On the global well-posedness for the compressible Navier–Stokes equations with slip boundary condition

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ScienceDirect J. Differential Equations ••• (••••) •••–••• www.elsevier.com/locate/jde

On the global well-posedness for the compressible Navier–Stokes equations with slip boundary condition Yoshihiro Shibata a,1 , Miho Murata b,∗,2 a Department of Mathematics and Research Institute of Science and Engineering, Waseda University, Ohkubo 3-4-1,

Shinjuku-ku, Tokyo 169-8555, Japan b Department of Mathematics, Waseda University, Ohkubo 3-4-1, Shinjuku-ku, Tokyo 169-8555, Japan

Received 15 August 2015; revised 4 December 2015

Abstract In this paper, we prove a global in time unique existence theorem for the compressible viscous fluids in a bounded domain with slip boundary condition in the maximal Lp -Lq regularity class with 2 < p < ∞ and N < q < ∞ under the assumption that initial data are small enough and orthogonal to rigid motions if domain is rotationally symmetric. To prove the global well-posedness, we use the prolongation argument based on the maximal Lp -Lq regularity estimate of exponentially decay type. The same problem was first treated by Kobayashi and Zaja¸czkowski [5] in the L2 framework by using the energy method; our approach is completely different from Kobayashi and Zaja¸czkowski [5]. © 2015 Elsevier Inc. All rights reserved.

1. Introduction We consider the motion of a compressible viscous fluid occupying a bounded domain  of the N dimensional Euclidean space RN (N ≥ 2) with slip boundary condition. Let  be the boundary of . The motion is described by the following equations: * Corresponding author.

E-mail addresses: [email protected] (Y. Shibata), [email protected] (M. Murata). 1 Partially supported by Top Global University Project and JSPS Grant-in-aid for Scientific Research (S) # 24224004. 2 Partially supported by grant for young researcher (Early Bird Program) from Waseda Research Institute for Science

and Engineering. http://dx.doi.org/10.1016/j.jde.2015.12.018 0022-0396/© 2015 Elsevier Inc. All rights reserved.

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⎧ ⎪ ⎪ ⎪ ⎨

∂t ρ + div(ρu) = 0 ρ(∂t u + u · ∇u) − Div S(u) + ∇p = 0 ⎪ D(u)n − D(u)n, nn = 0, u · n = 0 ⎪ ⎪ ⎩ (ρ, u)|t=0 = (ρ∗ + θ0 , u0 )

in  × (0, T ), in  × (0, T ), on  × (0, T ), in ,

(1.1)

where ∂t = ∂/∂t, t is the time variable, ρ = ρ(x, t), x = (x1 , . . . , xN ) ∈  the density, u = u(x, t) = (u1 (x, t), . . . , uN (x, t)) the velocity, p = p(x, t) the pressure, ρ∗ a positive constant describing the mass density of the reference body , and n the unit outward vector normal to . Moreover, let −pI + S(u) be the stress tensor with S(u) = μD(u) + (ν − μ) div uI,

(1.2)

where μ and ν are positive constants describing the first and second viscosity coefficients, respectively, D(u) denotes the deformation tensor whose (j, k) components are Dj k (u) = ∂j uk + ∂k uj with ∂j = ∂/∂xj , and I is the N × N identity matrix. We assume that the viscosity coefficients μ and ν satisfy the stability condition: μ > 0,

ν>

N −2 μ. N

(1.3)

Moreover, we consider the barotropic motion p = P (ρ)

(1.4)

where P (ρ) is a C ∞ function defined on ρ > 0 satisfying the condition: P (ρ) > 0 for ρ > 0. Finally, for any matrix field K with components Kj k , j, k = 1, . . . , N , the quantity Div K  is an N -vector with j -th component N , and also for any vector of functions v = k=1 ∂k Kj k N (v1 , . . . , vN ) and scalar function f , we set div v = N j =1 ∂j vj and v · ∇f = j =1 vj ∂j f with ∇ = (∂1 , . . . , ∂N ). The local well-posedness for (1.1) was first proved by Burnat and Zaja¸czkowski [2] in a bounded domain of 3-dimensional Euclidean space R3 , where the velocity field u and density 2+α,1+α/2 1+α,1/2+α/2 field ρ of the fluid belong to Sobolev–Slobodetskii spaces W2 and W2 with α ∈ (1/2, 1), respectively. Moreover, the global well-posedness was proved by Kobayashi and Zaja¸czkowski [5] in the same class as in the local in time unique existence theorem by the energy method. The proof is based on a local existence result from [2] and on the prolongation technique from [14]. They used an energy type inequality for the solution to problem in Euler coordinate. Recently, Murata [6] proved a local in time unique existence theorem in a uniform 3−1/q Wq domain3 in the maximal Lp -Lq regularity class. Namely, the velocity field u belongs to 2,1 1,1 Wq,p ( × (0, T )), while the density field ρ belongs to Wq,p ( × (0, T )), where 2 < p < ∞, N < q < ∞, and we have set ,m Wq,p ( × (0, T )) = Lp ((0, T ), Wq ()) ∩ Wpm ((0, T ), Lq ()).

3 The definitions of W 3−1/q and W 4−1/q domain are given in Definition 1.1, below. q q

(1.5)

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The purpose of this paper is to prove a global in time unique existence theorem in a bounded domain in the maximal Lp -Lq regularityclass with 2 < p < ∞ and N < q < ∞ under the assumption that initial data are small enough,  θ0 (x) dx = 0, and in addition, orthogonal to rigid motions if the domain is rotationally symmetric.4 Our approach has a new aspect to study the global well-posedness for the viscous compressible barotropic fluid flow. In fact, our setting is in Lp in time and Lq in space, while traditional approach like Kobayashi and Zaja¸czkowski [5] and Solonnikov and Tani [13] has been made in the H s setting. A global in time existence theorem follows the boot-strap argument to prolong the local in time solution to any time interval, which is achieved with the help of decay estimates of solutions to the linearized equations. To prove the decay estimates we combine the maximal Lp -Lq estimate and the exponential stability of the analytic semigroup, while the traditional arguments were based on the energy methods (cf. [5] and [13]). Our strategy follows Shibata [11], where the global well-posedness of free boundary problem was proved. Since the exponential stability for the linearized equation is obtained in the bounded domain case, that exponents p and q can be taken differently is not so big issue. But, if we consider the problem in unbounded domains, that exponents p and q can be taken rather freely will become relevant to prove the global well-posedness, because we can expect only polynomially decay properties, so that we have to choose p large enough freely to guarantee the global integrability in time (cf. H. Saito and Y. Shibata [7,8]). This is one of the important aspects of the Lp -Lq maximal regularity approach to the mathematical study of the viscous fluid flows, and it is future works the global well-posedness in unbounded domains like exterior domains, perturbed half spaces, layer domains and so on. In order to treat (1.1) in the maximal Lp -Lq regularity class, we reduce the problem to the parabolic systems by using the Lagrangian transformation. Let velocity fields v(ξ, t) and u(x, t) be known as vectors of functions of the Lagrange coordinates ξ and the Euler coordinates x of the same fluid particle, respectively. In this case, the connection between the Lagrange coordinate and the Euler coordinate is written in the form: 4−1/q Wq

t x=ξ +

v(ξ, s)ds ≡ Xv (ξ, t),

(1.6)

0

and v(ξ, t) = u(x,  tt). Let Av be the Jacobi matrix of the transformation x = Xv (ξ, t) with element aij = δij + 0 (∂vi /∂ξj )(ξ, s) ds. There exists a small number σ such that Av is invertible, that is det Av = 0, provided that T

(∂vi /∂ξj )(·, s) ds L∞ () < σ

max

i,j =1,...,N

(1.7)

0

t with sufficiently small σ > 0. In this case, we have ∇x = A−1 v ∇ξ = (I + V0 ( 0 ∇v(ξ, s) ds))∇ξ , where V0 (K) is an N × N matrix of C ∞ functions with respect to K = (kij ) defined on |K| < t 2σ and V0 (0) = 0, where kij are corresponding variables to 0 (∂vi /∂ξj )(·, s) ds. Assume that 4 The exact definition of the rotational symmetricity is given before the main results below and for example the ball is rotationally symmetric.

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ρ(x, t) and u(x, t) are solutions of (1.1) in the Euler coordinate. Setting ρ(Xv (ξ, t), t) = ρ∗ + θ (ξ, t) and u(Xv (ξ, t), t) = v(ξ, t), we write (1.1) in Lagrangian coordinate of the form: ⎧ ∂t θ + ρ∗ div v = F (θ, v) in  × (0, T ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ρ∗ ∂t v − Div S(v) + ∇(P (ρ∗ )θ ) = G(θ, v) in  × (0, T ), D(v)n − D(v)n, nn = H(v) on  × (0, T ), ⎪ ⎪ ⎪ v · n = H (v) on  × (0, T ), ⎪ ⎪ ⎪ ⎩ (θ, v)|t=0 = (θ0 , u0 ) in ,

(1.8)

where nˆ v = n(Xv (ξ, t)) and F (θ, v), G(θ, v), H(v) and H (v) are functions of the following forms: t F (θ, v) = −θ div v − (ρ∗ + θ )V1 ( ∇vds)∇v, 0

t G(θ, v) = −θ ∂t v + Div[μV2 (

t ∇vds)∇v + (ν − μ)V1 ( ∇vds)∇vI]

0

t + V3 (

0

t ∇vds)∇[μD(v) + μV2 (

0

1 − ∇(

t ∇vds)∇v + (ν − μ) div vI + (ν − μ)V1 ( ∇vds)∇v]

0

0

t

P (ρ∗ + τ θ)(1 − τ ) dτ θ 2 ) − P (ρ∗ + θ )V0 (

0

∇vds)∇θ, 0

H(v) = −D(v)(nˆ v − n) − {D(v)nˆ v , nˆ v nˆ v − D(v)n, nn} t t − {(V2 ( ∇v ds)∇v)nˆ v − (V2 ( ∇v ds)∇v)nˆ v , nˆ v nˆ v }, 0

0

H (v) = −v · (nˆ v − n).

(1.9)

Here, V1 (K), V2 (K) and V3 (K) are N × N matrices of C ∞ functions with respect to K = (kij ) defined on |K| < 2σ and V1 (0), V2 (0), V3 (0) = 0, where kij are corresponding variables to t 0 (∂vi /∂ξj )(·, s) ds. Before stating our results, we summarize several symbols and functional spaces used throughout the paper. N, R, and C denote the sets of all natural numbers, real numbers and complex numbers, respectively. We set N0 = N ∪ {0}. For 1 < q < ∞, let q be the dual exponent of q defined by q = q/(q − 1). For any multi-index κ = (κ1 , . . . , κN ) ∈ NN 0 , we κ write |κ| = κ1 + · · · + κN and ∂xκ = ∂1κ1 · · · ∂NN with x = (x1 , . . . , xN ). For scalar function f and N -vector of functions g, we set ∇f = (∂1 f, . . . , ∂N f ), ∇g = (∂i gj | i, j = 1, . . . , N), ∇ 2 f = {∂ α f | |α| = 2}, ∇ 2 g = {∂ α gi | |α| = 2, i = 1, . . . , N }.

