birkh¨ auser, 2007: 15–31 [18] Haspot B. Cauchy problem for the viscous shallow water equations with a term of capillarity. Math Models Methods Appl Sci, 2010, 20(7): 1049–1087 [19] Wang W K, Xu C J. The Cauchy problem for viscous shallow water equations. Rev Mat Iberoamericana, 2005, 21: 1–24 [20] Fan J S, Zhou Y. A blow-up criterion of strong solutions to the compressible viscous heat-conductive flows with zero heat conductivity. Acta Appl Math, 2011, 116: 317–327 [21] Fan J S, Ni L D, Zhou Y. Local well-posedness for the cauchy problem of the MHD equations with mass diffusion. Math Methods Appl Sci, 2011, 34(7): 792–797 [22] Gerebeau J F, Bris C L, Lelievre T. Mathematical Methods for the Magnetohydrodynamics of Liquid Metals. Oxford: Oxford University Press, 2006 [23] Li T, Qin H U. Physics and Partial Differential Equations, Vol I. 2nd ed. Beijing: Higher Education Press, 2005 [24] Moreau R. Magnetohydrodynamics. Dordredht: Kluwer Academic Publishers, 1990 [25] Mohamed A A, Jiang F, Tan Z. Decay estimates for isentropic compressible magnetohydrodynamic equations in bounded domain. Acta Math Sci, 2012, 32B(6): 2211–2220 [26] Barker B, Lafitte O, Zumbrun K. Existence and stability of viscous shock profiles for 2-D isentropic MHD with infinite electrical resistivity. Acta Math Sci, 2010, 30B(2): 447–498

1176

ACTA MATHEMATICA SCIENTIA

Vol.33 Ser.B

[27] Chen G Q, Wang D H. Global solutions of nonlinear magnetohydrodynamics with large initial data. J Differ Equ, 2002, 182: 344–376 [28] Chen G Q, Wang D H. Existence and continuous dependence of large solutions for the magnetohydrodynamics equations. Z Angew Math Phys, 2003, 54: 608–632 [29] Hoff D, Tsyganov E. Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics. Z Angew Math Phys, 2005, 56: 215–254 [30] Kato T. Quasi-linear equations of evolution, with applications to partial differential equations//Lecture Notes in Mathematics, Vol 448. Berlin: Springer-Verlag, 1975 [31] Kawashima S, Okada M. Smooth global solutions for the one-dimensional equations in magnetohydrodynamics. Proc Japan Acad Ser A Math Sci, 1982, 58: 384–387 [32] Chen Q L, Miao C X, Zhang Z F. Well-posedness in critical spaces for compressible Navier-Stokes equations with density dependent viscosities. Rev Mat Iberoamericana, 2010, 26: 915–946 [33] Zhou Y, Xin Z P, Fan J S. Well-posedness for the density-dependent incompressible Euler equations in the critical Besov spaces (in Chinese). Sci Sin Math, 2010, 40(10): 959–970 [34] Bian D F, Yuan B Q. Local well-posedness in critical spaces for compressible MHD equations. preprint. [35] Bian D F, Yuan B Q. Well-posedness in super critical Besov spaces for compressible MHD equations. Int J Dynamical Systems Differ Equ, 2011, 3: 383–399 [36] Xin Z P. Blow up of smooth solutions to the compressible Navier-Stokes equation with compact density. Commun Pure Appl Anal, 1998, 51: 229–240 [37] Cho Y, Jin B J. Blow-up of viscous heat-conducting compressible flows. J Math Anal Appl, 2006, 320(2): 819–826 [38] Rozanova O. Blow up of smooth solutions to the compressible Navier-Stokes equations with the data highly decreasing at infinity. J Differ Equ, 2008, 245: 1762–1774 [39] Fujita H, Kato T. On Navier-Stokes initial value problem. I. Arch Rational Mech Anal, 1964, 16: 269–315 [40] Meyer A. Wavelets, Paraproducts and Navier-Stokes equations//Current Developments in Mathematics. Cambridge: International Press, 1996 [41] Cannone M. Ondelettes, Paraproduits et Navier-Stokes equation//Nouveaux Essais. Paris: Diderot ´ Editeurs, 1995 [42] Cannone M. Harmonic analysis tools for solving the incompressible Navier-Stokes equations//Handbook of Mathematical Fluid Dynamics, Vol 3. Amsterdam: North-Holland, 2004 [43] Chemin J Y, Lerner N. Flot de champs de vecteurs non lipschitziens et ´ equations de Navier-Stokes. J Differ Equ, 1992, 121: 314–328 [44] Chemin J Y. Perfect Incompressible Fluids. New York: Oxford University Press, 1998 [45] Chemin J Y. Th´ eor` emes d’unicit´ e pour le syst` eme de Navier-Stokes tridimensionnel. J d’Analyse Math, 1999, 77: 27–50