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For Banach spaces X and Y , L(X, Y ) denotes the set of all bounded linear operators from X into Y , and Hol(U, L(X, Y )) the set of all L(X, Y ) valued holomorphic functions defined on a s (D) dedomain U in C. For any domain D in RN and 1 ≤ p, q ≤ ∞ Lq (D), Wqm (D) and Bq,p note the usual Lebesgue space, Sobolev space, and Besov space, while · Lq (D) , · Wqm (D) , and s (D) denote their norms, respectively. We set W 0 (D) = Lq (D) and W s (D) = B s (D).

· Bq,p q q q,q ∞ C (D) denotes the set of all C ∞ functions defined on D. Lp ((a, b), X) and Wpm ((a, b), X) denote the usual Lebesgue space and Sobolev space of X-valued function defined on an interval (a, b), respectively. We set m 

1/p

b

e f Wpm ((a,b),X) = ηt

=0

(e f ηt

( )

for 1 ≤ p < ∞

p

(t) X ) dt

a

with f ( ) = d f/dt and Wp0 ((a, b), X) = Lp ((a, b), X). The d-product space of X is defined by X d = {f = (f, . . . , fd ) | fi ∈ X (i = 1, . . . , d)}, while its norm is denoted by · X instead of · Xd for the sake of simplicity. We set Wqm, (D) = {(f, g) | f ∈ Wqm (D), g ∈ Wq (D)N }, (f, g) W m, (D) = f Wqm (D) + g Wq (D) . q

 For a = (a1 , . . . , aN ) and b = (b1 , . . . , bN ), we set a · b = a, b = N funcj =1 aj bj . For scalar   tions f , g and N -vectors of functions f, g we set (f, g) = f g dx, (f, g) = f · g dx, D D D D  (f, g) =  f g dσ , (f, g) =  f · gdσ , where σ is the surface element of . The letter C denotes generic constants and the constant Ca,b,... depends on a, b, . . . . The values of constants C and Ca,b,... may change from line to line. We use small boldface letters, e.g. u to denote vector-valued functions and capital boldface letters, e.g. H to denote matrix-valued functions, respectively. k−1/r Finally, we introduce the definition of a uniform Wr domain (k ≥ 2). Definition 1.1. Let 1 < r < ∞, let k be an integer ≥ 2, and let  be a domain in RN with k−1/r boundary . We say that  is a uniform Wr domain if there exists positive constants α, β k−1/r and K such that for any x0 = (x01 , . . . , x0N ) ∈  there exist a coordinate number j and a Wr function h(x ) (x = (x1 , . . . , xˆj , . . . , xN )) defined on Bα (x0 ) with x0 = (x01 , . . . , xˆ0j , . . . , x0N ) and h W k−1/r (B (x )) ≤ K such that r

α

0

 ∩ Bβ (x0 ) = {x ∈ RN | xj > h(x ) (x ∈ Bα (x0 ))} ∩ Bβ (x0 ),  ∩ Bβ (x0 ) = {x ∈ RN | xj = h(x ) (x ∈ Bα (x0 ))} ∩ Bβ (x0 ). Here, Bα (x0 ) = {x ∈ RN−1 | |x − x0 | < α}, Bβ (x0 ) = {x ∈ RN | |x − x0 | < β}. k−1/r

If the domain  is bounded and its boundary  is a Wr hyper-surface, then  is a uniform 3 domain. Since we estimate ∇ n which appears in boundary data for normal direction, 4−1/r we need that the boundary  is a Wr hyper-surface. Next, we introduce the definition of the rotationally symmetric domain.

k−1/r Wr

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Definition 1.2. By a hyperline in RN we mean an affine subspace of codimension two. Then,  is said to be rotationally symmetric with respect to a hyperline L if for any a ∈ L the two⊥ dimensional section L⊥ a ∩  of  by La is, if nonempty, symmetric with respect to a, where L⊥ denotes the plane through a and orthogonal to L. Moreover,  is said to be rotationally a symmetric if there exists a hyperline L such that  is rotationally symmetric with respect to L. The rotational symmetricity is deeply related to the existence of the rigid motion. Let Rd = {p | D(p) = 0 in , p · n = 0 on }.

(1.10)

As was seen in Ito [4], we know that Rd = ∅

when  is rotationally symmetric

Rd = ∅

when  is not rotationally symmetric.

(1.11)

For example,  = BR = {x ∈ RN | |x| < R} is rotationally symmetric for any R > 0, and in this case pij (x) = xi ej − xj ei , 1 ≤ i < j ≤ N , belong to Rd . Here, ei is the unit vectors of RN whose i-th component is 1. Let {p }M =1 be the orthogonal bases of Rd . Our global well-posedness theorem is the following. Theorem 1.3. Let N < q < ∞ and 2 < p < ∞. Let  be a bounded domain and assume that 4−1/r  is a Wr compact hyper-surface. Assume that the viscosity coefficients μ and ν satisfy the stability condition (1.3). Then, there exist positive numbers  and γ such that for any initial data 2(1−1/p) ()N satisfying the smallness condition: (θ0 , u0 ) ∈ Wq1 () × Bq,p

θ0 Wq1 () + u0 B 2(1−1/p) () ≤ ,

(1.12)

q,p

the compatibility condition: D(u0 )n − D(u0 )n, nn = 0,

u0 · n = 0

on ,

(1.13)

and the orthogonal conditions:

(θ0 , 1) = 0,

((ρ∗ + θ0 )u0 , p ) = 0 ( = 1, . . . , M)

(θ0 , 1) = 0

if  is rotationally symmetric, if  is not rotationally symmetric, (1.14)

problem (1.1) with T = ∞ admits unique solutions ρ and u with 1,1 ρ ∈ Wq,p ( × (0, ∞)),

possessing the estimate:

2,1 u ∈ Wq,p ( × (0, ∞)),

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eγ t ρ Wp1 ((0,∞),Lq ()) + eγ t ρ Lp ((0,∞),Wq1 ()) + eγ t ∂t u Lp ((0,∞),Lq ()) + eγ t u Lp ((0,∞),Wq2 ()) ≤ C with some positive number C independent of . Since the map x = Xv (ξ, t) is bijective (cf. [6, Proof of Theorem 2.7]), Theorem 1.3 immediately follows from Theorem 1.4. Let N < q < ∞ and 2 < p < ∞. Let  be a bounded domain and assume that 4−1/q compact hyper-surface. Assume that the viscosity coefficients μ and ν satisfy  is a Wq the stability condition (1.3). Then, there exist positive numbers  and γ such that for any initial 2(1−1/p) ()N satisfying the smallness condition (1.12), compatibility data (θ0 , u0 ) ∈ Wq1 () × Bq,p condition (1.13) and the orthogonal condition (1.14), problem (1.8) with T = ∞ admits unique solutions θ and v with θ ∈ Wp1 ((0, ∞), Wq1 ()),

v ∈ Wp1 ((0, ∞), Lq ()N ) ∩ Lp ((0, ∞), Wq2 ()N ),

possessing the estimate:

eγ t θ Wp1 ((0,∞),Wq1 ()) + eγ t ∂t v Lp ((0,∞),Lq ()) + eγ t v Lp ((0,∞),Wq2 ()) ≤ C with some positive number C independent of . Remark 1.5. The reason why we assume that 2 < p < ∞ is to use the inequality (4.19) in Sect. 4 below. 2. Some decay properties of solutions to the linearized problem In this section, we show some decay properties of solutions to the following problem: ⎧ ∂t κ + ρ∗ div w = f ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ρ∗ ∂t w − Div S(w) + P (ρ∗ )∇κ = g D(w)n − D(w)n, nn = h − h, nn, ⎪ ⎪ ⎪ w·n=h ⎪ ⎪ ⎪ ⎩ (κ, w)|t=0 = (θ0 , u0 )

in  × (0, T ), in  × (0, T ), on  × (0, T ),

(2.1)

on  × (0, T ), in .

3−1/r

Let  be a uniform Wr domain with N < r < ∞. To state our decay theorem for (2.1), according to Appendix in Schade and Shibata [9], we introduce the space W−1 q (), which is deN fined as follows: Let ι be the extension map from L1,loc () into L1,loc (R ) having the following properties: (e-1) For any 1 < q < ∞ and f ∈ Wq1 (), ιf ∈ Wq1 (RN ), ιf = f in  and ιf Wq (RN ) ≤ Cq f Wq () for = 0, 1 with some constant Cq depending on q, r and .

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(e-2) For any 1 < q < ∞ and f ∈ Wq1 (), ι(∇f ) W −1 (RN ) ≤ Cq f Lq () with some constant q Cq depending on q, r and . Here, Wq−1 (RN ) is the dual space of Wq1 (RN ). In the sequel, such extension map ι is fixed. Let W−1 q () = {f ∈ L1,loc () | f W−1 () = ιf W −1 (RN ) < ∞}. q

q

Then, we have Theorem 2.1. Let 1 < p, q < ∞, N < r < ∞ and T > 0. Assume that max(q, q ) ≤ r 3−1/r compact (q = q/(q − 1)). Let  be a bounded domain and assume that  is a Wr hyper-surface. Then, there exists a positive number γ0 such that for any initial data (θ0 , u0 ) ∈ 2(1−1/p) ()N and f , g, h and h with Wq1 () × Bq,p f ∈ Lp ((0, T ), Wq1 ()), g ∈ Lp ((0, T ), Lq ()N ), N h ∈ Lp ((0, T ), Wq1 ()N ) ∩ Wp1 ((0, T ), W−1 q () ),

h ∈ Lp ((0, T ), Wq2 ()) ∩ Wp1 ((0, T ), Lq ())

(2.2)

satisfying the compatibility conditions: D(u0 )n − D(u0 )n, nn = h|t=0 − h|t=0 , nn,

u0 · n = h|t=0

on ,

(2.3)

problem (2.1) admits unique solutions κ and w with κ ∈ Wp1 ((0, T ), Wq1 ()), w ∈ Lp ((0, T ), Wq2 ()N ) ∩ Wp1 ((0, T ), Lq ()N ) possessing the estimate:

eγ s κ Wp1 ((0,t),Wq1 ()) + eγ s ∂s w Lp ((0,t),Lq ()) + eγ s w Lp ((0,t),Wq2 ()) ≤ Cγ θ0 Wq1 () + u0 B 2(1−1/p) () + eγ s (f, g) L ((0,t),W 1,0 ()) + eγ s h Lp ((0,t),Wq1 ()) p

q,p

q

+ eγ s ∂s h Lp ((0,t),W−1 ()) + eγ s h Lp ((0,t),Wq2 ()) + eγ s h Wp1 ((0,t),Lq ()) q

+

 t 0

(eγ s |(κ(·, s), 1) |)p ds

1/p

⎞1/p ⎛ t   M ⎝ (eγ s |(w(·, s), p ) |)p ds ⎠ + δ() =1

0

for any t ∈ (0, T ] and γ ∈ (0, γ0 ) with some constant Cγ depending on γ but independent of t , where δ() is number defined by δ() = 1 if  is rotationally symmetric and δ() = 0 if  is not rotationally symmetric.

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In the sequel, we prove Theorem 2.1. For this purpose, first we analyze the corresponding generalized resolvent problem: ⎧ ⎪ ⎪ ⎪ ⎨

λκ + ρ∗ div w = f ρ∗ λw − Div S(w) + P (ρ∗ )∇κ = g

⎪ ⎪ ⎪ ⎩

in , in ,

D(w)n − D(w)n, nn = h − h, nn, on , w·n=h on .

(2.4)

To solve (2.4) in the case λ = 0, inserting the formula: κ = λ−1 (f − ρ∗ div w) into the second equation in (2.4), we have ρ∗ λw − Div S(w) − λ−1 ρ∗ P (ρ∗ )∇ div w = g − λ−1 P (ρ∗ )∇f.

(2.5)

Thus, instead of (2.4), we consider the equations: ⎧ in , ⎪ ⎨ ρ∗ λw − Div S(w) − δ∇ div w = f D(w)n − D(w)n, nn = h − h, nn on , ⎪ ⎩ w·n=h on ,

(2.6)

with f = g − λ−1 P (ρ∗ )∇f . In the sequel, δ and λ in (2.6) satisfy one of the following three cases: (C1) δ = λ−1 ρ∗ P (ρ∗ ) and λ ∈ ,λ0 ;

   δ (C2) δ ∈  with Re δ < 0 and λ ∈ C with Re λ ≥  Re Im δ |Imλ| and |λ| ≥ λ0 ; (C3) Re δ ≥ 0 and λ ∈ C with Re λ ≥ λ0 , where  = {λ ∈ C \ {0} | | arg λ| ≤ π − },

,λ0 = {λ ∈  | |λ| ≥ λ0 }

with 0 <  < π/2 and λ0 > 0. In (C2), we note that Im δ = 0, because δ ∈  and Re δ < 0. We may include the case where δ = 0 in (C1), which is corresponding to the Lamé system. For the sake of simplicity, we introduce the set ,λ0 defined by

,λ0

⎧ for (C1),  ⎪ ⎨ ,λ0    Re δ  = {λ ∈ C | Re λ ≥  Im δ |Im λ|, |λ| ≥ λ0 } for (C2), ⎪ ⎩ for (C3). {λ ∈ C | Re λ ≥ λ0 }

(2.7)

To quote a result due to Murata [6], we introduce the R-boundedness of operator families and the Weis operator valued Fourier multiplier theorem. Definition 2.2. A family of operators T ⊂ L(X, Y ) is called R-bounded on L(X, Y ), if there exist constants C > 0 and p ∈ [1, ∞) such that for any n ∈ N, {Tj }nj=1 ⊂ T , {fj }nj=1 ⊂ X and

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10

sequences {rj }nj=1 of independent, symmetric, {−1, 1}-valued random variables on [0, 1], we have the inequality: 1/p 1 1/p 1 n n p p

rj (u)Tj fj Y du ≤C

rj (u)fj X du . 0

j =1

0

j −1

The smallest such C is called R-bound of T , which is denoted by RL(X,Y ) (T ). Let D(R, X) and S(R, X) be the set of all X valued C ∞ functions having compact support and the Schwartz space of rapidly decreasing X valued functions, respectively, while S (R, X) = L(S(R, C), X). Given M ∈ L1,loc (R\{0}, X), we define the operator TM : F −1 D(R, X) → S (R, Y ) by TM φ = F −1 [MF[φ]],

(F[φ] ∈ D(R, X)).

(2.8)

The following theorem is obtained by Weis [16]. Theorem 2.3. Let X and Y be two UMD Banach spaces and 1 < p < ∞. Let M be a function in C 1 (R\{0}, L(X, Y )) such that RL(X,Y ) ({(τ

d ) M(τ ) | τ ∈ R\{0}}) ≤ κ < ∞ dτ

( = 0, 1)

with some constant κ. Then, the operator TM defined in (2.8) is extended to a bounded linear operator from Lp (R, X) into Lp (R, Y ). Moreover, denoting this extension by TM , we have

TM L(Lp (R,X),Lp (R,Y )) ≤ Cκ for some positive constant C depending on p, X and Y . Remark 2.4. For the definition of UMD space, we refer to a book due to Amann [1]. For 1 < q < ∞, Lebesgue space Lq () and Sobolev space Wqm () are both UMD spaces. According to Theorem 2.2 in Murata [6], we know Theorem 2.5. Let 1 < q < ∞, N < r < ∞, max(q, q ) ≤ r, 0 <  < π/2, δ0 > 0 and λ0 ≥ 1. 3−1/r domain and that |δ| ≤ δ0 . Let ,λ0 be the set defined in (2.7). Assume that  is a uniform Wr Set Yq () = {(f, h, h) | f ∈ Lq ()N , h ∈ Wq1 ()N , h ∈ Wq2 ()}, 2

Yq () = {F = (F2 , . . . , F7 ) | F2 , F3 , F6 ∈ Lq ()N , F4 , F7 ∈ Lq ()N , F5 ∈ Lq ()}.

(2.9)

Then, there exist a positive constant λ0 and an operator family A(λ) ∈ Hol(,λ0 , L(Yq (), Wq2 ()N )) such that for any (f, h, h) ∈ Yq () and λ ∈ ,λ0 , w = A(λ)Fλ (f, h, h) is a unique solution to problem (2.6), and A(λ) satisfies the following estimates:

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R L (Y

q (),Lq ()

N˜ )

({(τ

d ) (Gλ A(λ)) | λ ∈ ,λ0 }) ≤ C dτ

11

( = 0, 1)

with some constant C depending on , λ0 , δ0 , μ, ν, q and N . Here and in the following, we set N˜ = N 3 + N 2 + 2N , Gλ w = (λw, γ w, λ1/2 ∇w, ∇ 2 w), and Fλ (f, h, h) = (f, λ1/2 h, ∇h, λh, λ1/2 ∇h, ∇ 2 h). In view of (2.5), using the R boundedness of the composition and summation of the R bounded operators and Theorem 2.5, we have Theorem 2.6. (Cf. Theorem 2.1 in Murata [6].) Let 1 < q < ∞, N < r < ∞, max(q, q ) ≤ r, 3−1/r domain and that |δ| ≤ δ0 . 0 <  < π/2, δ0 > 0 and λ0 ≥ 1. Assume that  is a uniform Wr Let ,λ0 be the set defined in (2.7). Set Xq () = {(f, g, h, h) | (f, g) ∈ Wq1,0 (), h ∈ Wq1 ()N , h ∈ Wq2 ()}, Xq () = {F = (F1 , F2 , . . . , F7 ) | F1 ∈ Wq1 (), F2 , F3 , F6 ∈ Lq ()N , 2

F4 , F7 ∈ Lq ()N , F5 ∈ Lq ()}.

(2.10)

Then, there exist a positive constant λ0 and an operator family B(λ) ∈ Hol(,λ0 , L(Xq (), Wq1,2 ()) such that for any (f, g, h, h) ∈ Xq () and λ ∈ ,λ0 , (κ, w) = B(λ)F¯λ (f, g, h, h) is a unique solution to problem (2.4), and B(λ) satisfies the following estimates: d ) λB(λ)) | λ ∈ ,λ0 }) ≤ C ( = 0, 1), dτ d RL(Xq (),Lq ()N ) ({(τ ) λ1/2 ∇Pv B(λ)) | λ ∈ ,λ0 }) ≤ C ( = 0, 1), dτ d RL(X (),L ()N 2 ) ({(τ ) ∇ 2 Pv B(λ)) | λ ∈ ,λ0 }) ≤ C ( = 0, 1) q q dτ

R L (X

1,0 q (),Wq ())

({(τ

with some constant C depending on , λ0 , δ0 , μ, ν, q and N . Here Pv is the projection defined by Pv (κ, w) = w and we have set F¯λ (f, g, h, h) = (f, Fλ (g, h, h)). Since the definition of R boundedness implies the uniform boundedness of the operator family, it follows from Theorem 2.6 that for any λ ∈ ,λ0 and (f, g, h, h) ∈ Xq () problem (2.4) admits a unique solution (κ, w) ∈ Wq1,2 () possessing the estimate: |λ| κ Wq1 () + (λw, λ1/2 ∇w, ∇ 2 w) Lq () ≤ C( (f, g) W 1,0 () + (λ1/2 h, ∇h) Lq () + (λh, λ1/2 ∇h, ∇ 2 h) Lq () ). q

Let S be a linear operator defined by S(κ, w) = (−ρ∗ div w, ρ∗−1 Div S(w) − ρ∗−1 P (ρ∗ )∇κ) with

for (κ, w) ∈ Dq ()

(2.11)

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12

Dq () = {(κ, w) ∈ Wq1,2 () | D(w)n − D(w)n, nn = 0, w · n = 0 on }.

(2.12)

Using the operator S, problem (2.4) with h = 0 and h = 0 is written in the form: λ(κ, w) − S(κ, w) = (f, g). Then, by (2.11) we have Theorem 2.7. Let 1 < q < ∞, N < r < ∞ and max(q, q ) ≤ r. Assume that  is a uniform 3−1/r domain. Then, the operator S generates a continuous semigroup {T (t)}t≥0 on Wq1,0 (), Wr which is analytic. The next theorem is the core of our approach. Theorem 2.8. Let 1 < q < ∞, N < r < ∞ and max(q, q ) ≤ r. Assume that  is a bounded 3−1/r hyper-surface. Let domain whose boundary  is a Wr  L˙ q () =

{g ∈ Lq ()N | (g, p ) = 0 Lq ()N

( = 1, . . . , M)}

if  is rotationally symmetric, if  is not rotationally symmetric,

 and let W˙ q1 () = {κ ∈ Wq1 () |  κ dx = 0}. Let {T (t)}t≥0 be the C0 semigroup given in Theorem 2.7. Then, it is exponentially stable on W˙ q1 () × L˙ q (). Namely, there exist constants C and η1 > 0 such that

T (t)(f, g) W 1,0 () ≤ Ce−η1 t (f, g) W 1,0 () q

q

for any (f, g) ∈ W˙ q1 () × L˙ q () and t > 0.

Postponing the proof of Theorem 2.8 to Sect. 3, we give a Proof of Theorem 2.1. First of all, we reduce the problem to the case where initial data and boundary data are both zero. For this purpose, we consider the time shifted equations: ⎧ ∂t ρ1 + λ1 ρ1 + ρ∗ div u1 = f ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ρ∗ (∂t u1 + λ1 u1 ) − Div S(u1 ) + P (ρ∗ )∇ρ1 = g D(u1 )n − D(u1 )n, nn = h − h, nn, ⎪ ⎪ ⎪ u1 · n = h ⎪ ⎪ ⎪ ⎩ (ρ1 , u1 )|t=0 = (θ0 , u0 )

in  × (0, T ), in  × (0, T ), on  × (0, T ),

(2.13)

on  × (0, T ), in 

with large λ1 > λ0 . By Theorem 2.6 with the help of Theorem 2.3, we have the following theorem. Theorem 2.9. Let 1 < p, q < ∞, N < r < ∞, max(q, q ) ≤ r and T > 0. Let  be a bounded 4−1/r compact hyper-surface. Then, for any initial domain in RN , whose boundary  is a Wr 2(1−1/p) ()N and right-hand sides f , g, h, and h satisfying (2.2) data (θ0 , u0 ) ∈ Wq1 () × Bq,p and (2.3), problem (2.13) admits a unique solution (ρ, u1 ) with

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13

ρ1 ∈ Wp1 ((0, T ), Wq1 ()), u1 ∈ Lp ((0, T ), Wq2 ()N ) ∩ Wp1 ((0, T ), Lq ()N ). Moreover, there exists an η2 > 0 such that for any 0 < δ ≤ η2 there exists a constant C depending on δ but independent of T such that ρ1 and u1 possess the estimate:

eδs ρ1 Wp1 ((0,t),Wq1 ()) + eδs ∂s u1 Lp ((0,t),Lq ()) + eδs u1 Lp ((0,t),Wq2 ()) ≤ C θ0 Wq1 () + u0 B 2(1−1/p) () + eδs (f, g) L ((0,t),W 1,0 ()) + eδs h Lp ((0,t),Wq1 ()) p

q,p

q



+ e ∂s h Lp ((0,t),W−1 ()) + e ∂s h Lp ((0,t),Lq ()) + e h Lp ((0,t),Wq2 ()) δs

δs

δs

q

(2.14)

for any t ∈ (0, T ]. 4−1/r

Proof. Since  is a bounded domain whose boundary is a compact hyper-surface of Wr 2(1−1/p) class, there exist θ˜0 ∈ Wq1 (RN ) and u˜ 0 ∈ Bq,p (RN ) such that θ˜0 = θ0 and u˜ 0 = u0 in , and

θ˜0 Wq1 (RN ) ≤ C θ0 Wq1 () and u˜ 0 B 2(1−1/p) (RN ) ≤ C u0 B 2(1−1/p) () . And now, let (x, t) = q,p

q,p

e−λ1 t θ˜0 (x) and let U(x, t) be a solution to the time shifted heat equation: ∂t U + λ1 U − U = 0 in RN × (0, ∞),

U|t=0 = u˜ 0

in RN .

We know that  and U satisfy the estimate:

eδs  Wp1 (R+ ,Wq1 (RN )) + eδs ∂s U Lp (R+ ,Lq (RN )) + eδs U Lp (R+ ,Wq2 (RN )) ≤ C( θ0 Wq1 () + u0 B 2(1−1/p) () )

(2.15)

q,p

with some positive constants δ and C. Let ρ1 =  + ρ2 and u1 = U + u2 , and then ρ2 and u2 satisfy the zero initial value problem: ⎧ ∂t ρ2 + λ1 ρ2 + ρ∗ div u2 = F ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ρ∗ (∂t u2 + λ1 u2 ) − Div S(u2 ) + P (ρ∗ )∇ρ2 = G D(u2 )n − D(u2 )n, nn = H ⎪ ⎪ ⎪ u2 · n = H ⎪ ⎪ ⎪ ⎩ (ρ2 , u2 )|t=0 = (0, 0)

in  × (0, T ), in  × (0, T ), on  × (0, T ), on  × (0, T ), in ,

where F = f − ρ∗ div U, G = g − ρ∗ U + Div S(U) − P (ρ∗ )∇, H = h − hh, nn − (D(U)n − D(U)n, nn), H = h − U · n.

(2.16)

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Given g(·, s) defined for s ≥ 0, g0 (·, s) denotes the zero extension of g to s < 0, that is g0 (·, s) = g(·, s) for s ≥ 0 and g0 (·, s) = 0 for s < 0. Let g¯ be the extension of g defined by g¯ =

g0 (·, s) g0 (·, 2t − s)

for s ≤ t, for s ≥ t,

(2.17)

for any t ∈ (0, T ]. Let φ(s) = 1 for s ≤ 0 and φ(s) = 0 for s ≥ 1, and let φt (s) = φ(s − t). Using Theorem 2.6 with the help of Theorem 2.3, there exist solutions ρ¯1 and u¯ 1 satisfying the following equations: ⎧ ∂t ρ¯1 + λ1 ρ¯1 + ρ∗ div u¯ 1 = φt F¯ ⎪ ⎪ ⎪ ⎪ ⎪ ¯ ⎪ ⎨ ρ∗ (∂t u¯ 1 + λ1 u¯ 1 ) − Div S(u¯ 1 ) + P (ρ∗ )∇ ρ¯1 = φt G ¯ D(u¯ 1 )n − D(u¯ 1 )n, nn = φt H ⎪ ⎪ ⎪ ⎪ u¯ 1 · n = φt H¯ ⎪ ⎪ ⎩ (ρ¯1 , u¯ 1 )|t=0 = (0, 0)

in  × (0, ∞), in  × (0, ∞), on  × (0, ∞),

(2.18)

on  × (0, ∞), in .

In fact, using the R bounded solution operator B(λ) in Theorem 2.6, the solution (ρ¯1 , u¯ 1 ) of (2.18) is written in the form of ¯ ¯ ¯ ¯ ¯ (ρ¯1 (·, t), u¯ 1 (·, t)) = L−1 λ [B(λ + λ1 )Fλ (φt F (λ), φt G(λ), φt H(λ), φt H (λ))](t), ¯ where L−1 denotes λ denotes the Laplace inverse transform with respect to λ variable and φt g(λ) the Laplace transform of g with g = F , G, H and H . Applying Theorem 2.3 to ¯ ¯ ¯ ¯ ¯ e−(Re λ)t (ρ¯1 (·, t), u¯ 1 (·, t)) = e−(Re λ)t L−1 λ [B(λ + λ1 )Fλ (φt F (λ), φt G(λ), φt H(λ), φt H (λ))](t) ¯ φt H, ¯ φt H¯ )]](t), = F −1 [B(λ + λ1 )F[e−(Re λ)t F¯λ (φt F¯ , φt G, we see that there exists a unique solution for problem (2.18). By uniqueness of solution to (2.18) and g(·, ¯ s) = g(·, s) for 0 < s < T , (ρ¯1 , u¯ 1 ) is a unique solution to (2.16). Moreover, there exists an η2 > 0 with 0 < δ ≤ η2 such that there exists a constant C depending on δ but independent of T such that ρ2 and u2 possess the estimate:

eδs ρ2 Wp1 ((0,t),Wq1 ()) + eδs ∂s u2 Lp ((0,t),Lq ()) + eδs u2 Lp ((0,t),Wq2 ()) ¯ ¯ L (R,W 1 ()) ≤ C eδs (φt F¯ , φt G) + eδs φt H p L (R,W 1,0 ()) q p

q

δs δs ¯ ¯ ¯ + eδs ∂s φt H Lp (R,W−1 ()) + e ∂s φt H Lp (R,Lq ()) + e φt H Lp (R,Wq2 ()) q

 (2.19)

for any t ∈ (0, T ]. Since we can estimate right hand side of (2.19) by (2.17) and (2.15), we have Theorem 2.9. 2 We look for a solution (κ, w) of the form κ = ρ1 + ρ3 and w = u1 + u3 where (ρ1 , u1 ) is a solution of (2.13) and (ρ3 , u3 ) is a solution to the following problems:

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⎧ ∂t ρ3 + ρ∗ div u3 = −λ1 ρ1 ⎪ ⎪ ⎪ ⎪ ⎪ ∂ u − Div S(u ρ ⎪ 3 ) + P (ρ∗ )∇ρ3 = −ρ∗ λ1 u1 ⎨ ∗ t 3 D(u3 )n − D(u3 )n, nn = 0 ⎪ ⎪ ⎪ u3 · n = 0 ⎪ ⎪ ⎪ ⎩ (ρ3 , u3 )|t=0 = (0, 0)

in  × (0, T ), in  × (0, T ), on  × (0, T ), on  × (0, T ),

15

(2.20)

in ,

respectively. For the sake of simplicity, we set Jη,p,q = θ0 Wq1 () + u0 B 2(1−1/p) () + eηs (f, g) L

1,0 p ((0,t),Wq ())

q,p

+ eηs h Lp ((0,t),Wq1 ())

+ ∂s h Lp ((0,t),W−1 ()) + eηs ∂s h Lp ((0,t),Lq ()) + eηs h Lp ((0,t),Wq2 ()) . q

In the sequel, let γ = (1/2) min(η1 , η2 ), where η1 and η2 are positive numbers appearing in Theorem 2.8 and Theorem 2.9, respectively. Let {T (t)}t≥0 be the semigroup given in Theorem 2.8 and let ω(x, s) = ρ1 (x, s) − ||−1

 ρ1 (x, s) dx,

z(x, s) = u1 (x, s) − δ()

M (u1 (·, s), p ) p , =1



where || is the volume of  and the δ() is the same number as in Theorem 2.1. Defining ρ4 and u4 by t (ρ4 (·, t), u4 (·, t)) =

T (t − s)(−λ1 ω(·, s), −ρ∗ λ1 z(·, s))ds

(2.21)

0

by the Duhamel principle, ρ4 and u4 satisfy the equations ⎧ ∂t ρ4 + ρ∗ div u4 = −λ1 ω ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ρ∗ ∂t u4 − Div S(u4 ) + P (ρ∗ )∇ρ4 = −ρ∗ λ1 z D(u4 )n − D(u4 )n, nn = 0 ⎪ ⎪ ⎪ u4 · n = 0 ⎪ ⎪ ⎪ ⎩ (ρ4 , u4 )|t=0 = (0, 0)

in  × (0, T ), in  × (0, T ), on  × (0, T ),

(2.22)

on  × (0, T ), in .

Since (ω(·, s), 1) = 0 for any s ∈ (0, T ), and since (z(·, s), p ) = 0 ( = 1, . . . , M) for any s ∈ (0, T ) when  is rotationally symmetric, by Theorem 2.8 we have t

(ρ4 (·, t), u4 (·, t)) W 1,0 () ≤ C q

e−η1 (t−s) (ω(·, s), z(·, s)) W 1,0 () ds. q

0

Thus, by 0 < γ ≤ (1/2) min(η1 , η2 ), Hölder’s inequality and the change of the integral order, we have

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16

t (eγ τ (ρ4 (·, τ ), u4 (·, τ )) W 1,0 () )p dτ q

0

 t τ ≤C

e 0

e (ω(·, s), z(·, s)) W 1,0 () ds q

dτ

0

 t τ ≤C

p

−(η1 −γ )(τ −s) γ s

e

−(η1 −γ )(τ −s)

(e (ω(·, s), z(·, s)) W 1,0 () ) ds γs

p

τ

q

0 0

e−(η1 −γ )(τ −s) ds

p/p dτ

0

≤ C(η1 − γ )−p/p



t

t (eγ s (ω(·, s), z(·, s)) W 1,0 () )p q

s

0

≤ C(η1 − γ )−p

e−(η1 −γ )(τ −s) dτ ds

t (eγ s (ω(·, s), z(·, s)) W 1,0 () )p ds, q

0

which, combined with (2.14), furnishes that

eγ s (ρ4 , u4 ) L

1,0 p ((0,t),Wq ())

≤ CJγ ,p,q .

(2.23)

Since ρ4 and u4 satisfy the shifted equations: ⎧ in  × (0, T ), ∂t ρ4 + λ1 ρ4 + ρ∗ div u4 = λ1 (ρ4 − ω) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ρ∗ (∂t u4 + λ1 u4 ) − Div S(u4 ) + P (ρ∗ )∇ρ4 = ρ∗ λ1 (u4 − z) in  × (0, T ), on  × (0, T ), D(u4 )n − D(u4 )n, nn = 0, ⎪ ⎪ ⎪ on  × (0, T ), u4 · n = 0 ⎪ ⎪ ⎪ ⎩ (ρ4 , u4 )|t=0 = (0, 0) in .

(2.24)

By Theorem 2.9 and (2.23) we have

eγ s ρ4 Wp1 ((0,t),Wq1 ()) + eγ s ∂s u4 Lp ((0,t),Lq ()) + eγ s u4 Lp ((0,t),Wq2 ()) ≤ CJγ ,p,q .

(2.25)

In view of (2.21), we have −1

t

ρ3 = ρ4 − λ1 ||

 T (t − s) 

0

u3 = u4 − ρ∗ δ()λ1

ρ1 (x, s) dx ds,

M t

T (t − s)(u1 (·, s), p ) ds p .

=1 0

Since p satisfies D(p ) = 0, we see that div p = 0. In fact, D(p ) = 0 implies that p is written of the form p = Ax + b with some N × N anti-symmetric matrix A and b ∈ RN , and therefore

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17

div p = 0 in . Since p ∈ Rd , n · p | = 0. Thus, ρ2 and u2 satisfy the equations (2.20). Since κ = ρ1 + ρ3 and w = u1 + u3 , κ and w satisfy the equations (2.1). Since ∂t κ = ∂t ρ1 + ∂t ρ4 − λ1 ||−1

 ρ1 (x, t) dx, 

M ∂t w = ∂t u1 + ∂t u4 − ρ∗ δ()λ1 (u1 (·, t), p ) p , =1

∇κ = ∇ρ1 + ∇ρ4 ,

D(w) = D(u1 ) + D(u4 ),

∇ 2 w = ∇ 2 u1 + ∇ 2 u4 ,

by (2.14) and (2.25),

eγ s ∂s κ Lp ((0,t),Wq1 ()) + eγ s ∇κ Lp ((0,t),Lq ()) + eγ s ∂s w Lp ((0,t),Lq ()) + eγ s D(w) Lp ((0,t),Lq ()) + eγ s ∇ 2 w Lp ((0,t),Lq ()) ≤ CJγ ,p,q .

(2.26)

Using the Poincaré inequality, we have

τ Lq () ≤ C( ∇τ Lq () + |(τ, 1) |)

for any τ ∈ Wq1 (),

(2.27)

which, combined with (2.26), furnishes that

 t 1/p 

eγ s κ Lp ((0,t),Wq1 ()) ≤ C Jγ ,p,q + (eγ s |(κ(·, s), 1) |)p ds . 0

Moreover, using the second Korn inequality and the contradiction argument, we have

a(·, s) Wq1 () ≤ C D(a(·, s)) Lq () + δ()

M

 |(a(·, s), p ) |

=1

for any a ∈ Wq1 () with a · n| = 0, which, combined with (2.26), furnishes that

eγ s w Lp ((0,t),Wq1 ()) ≤ C e D(w) Lp ((0,t),Lq ()) + δ() γs

M  =1

≤ C Jγ ,p,q + δ()

M t =1

1/p 

t

(e |(w(·, s), p ) |) ds γs

p

0

1/p  (e |(w(·, s), p ) |) ds γs

0

Thus, combining (2.26) and (2.28), we have

p

.

(2.28)

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18

eγ s κ Wp1 ((0,t),Wq1 ()) + eγ s ∂s w Lp ((0,t),Lq ()) + eγ s w Lp ((0,t),Wq2 ()) ≤ C Jγ ,p,q +

 t

(eγ s |(κ(·, s), 1) |)p ds

1/p

0

+ δ()

M t =1

1/p  (e |(w(·, s), p ) |) ds γs

p

.

0

This completes the proof of Theorem 2.1.

2

3. Exponential stability of the semigroup {T (t)}t≥ ≥0 In this section, we prove Theorem 2.8. For this purpose, first we consider problem (2.4) with λ = 0, that is ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

ρ∗ div w = f − Div S(w) + P (ρ∗ )∇κ = g

in , in , D(w)n − D(w)n, nn = h − h, nn, on , w·n=h

(3.1)

on .

When κ satisfies equations (3.1) with some w, then κ + c also satisfies equations (3.1) with the same u for any constant c. Thus, we look for θ ∈ W˙ q1 () in (3.1). Moreover, when  is rotationally symmetric, to guarantee the uniqueness for w, we need the orthogonal condition (w, p ) = 0 for any = 1, . . . , M, so that we introduce the spaces: W˙ q2 () = {w ∈ L˙ q () | w ∈ Wq2 ()N }, where L˙ q () has been defined in Theorem 2.8. We start with Lemma 3.1. Let 1 < p, q < ∞, N < r < ∞, max(q, q ) ≤ r and T > 0. Let  be a bounded do2−1/r compact hyper-surface. Then, for any (f, g, h, h) ∈ main in RN , whose boundary  is a Wr Xq () satisfying the condition: ρ∗−1 (f, 1) = (h, 1)

(3.2)

and in addition, (g, p ) + μ(h, p ) = 0

( = 1, . . . , M)

(3.3)

when  is rotationally symmetric, problem (3.1) admits unique solutions κ ∈ W˙ q1 () and w ∈ W˙ q2 () possessing the estimate:

(κ, w) W 1,2 () ≤ C{ (f, g) W 1,0 () + h Wq1 () + h Wq2 () }. q

q

(3.4)

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19

Proof. Inserting the relation div w = ρ∗−1 f in the second equation and boundary condition, we have ⎧ ⎪ ⎪ ⎪ ⎨

ρ∗ div w = f −μ Div D(w) + P (ρ∗ )∇κ = g

⎪ ⎪ ⎪ ⎩

in , in , on ,

D(w)n − D(w)n, nn = h − h, nn, w·n=h

(3.5)

on .

with g = g + (ν − μ)ρ∗−1 ∇f . In the sequel, we use the estimate:

hn Wqi () ≤ C h Wqi ()

(i = 0, 1),

(3.6)

which follows from (6.2) in Murata [6]. By (3.6),

g Lq () ≤ C (f, g) W 1,0 () .

(3.7)

q

As we know well, under the condition (3.2) there exists a w ∈ Wq3 () such that w = ρ∗−1 f in  with ∇w · n| = h satisfying the estimate:

w Wq3 () ≤ C( f Wq1 () + h Wq2 () ).

(3.8)

Let w = ∇w + w1 in (3.5), and then κ and w1 satisfy the equations: ⎧ ⎪ ⎪ ⎪ ⎨

ρ∗ div w1 = 0 −μ Div D(w1 ) + P (ρ∗ )∇κ = g ⎪ D(w1 )n − D(w1 )n, nn = h ⎪ ⎪ ⎩ w1 · n = 0

in , in , on , on ,

(3.9)

where g = g + (μ + ν)ρ∗−1 ∇f,

h = h − h, nn − 2((∇ 2 w)n − (∇ 2 w)n, nn).

Thus, in the sequel, we consider equations (3.9). Instead of (3.9), first we consider the equations: ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

ρ∗ div w2 λ0 w2 − μ Div D(w2 ) + P (ρ∗ )∇κ2

=0

in ,

= g

in , on , on .

D(w2 )n − D(w2 )n, nn = h w2 · n = 0

(3.10)

As was proved in Shibata and Shimada [12], the generalized Stokes problem (3.10) is uniquely solvable for any λ0 ∈  with large |λ0 |, so that noting (3.7) and (3.8), we see that problem (3.10) admits unique solutions κ ∈ W˙ q1 () and w2 ∈ Wq2 ()N possessing the estimate:

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κ2 Wq1 () + (λ0 w2 , λ0 ∇w2 , ∇ 2 w2 ) Lq () ≤ C( g Lq () + (λ0 h , ∇h ) Lq () ). 1/2

1/2

In the following, λ0 is a fixed large positive number. Thus, we have

κ2 Wq1 () + w2 Wq2 () ≤ C( g Lq () + h Wq1 () ).

(3.11)

Moreover, when  is rotationally symmetric, multiplying the second equation of (3.10) by p and integrating the resultant equation over , by (3.3), (1.10), and the divergence theorem of Gauß, we have λ0 (w2 , p ) = (g + (μ + ν)ρ∗−1 ∇f, p ) + μ(Div D(w2 ), p ) − P (ρ∗ )(∇κ2 , p ) = (g, p ) + (μ + ν)ρ∗−1 (f, n · p ) + μ(D(w2 )n − D(w2 )n, nn, p ) + μ(D(w2 )n, n, n · p ) − (μ + ν)ρ∗−1 (f, div p ) μ − (D(w2 ), D(p )) + P (ρ∗ )(κ2 , div p ) 2 = (g, p ) + μ(h, p ) = 0, and therefore, w2 ∈ W˙ q2 (). Now, we solve equations (3.9). Let w2 ∈ W˙ q2 () and κ2 ∈ W˙ q1 () be solutions to the equations (3.10) with some large λ0 > 0, which possess the estimate:

w2 Wq2 () + κ2 Wq1 () ≤ C( (f, g) W 1,0 () + h Wq1 () ). q

(3.12)

Let w1 = w2 + w3 and κ = κ2 + κ3 in (3.9). And then, w3 and κ3 should satisfy the equations: ⎧ ρ∗ div w3 = 0 in , ⎪ ⎪ ⎪ ⎨ −μ Div D(w ) + P (ρ )∇κ = f in , 3 ∗ 3 ⎪ )n − D(w )n, nn = 0, on , D(w 3 3 ⎪ ⎪ ⎩ w3 · n = 0 on ,

(3.13)

with f = λ0 w1 ∈ L˙ q (). According to a result due to Shibata and Shimada [12], for any f ∈ L˙ q (), problem: ⎧ ρ∗ div w3 = 0 ⎪ ⎪ ⎪ ⎨ λ w − μ Div D(w ) + P (ρ )∇κ = f 0 3 3 ∗ 3 ⎪ D(w3 )n − D(w3 )n, nn = 0, ⎪ ⎪ ⎩ w3 · n = 0

in , in , on , on 

admits unique solutions w3 ∈ W˙ q2 () and κ3 ∈ W˙ q1 () possessing the estimate

κ3 Wq1 () + w3 Wq2 () ≤ C f Lq () .

(3.14)

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21

Let T : L˙ q () → W˙ q2 () be the solution operator for problem (3.14) defined by T f = w3 . Since  is compact, by the Rellich compactness theorem, the inverse of T is a compact operator from L˙ q () into itself, so that by the Riesz–Schauder theorem, the uniqueness implies the existence. This fact is called the homotopic argument in the sequel. Thus, to solve the unique existence of solutions of problem (3.13), we examine its uniqueness. Let w3 ∈ W˙ q2 () and κ3 ∈ W˙ q1 () be solutions of the homogeneous equation: ⎧ ⎪ ⎪ ⎪ ⎨

ρ∗ div w3 = 0 in , −μ Div D(w3 ) − P (ρ∗ )∇κ3 = 0 in , ⎪ D(w3 )n − D(w3 )n, nn = 0 on , ⎪ ⎪ ⎩ w3 · n = 0 on .

(3.15)

First we consider the case where 2 ≤ q < ∞. Then, w3 ∈ W˙ 22 () and κ3 ∈ W˙ 21 (), so that multiplying the second equation of (3.15) by w3 , integrating the resultant formula over , and using the divergence theorem of Gauß, we have μ

D(w3 ) 2L2 () = 0 2 because div w3 = 0. Thus, D(w3 ) = 0 in  and w3 · n = 0 on , namely w3 ∈ Rd . When  is rotationally symmetric, what w3 satisfies the condition: (w3 , p ) = 0 ( = 1, . . . , M) implies w3 = 0. On the other hand, when  is not rotationally symmetric, Rd = ∅, so that w3 = 0. Thus, ∇κ3 = 0. Since κ3 ∈ W˙ q1 (), we have κ3 = 0 with the help of the Poincaré inequality (2.27). Thus, we have the uniqueness, so that the Riesz–Schauder theorem implies the unique existence of solutions to problem (3.13). When 1 < q < 2, the uniqueness follows from the existence of the dual problem with exponent q > 2. Then, we also have the uniqueness when 1 < q < 2, and therefore the Riesz–Schauder theorem implies the unique existence of solutions to problem (3.13) when 1 < q < 2. Moreover, the estimate of the solutions w3 and κ3 of problem (3.13) is

w3 Wq2 () + κ3 Wq1 () ≤ C f Lq () . Summing up, we have proved the unique existence of solutions κ ∈ W˙ q1 (), w ∈ W˙ q2 ()N for problem (3.1) possessing the estimate (3.4), which completes the proof of Lemma 3.1. 2 Now, we are in a position to prove Theorem 2.8. To prove Theorem 2.8, we consider the resolvent problem: ⎧ ⎪ ⎨ ⎪ ⎩

λκ + ρ∗ div w = f ρ∗ λw − Div S(w) + P (ρ∗ )∇κ = g

in , in ,

D(w)n − D(w)n, nn = 0,

w · n = 0 on .

(3.16)

Then, to prove Theorem 2.8, in view of Theorem 2.7 or (2.11), it is sufficient to prove Lemma 3.2. Let λ2 > 0, 1 < q < ∞, N < r < ∞ and max(q, q ) ≤ r. Assume that  is a 3−1/r hyper-surface. If μ and ν satisfy the stability bounded domain whose boundary  is a Wr

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condition (1.3), then, for any λ ∈ C with Re λ ≥ 0 and |λ| ≤ λ2 and (f, g) ∈ W˙ q1 () × L˙ q (), problem (3.16) admits unique solutions (κ, w) ∈ W˙ q1 () × W˙ q2 () possessing the estimate:

κ Wq1 () + w Wq2 () ≤ C( f Wq1 () + g Lq () ).

(3.17)

Proof. To prove Lemma 3.2, we follow the idea due to Shibata and Tanaka [13] (cf. also Enomoto and Shibata [3]). In view of Lemma 3.1, by the small perturbation argument there exists a small σ > 0 such that for any λ ∈ C with |λ| ≤ σ and (f, g) ∈ W˙ q1 () × L˙ q (), problem (3.16) admits a unique solution (κ, w) ∈ W˙ q1 () × W˙ q2 () possessing the estimate:

κ Wq1 () + w Wq2 () ≤ C( f Wq1 () + g Lq () ) with some constant C depending on σ . Thus, in the sequel, we consider the case where λ ∈ , where  = {λ ∈ C | Re λ ≥ 0, σ ≤ |λ| ≤ λ2 }. In this case, setting κ = λ−1 (f − ρ∗ div w) in (3.16), we have 

ρ∗ λw − Div S(w) − ρ∗ P (ρ∗ )λ−1 ∇ div w = g D(w)n − D(w)n, nn = 0,

w·n=0

in , on ,

(3.18)

in , on ,

(3.19)

with g = g − λ−1 P (ρ∗ )∇f . Since σ ≤ |λ| ≤ λ2 ,

g Lq () ≤ Cσ,λ2 (f, g) W 1,0 () . q

To solve (3.18), first for fixed λ we consider the equations: 

ρ∗ λ˜ w − Div S(w) − ρ∗ P (ρ∗ )λ−1 ∇ div w = g D(w)n − D(w)n, nn = 0, w · n = 0

with new resolvent parameter λ˜ ∈ R. By Theorem 2.5, for each λ there exists a large λ˜ > 0 such that problem (3.19) admits a unique solution w ∈ Wq2 ()N , so that by the homotopic argument the uniqueness of solutions of (3.18) implies the existence of solutions of (3.18). Moreover, when  is rotationally symmetric, g satisfies the orthogonal condition: (g , p ) = (g, p ) − λ−1 P (ρ∗ )(∇f, p ) = −λ−1 P (ρ∗ ){(f, n · p ) − (f, div p ) } = 0. Thus, the solution w of (3.19) belongs to w ∈ W˙ q2 (), because ρ∗ λ˜ (w, p ) = (Div S(w) + ρ∗ P (ρ∗ )λ−1 ∇ div w, p ) = μ(D(w)n, n, n · p ) + (ν − μ + ρ∗ P (ρ∗ )λ−1 )(div w, n · p ) μ − (D(w), D(p )) − (ν − μ + ρ∗ P (ρ∗ )λ−1 )(div w, div p ) = 0. 2

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23

Thus, the uniqueness can be discussed for w ∈ W˙ q2 (). Let w ∈ W˙ q2 () satisfy the homogeneous equations: 

ρ∗ λw − Div S(w) − ρ∗ P (ρ∗ )λ−1 ∇ div w = 0 in , D(w)n − D(w)n, nn = 0, w · n = 0 on .

(3.20)

First we consider the case 2 ≤ q < ∞. In this case, w ∈ W˙ 22 (). Thus, multiplying the first equation (3.20) by w and using the divergence theorem of Gauß, we have 0 = ρ∗ λ w 2L2 () +

μ

D(w) 2L2 () + (ν − μ + ρ∗ P (ρ∗ )λ−1 ) div w 2L2 () . 2

(3.21)

Since we consider the case where Re λ ≥ 0, we have also Re ρ∗ P (ρ∗ )λ−1 ≥ 0, so that taking the real part of (3.21), we have μ

D(w) 2L2 () + (ν − μ) div w 2L2 () ≤ 0. 2

(3.22)

Since

div w 2L2 ()

≤N

N j =1

∂j uj 2L2 () ≤

N

D(w) 2L2 () , 4

we have μ 2μ

D(w) 2L2 () + (ν − μ) div w 2L2 () ≥ ( + ν − μ) div w 2L2 () . 2 N By the stability condition (1.3), 2μ N + ν − μ > 0, so that div w = 0. Thus, by (3.22) we have D(w) = 0. Since w · n = 0 on , we have w ∈ Rd . When  is not rotationally symmetric, Rd = ∅, which means that u = 0. When  is rotationally symmetric, the orthogonal condition implies that w = 0. Summing up, we have w = 0, that is the uniqueness holds for problem (3.18) when 2 ≤ q < ∞. Thus, we have the unique existence of solutions to problem (3.18) for 2 ≤ q < ∞. When 1 < q < 2, the unique existence theorem for the dual problem q > 2 implies the uniqueness, so that when 1 < q < 2 the unique existence theorem for problem (3.18) also holds. If we consider problem: 

ρ∗ λ w − Div S(w) − ρ∗ P (ρ∗ )λ−1 ∇ div w = g D(w)n − D(w)n, nn = 0,

w·n=0

in , on ,

(3.23)

near λ, then by the small perturbation arguments, for each λ ∈ , there exists a σ = σ (λ) such that for any g ∈ L˙ q () and λ ∈ C with |λ − λ| ≤ σ (λ) problem (3.23) admits a unique solution w ∈ W˙ q2 () possessing the estimate:

w Wq2 () ≤ C g Lq ()

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24

with some constant C depending on λ. Since  is compact, we can conclude that for any λ ∈  and g ∈ L˙ q () problem (3.18) admits a unique solution w ∈ W˙ q2 () possessing the estimate:

w Wq2 () ≤ C g Lq () with some constant C depending on . This completes the proof of Lemma 3.2.

2

4. A proof of Theorem 1.4 In this section, we prove the global in time unique existence theorem for (1.8), that is we prove Theorem 1.4. For this purpose, first we quote the result of the local well-posedness for (1.8) due to Murata. Theorem 4.1. (Cf. Theorem 7.1 in Murata [6].) Let 2 < p < ∞, N < q < ∞, R > 0 and assume 3−1/q domain in RN . Let ρ∗ be a positive constant and let P (ρ) be a C ∞ that  is a uniform Wq function defined on ρ > 0 such that ρ1 < P (ρ) < ρ2 with some positive constants ρ1 and ρ2 for any ρ ∈ (ρ∗ /4, 4ρ∗ ). Then, there exists a time T depending on R such that for any initial data 2(1−1/p) ()N with θ0 Wq1 () + u0 B 2(1−1/p) () ≤ R satisfying the range (θ0 , u0 ) ∈ Wq1 () × Bq,p q,p condition: ρ∗ /2 < ρ∗ + θ0 (x) < 2ρ∗

(x ∈ )

(4.1)

and compatibility condition (1.13), problem (1.8) admits a unique solution (θ, v) with θ ∈ Wp1 ((0, T ), Wq1 ()),

2,1 v ∈ Wq,p ( × (0, T )).

possessing the estimates: T ρ∗ /4 < ρ∗ + θ (x, t) < 4ρ∗ ((x, t) ∈  × (0, T )),

∇v(·, s) L∞ () ds ≤ σ. 0

In order to estimate the nonlinear terms to (1.8), we use the following lemma. 2−1/r

Lemma 4.2. Let 1 < p, q < ∞, T be any positive number and  be a uniform Wr with N < r < ∞. Then, the following two assertions hold:

domain

(1) We have sup v(t) B 2(1−1/p) () ≤ C{ v(·, 0) B 2(1−1/p) () + ev (T )}.

t∈(0,T )

q,p

q,p

(4.2)

for any v ∈ Wp1 ((0, T ), Lq ()) ∩ Lp ((0, T ), Wq2 ()) with some constant C independent of T , where ev (T ) = ∂t v Lp ((0,T ),Lq ()) + v Lp ((0,T ),Wq2 ()) .

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(2) In addition, we assume that N < q < ∞. Then, for any γ ∈ R we have

eγ t ∂t ((∇f )g) Lp ((0,T ),W−1 ()) q



T

≤C

(eγ t ∂t f (·, t) Lq () g(·, t) Wq1 () )p dt

1/p

0

+

T

(eγ t ∇f (·, t) Lq () ∂t g(·, t) Lq () )p dt

1/p 

(4.3)

0

with some constant C independent of γ and T . Lemma 4.2 can be proved in the same manner as in the proofs of the formula (3.17) and Lemma 3.3 in Shibata [10], so that we may omit its proof. In the sequel, we assume that 2 < p < ∞, N < q < ∞, and that the boundary  of  is a 4−1/q Wq hyper-surface. By Sobolev’s embedding theorem we have Wq1 () ⊂ L∞ (),

m  j =1

fj Wq1 () ≤ C

m  j =1

fj Wq1 () .

(4.4)

Let T be a positive number and let θ and v be solutions of problem (1.8) satisfying θ ∈ Wp1 ((0, T ), Wq1 ()),

v ∈ Wp1 ((0, T ), Lq ()N ) ∩ Lp ((0, T ), Wq2 ()N ),

(4.5)

and T ρ∗ /4 < ρ∗ + θ (x, t) < 4ρ∗

((x, t) ∈  × (0, T )),

∇v(·, s) L∞ () ds ≤ σ.

(4.6)

0

We may assume that 0 < σ < 1 without loss of generality, and therefore T sup θ (·, t) L∞ () ≤ 3ρ∗ ,

0
∇v(·, s) L∞ () ≤ 1.

(4.7)

0

Let I(t) = eγ s θ Wp1 ((0,t),Wq1 ()) + eγ s ∂s v Lp ((0,t),Lq ()) + eγ s v Lp ((0,t),Wq2 ()) with some positive constant γ for which Theorem 2.1 holds. To prolong θ and v beyond T , we show the estimate: I(t) ≤ K1 ( + I(t)2 + I(t)3 )

(4.8)

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26

with some constant K1 independent of  provided that

θ0 Wq1 () + u0 B 2(1−1/p) () ≤ 

(4.9)

q,p

with 0 <  < 1. We have

θ (·, t) Wq1 () ≤ C( + I(t)),

v(·, t) B 2(1−1/p) () ≤ C( u0 B 2(1−1/p) () + I(t)). q,p

In fact, using θ (x, t) = θ0 + t

(4.10)

q,p

t

0 ∂s θ (·, s)ds

and

⎛ t ⎞1/p ⎛ t ⎞1/p   ⎝ (eγ s ∂s θ (·, s) )p 1 ds ⎠

∂s θ (·, s) Wq1 () ds ≤ ⎝ e−p γ s ds ⎠ W () q

0

0

0

≤ C eγ s ∂s θ Lp ((0,t),Wq1 ()) ≤ I(t), we have the first inequality in (4.10). On the other hand, by (4.2) we have

v(·, t) B 2(1−1/p) () ≤ C{ u0 B 2(1−1/p) () + ev (t)} ≤ C{ + I(t)}. q,p

q,p

Applying Theorem 2.1 to problem (1.8), we have I(t) ≤ Cγ { θ0 Wq1 () + u0 B 2(1−1/p) () + eγ s (F, G) L

1,0 p ((0,t),Wq ())

q,p

+ eγ s H Lp ((0,t),Wq1 ())

+ eγ s ∂s H Lp ((0,t),W−1 ()) + eγ s H Lp ((0,t),Wq2 ()) + eγ s H Wp1 ((0,t),Lq ()) q

+

 t

(e |(θ (·, s), 1) |) ds γs

p

1/p

+ δ()

M  t

(eγ s |(v(·, s), p ) |)p ds

1/p

}.

(4.11)

=1 0

0

Here, F = F (θ, v), G = G(θ, v), H = H(v) and H = H (v) have been defined in (1.9). We consider the estimate of right hand side to (4.11). Using t

v(·, s) Wqm () ds ≤ 0

 t 0



e−p γ s ds

1/p  t

(eγ s ∇v(·, s) Wqm () )p ds

1/p

≤ CI(t)

(4.12)

0

for m = 0, 1, 2 and combining (4.6), (4.4) and (4.10), we have

eγ s (F (θ, v), G(θ, v)) L

1,0 p ((0,t),Wq ())

≤ C( + I(t)2 ).

In fact, for example, we consider the estimate to the second term of F (θ, v), and then

(4.13)

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27

s

eγ s θ (·, s)V1 ( ∇v(·, τ )dτ )∇v(·, s) Lq () 0

s ≤ θ (·, s) L∞ () V1 ( ∇v(·, τ ))dτ eγ s ∇v(·, s) Lq () . 0

s s Since V1 (0) = 0, we have V1 ( ∇v(·, τ )dτ ) L∞ () ≤ C ∇v(·, τ ) Wq1 () dτ , and therefore 0

0

by (4.7) and (4.12) we have s s

e θ (·, s)V1 ( ∇v(·, τ )dτ )∇v(·, s) Lq () ≤ 3ρ∗ C v(·, τ ) Wq2 () dτ eγ s v(·, s) Wq1 () γs

0

0

≤ CI(t) eγ s v(·, s) Wq1 () . Therefore, we have s

e θ (·, s)V1 ( ∇v(·, τ )dτ )∇v(·, s) Lp ((0,t),Lq ()) ≤ CI(t)2 . γs

0

Next, we consider the estimate to H(v). To estimate nˆ v − n Wq1 () , we write 1 nˆ v − n =

t (∇n)(ξ + τ

0

t v(ξ, s)ds)dτ ( v(ξ, s)ds)

0

0

and ∇(nˆ v − n) = A1 + A2 with 1 A1 =

t

t v(ξ, s) ds) dτ ( ∇v(ξ, s) ds),

0

0

(∇n)(ξ + τ 0

1 A2 =

t (∇ n)(ξ + τ

0

t v(ξ, s) ds)(I + τ

2

0

t ∇v(ξ, s) ds) dτ ( v(ξ, s) ds).

0

0

Using t

(∇n)(· + τ

t v(·, s) ds) L∞ () ≤ C,

0

for τ ∈ [0, 1] (cf. (7.28) in Murata [6]) we have

(∇ n)(· + τ

v(·, s) ds) Lq () ≤ C

2

0

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t

nˆ v − n Lq () ≤ (∇n)(· + τ

t v(·, s) ds) L∞ ()

0

0

1

A1 Lq () ≤

v(·, s) ds Lq () ≤ CI(t),

t (∇n)(· + τ

0

t v(·, s) ds) dτ L∞ ()

0

∇v(·, s) ds) Lq () ≤ CI(t), 0

t

A2 Lq () ≤ (1 + σ ) (∇ n)(· + τ

t v(·, s) ds) Lq ()

2

0

v(·, s) ds L∞ () ≤ CI(t). 0

Thus, we have

nˆ v − n Wq1 () ≤ CI(t).

(4.14)

eγ s D(v)(nˆ v − n) Lp ((0,t),Wq1 ()) ≤ C eγ s D(v) Lp ((0,t),Wq1 ()) I(t) ≤ CI(t)2 .

(4.15)

Using (4.4) and (4.14), we have

Since D(v)T = D(v), we write D(v)nˆ v , nˆ v nˆ v − D(v)n, nn = D(v)nˆ v , nˆ v (nˆ v − n) + D(v)(nˆ v − n), nˆ v nˆ v + D(v), n(nˆ v − n). By (4.14) and nˆ v W∞ 1 () , n W 1 () ≤ C we have ∞

eγ s {D(v)nˆ v , nˆ v nˆ v − D(v)n, nn} Lp ((0,t),Wq1 ()) ≤ CI(t)2 .

(4.16)

Employing the same argument as in proving the estimate of F (θ, v), we have s

V2 ( ∇vdτ )(∇v)nˆ v Lp ((0,t),Wq1 ()) ≤ CI(t), 0

s

V2 ( ∇vdτ )(∇v)nˆ v , nˆ v nˆ v Lp ((0,t),Wq1 ()) ≤ CI(t).

(4.17)

0

Combining (4.15), (4.16) and (4.17), we have

eγ s H(v) Lp ((0,t),Wq1 ()) ≤ CI(t)2 .

(4.18)

To estimate H(v) W−1 () , we need q

f Wq1 () ≤ C f B 2(1−1/p) () , q,p

(4.19)

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which holds provided that 2 < p < ∞. In fact, for example, we consider the estimate of the third term of H(v). By Lemma 4.2 we have s

e ∂s [V2 ( ∇v(·, τ )dτ )∇v(·, s)nˆ v ] Lp ((0,t),W−1 ()) γs

q

0

⎛ ⎞1/p t s p p ≤ C ⎝ epγ s ∂s v(·, s) Lq () V2 ( ∇v(·, τ )dτ )nˆ v W 1 () ds ⎠ q

0

0

⎛ t ⎞1/p   s p p pγ s ⎝ ⎠ + e ∇v(·, s) Lq () ∂s V2 ( ∇v(·, τ )dτ )nˆ v Lq () ds . 0

0

Using (4.12), we have ⎞1/p ⎛ t  s p ⎝ epγ s ∂s v(·, s) p ˆ v W 1 () ds ⎠ Lq () V2 ( ∇v(·, τ )dτ )n q

0

⎛

≤ CI(t) ⎝

0

t

⎞1/p

epγ s ∂s v(·, τ ) Lq () ds ⎠ p

0

≤ CI(t)2 . s Since ∂s ( 0 ∇v(·, τ )dτ ) = ∇v(ξ, s), (4.4) and (4.19), we have s s

∂s V2 ( ∇v(·, τ )dτ )nˆ v Lq () ≤ ∇v(·, s) Lq () V2 ( ∇v(·, τ )dτ ) L∞ () 0

0

≤ C v(·, s) Wq1 () ≤ C v(·, s) B 2(1−1/p) () ≤ C( + I(t)), q,p

s s where V 2 ( 0 ∇v(ξ, τ )dτ ) is matrix of C ∞ functions with respect to 0 ∇v(ξ, τ )dτ . Thus, we have ⎞1/p ⎛ t  s p ⎝ epγ s ∇v(·, s) p ˆ v Lq () ds ⎠ Lq () ∂s V2 ( ∇v(·, τ )dτ )n 0

⎛

≤ C( + I(t)) ⎝

0

t 0

Therefore, we have

⎞1/p

p epγ s ∇v(·, s) Lq () ds ⎠

≤ C( + I(t)2 ).

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s

eγ s ∂s [V2 ( ∇v(·, τ )dτ )∇v(·, s)nˆ v ] Lp ((0,t),W−1 ()) ≤ C( + I(t)2 ). q

0 2(1−1/p)

The assumption 2 < p < ∞ has been used to obtain Bq,p plies (4.19). Similarly, using (4.3), (4.14) and (4.19), we obtain

() ⊂ Wq1 (), which im-

eγ s ∂s H(v) Lp ((0,t),W−1 ()) ≤ C( + I(t)2 ).

(4.20)

q

Finally, we estimate H (v) = v · (nˆ v − n). Since

∂s H (v) Lq () ≤ C( ∂s v Lq () nˆ v − n L∞ () + v Lq () v L∞ () ), by (4.2), (4.4), (4.14) and (4.19) we have

eγ s H Wp1 ((0,t),Lq ()) ≤ C( + I(t)2 ). To estimate eγ s H Lq ((0,t),Wq2 ()) , we write ∇ 2 (nˆ v − n) = 1 B1 = 2

t (∇ n)(ξ + τ

0

0

1 B2 = 0

B3 =

0

∇v(ξ, s) ds) dτ

∇v(ξ, s) ds, 0

0

(∇ n)(ξ + τ

t v(ξ, s) ds)(I + τ

0

0

1

t (∇ 2 n)(ξ + τ

0

t

∇ 2 v(ξ, s) ds,

t 3

B4 =

where

t v(ξ, s) ds) dτ

1

i=1 Bi ,

0

t (∇n)(ξ + τ

4

t v(ξ, s) ds)(I + τ

2

(4.21)

t ∇v(ξ, s) ds) dτ 2

0

t

0

0

t ∇ 2 v(ξ, s) ds) dτ

v(ξ, s) ds)(τ 0

v(ξ, s) ds,

v(ξ, s) ds. 0

By (7.30) in Murata [6], we see that

f (∇ 2 n) Lq () ≤ C f Lq () ,

f (∇ 2 nˆ v ) Lq () ≤ C f Wq1 () .

(4.22)

4−1/r

Since  is assumed to be a Wr compact hyper-surface and since f g Lq () ≤ C f Lr () g Wq1 () provided that N < r < ∞ and 1 < q ≤ r, we have t

f (∇ n)(· + τ

v(·, s)ds) Lq () ≤ C f Wq1 () .

3

0

(4.23)

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Using (4.4), (4.6), (4.22), (4.12), and (4.23), t

B1 Lq () ≤ C(1 +

t

∇v(·, s) L∞ () ds)

0

∇v(·, s) Lq () ds ≤ CI(t), 0

t

B2 Lq () ≤ C

∇ 2 v(·, s) Lq () ds ≤ CI(t), 0

t

B3 Lq () ≤ C(1 +

t

∇v(·, s) L∞ () ds)

0

0

t

B4 Lq () ≤ C

v(·, s) Lq () ds ≤ CI(t),

2

t

∇ 2 v(·, s) Lq () ds

0

v(·, s) L∞ () ds ≤ CI(t)2 . 0

Therefore, by (4.14), we have

eγ s H Lq ((0,t),Wq2 ()) ≤ C(I(t)2 + I(t)3 ).

(4.24)

t t Next, we consider the estimates of 0 (eγ s |(θ (·, s), 1) |)p ds and 0 (eγ s |(v(·, s), p ) |)p ds. When the second assumption in (4.6) holds with suitably small σ > 0, using the same argumentation as in Ströhmer [15] we see that the Lagrange transformation x = Xv (ξ, t) is a bijective map from  onto itself for any t ∈ (0, T ). Let ξ = ξ(x, t) be its inverse map, and then this is also a bijective map from  onto itself with suitable regularity. Let u(x, t) = v(ξ(x, t), t),

ρ(x, t) = ρ∗ + θ (ξ(x, t), t),

(4.25)

and then u and ρ satisfy the equation (1.1). Let A be the Jacobi matrix of the transformation x = Xv (ξ, t) and set J (ξ, t) = det A. We know that ∂t J (ξ, t) = (div u)(x, t)J (ξ, t). Using the change of variable: x = Xv (ξ, t) and (1.1), we have d dt



 ρ(x, t) dx = 

 ∂t (ρJ ) dξ =



which, combined with the condition:

(ρt + div(ρu))J dξ = 0, 



 θ0 (x) dx



= 0, furnishes that

 ρ(x, t) dx =



(ρ∗ + θ0 (x)) dx = ρ∗ ||. 

Thus,  ρ∗ || =

 ρ(x, t) dx =



 (ρ∗ + θ (ξ, t))J (ξ, t) dξ = ρ∗ || +



θ (ξ, t)J (ξ, t) dξ, 

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which implies that  θ (ξ, t)J (ξ, t) dξ = 0. 

t t Since J (ξ, t) = det(I + V0 ( 0 ∇v(ξ, s) ds)) = 1 + V1 ( 0 ∇v(ξ, s) ds) with some function V1 = V1 (K) such that V1 (0) = 0, we have |(θ (·, t), 1) | ≤ C θ (·, t) Lq () I(t), which furnishes that

 t

(eγ s |(θ (·, s), 1) |)p ds

1/p

≤ CI(t)2 .

(4.26)

0

Next, we consider the case where  is rotationally symmetric. Since D(p ) = 0, p is written in the form: p = Ax + b with some anti-symmetric matrix A and some constant vector b. Note that D(p ) = 0 and n · p | = 0. We have d dt



 ρu, p  dx =



∂t ρu, p J dξ 





ρt + div(ρu))u, p J dξ +

= 

ρ(ut + u · ∇u), p J dξ 

= (Div S(u) − ∇p, p ) μ = − (D(u), D(p )) − (ν − μ)(div u, div p ) + (p, div p ) = 0, 2 so that we have d dt

 ρ(x, t)u(x, t), p (x) dx = 0, 

which furnishes that   ρ(x, t)u(x, t), p (x) dx = (ρ∗ + θ0 (x))u0 (x), p (x) dx = 0. 



Therefore, coming back to the Lagrange coordinate, we have 

t (ρ∗ + θ (ξ, t))v(ξ, t), p (ξ +



v(ξ, s)ds)J (ξ, t) dξ = 0. 0

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t Recalling that J = 1 + V1 ( 0 ∇v(ξ, s) ds) with V1 (0) = 0, we have

 t

(eγ s |(v(·, s), p ) |)p ds

1/p

≤ CI(t)2 .

(4.27)

0

Combining with (4.13), (4.18), (4.20), (4.21), (4.24), (4.26) and (4.27), we have (4.8). Next, we show that there exists a small 0 > 0 such that I(t) ≤ 2K1 

(4.28)

for any t ∈ (0, T ) and 0 <  < 0 . In fact, if we consider the cubic equation: K1 (x 3 + x 2 + ) = x, then this equation admits three real roots ri () (i = 1, 2, 3) such that r1 () < 0 < r2 () < r3 () provided that  is small enough. And moreover, we see that r2 () = K1  + K13  2 + O( 3 ) as  → 0+. From this observation, in order that I(t) satisfies (4.8), we have two possibilities: r1 () ≤ I(t) ≤ r2 () or I(t) ≥ r3 (). Since I(t) → 0 as t → 0 and I(t) is continuous with respect to t , we have 0 ≤ I(t) ≤ r2 () for any t ∈ (0, T ), because r2 () < r3 (). This implies (4.28). Finally, using (4.28), we prolong θ and v beyond T . By (4.10)

θ (·, T ) Wq1 () + v(·, T ) B 2(1−1/p) () ≤ K2  ≤ K2 q,p

(4.29)

with some constant K2 independent of . By (4.4), (4.10) and (4.28), we have θ (·, T ) L∞ () ≤ K3  with some constant K3 independent of , so that choosing  > 0 so small that K3  < ρ∗ /2, we have ρ∗ /2 < ρ∗ + θ (x, T ) < 2ρ∗ .

(4.30)

In order to prolong θ and v beyond T , we consider the following problem: ⎧ ⎪ ⎪ ⎪ ⎨

∂t θ¯ + ρ∗ div v¯ = F¯ (θ¯ , v¯ ) ¯ θ¯ , v¯ ) ρ∗ ∂t v¯ − Div S(¯v) + P (ρ∗ )∇ θ¯ = G( ¯ ⎪ D(¯v)n − D(¯v)n, nn = H(¯v), v¯ · n = H¯ (¯v) ⎪ ⎪ ⎩ (θ¯ , v¯ )|t=T = (θ (·, T ), v(·, T ))

in  × (T , T + T1 ), in  × (T , T + T1 ), on  × (T , T + T1 ),

(4.31)

in ,

¯ θ¯ , v¯ ), H(¯ ¯ v) and H¯ (¯v) are nonlinear functions defined by replacing θ , v where F¯ (θ, v¯ ), G( T t t ¯ and 0 ∇vds by θ , v¯ and 0 ∇vds + T ∇ v¯ ds for T < t < T + T1 in (1.9). Noting that T 0 ∇v Wq2 () ds ≤ K4 I(t) ≤ 2K1 K4  with some constant K4 independent of , by the same argumentation as in the proof of Theorem 7.1 in Murata [6], choosing positive number  and T1 small enough, we can show the unique existence of solutions θ¯ and v¯ of problem (4.31) with θ¯ ∈ Wp1 ((T , T + T1 ), Wq1 ()), satisfying the estimates:

v¯ ∈ Wp1 ((T , T + T1 ), Lq ()N ) ∩ Lp ((T , T + T1 ), Wq2 ()N )

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34

ρ∗ /4 < ρ∗ + θ¯ (x, t) < 4ρ∗ ((x, t) ∈  × (T , T + T1 )),

T+T1

∇ v¯ (·, s) L∞ () ds ≤ σ/2. T

(4.32) If we define θ1 and v1 by

θ (x, t) 0 < t < T , θ1 (x, t) = θ¯ (x, t) T < t < T + T1 ,

v1 (x, t) =

v(x, t) 0 < t < T , v¯ (x, t) T < t < T + T1 ,

then problem (1.8) in (0, T + T1 ) admits a solution (θ1 , v1 ) with θ1 ∈ Wp1 ((0, T + T1 ), Wq1 ()),

v1 ∈ Wp1 ((0, T + T1 ), Lq ()N ) ∩ Lp ((0, T + T1 ), Wq2 ()N ).

Moreover, by (4.4), (4.12) and (4.28), we have T

T

∇v(·, s) L∞ () ds ≤ C

0

∇v(·, s) Wq1 () ds ≤ CI(T ) ≤ K5  0

with some constant K5 independent of , so that choosing  so small that K5  ≤ σ/2, by (4.32) T+T1

∇v1 (·, s) L∞ () ds ≤ 0

T+T1

T

∇ v¯ L∞ () ds ≤ σ.

∇v L∞ () ds + 0

T

Therefore, we can prolong θ and v to (0, T + T1 ). Repeating the same argument on (0, T + T1 ), we can prolong θ and v to (0, T + 2T1 ). Therefore, finally we can define θ and v on the whole time interval (0, ∞) with I(t) ≤ 2K1  for any t ∈ (0, ∞). The uniqueness of solutions follows from the uniqueness of the local in time solutions. This completes the proof of Theorem 1.4. References [1] H. Amann, Linear and Quasilinear Parabolic Problems, vol. I, Birkhäuser, Basel, 1995. ˛ On local motion of a compressible barotropic viscous fluid with boundary slip condi[2] M. Burnat, W. Zajaczkowski, tion, Topol. Methods Nonlinear Anal. 10 (1997) 195–223. [3] Y. Enomoto, Y. Shibata, On the R-sectoriality and its application to some mathematical study of the viscous compressible fluids, Funkcial. Ekvac. 56 (2013) 441–505. [4] H. Ito, Remark on Korn’s inequality for vector fields which are tangent or normal to the boundary, preprint. ˛ On global motion of a compressible barotropic viscous fluid with boundary slip [5] T. Kobayashi, W. Zajaczkowski, condition, Appl. Math. 26 (2) (1999) 159–194. [6] M. Murata, On a maximal Lp -Lq approach to the compressible viscous fluid flow with slip boundary condition, Nonlinear Anal. 106 (2014) 86–109. [7] H. Saito, Y. Shibata, On decay properties of solutions to the Stokes equations with surface tension and gravity in the half space, J. Math. Soc. Japan (2015), in press. [8] H. Saito, Y. Shibata, Global well-posedness of a free boundary problem for the Navier–Stokes equations in some unbounded domain, preprint, 2015. [9] K. Schade, Y. Shibata, On strong dynamics of compressible Nematic Liquid Crystals, SIAM J. Math. Anal. 47 (5) (2015) 3963–3992.

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[10] Y. Shibata, On some free boundary problem for the Navier–Stokes equations in the maximal Lp -Lq regularity class, J. Differential Equations 253 (2015) 4127–4155. [11] Y. Shibata, On the global well-posedness of some free boundary problem for compressible barotropic viscous fluid flow, in: V. Radulescu, A. Sequeira, V. Solonnikov (Eds.), Recent Advances in Partial Differential Equations and Applications, in: Contemp. Math., American Mathematical Society, 2016, in press. [12] Y. Shibata, R. Shimada, On a generalized resolvent estimate for the Stokes system with Robin boundary condition, J. Math. Soc. Japan 59 (2) (2007) 469–519. [13] Y. Shibata, K. Tanaka, On a resolvent problem for the linearized system from the dynamical system describing the compressible viscous fluid motion, Math. Methods Appl. Sci. 27 (2004) 1579–1606. [14] V.A. Solonnikov, A. Tani, Evolution free boundary problem for equations of motion of a viscous compressible barotropic liquid, in: The Navier–Stokes Equations II – Theory and Numerical Methods, in: Lecture Notes in Math., vol. 1530, 1992, pp. 30–55; Tr. Mat. Inst. Steklova 83 (1965) 3–163 (in Russian); English transl.: Proc. Steklov Inst. Math. 83 (1967) 1–184. [15] G. Ströhmer, About a certain class of parabolic–hyperbolic systems of differential equation, Analysis 9 (1989) 1–39. [16] L. Weis, Operator-valued Fourier multiplier theorems and maximal Lp -regularity, Math. Ann. 319 (2001) 735–758.