On a maximal Lp–Lq approach to the compressible viscous fluid flow with slip boundary condition

Nonlinear Analysis 106 (2014) 86–109

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On a maximal Lp –Lq approach to the compressible viscous fluid flow with slip boundary condition Miho Murata ∗ Department of Pure and Applied Mathematics, Graduate School of Waseda University, Ohkubo 3-4-1, Shinjuku-ku, Tokyo 169-8555, Japan

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Article history: Received 13 March 2014 Accepted 16 April 2014 Communicated by Enzo Mitidieri MSC: 35Q35 76D07

abstract In this paper, we prove a local in time unique existence theorem for the compressible viscous fluids in the general domain with slip boundary condition. For the purpose, we use the contraction mapping principle based on the maximal Lp –Lq regularity by means of the Weis operator valued Fourier multiplier theorem for the corresponding time dependent problem. To obtain the maximal Lp –Lq regularity, we prove the sectorial R-boundedness of the solution operator to the generalized Stokes equations. © 2014 Elsevier Ltd. All rights reserved.

Keywords: Compressible viscous fluid Slip condition Local in time existence theorem R-boundedness Analytic semigroup Maximal Lp –Lq regularity

1. Introduction We consider the motion of a compressible barotropic viscous fluid occupying a domain Ω of the N dimensional Euclidean space RN (N ≥ 2) with the boundary slip condition. Let ρ = ρ(x, t ) be the density of the fluid, v = v(x, t ) = (v1 (x, t ), . . . , vN (x, t )) the velocity vector field, and P (ρ) the pressure function with x = (x1 , . . . , xN ) ∈ Ω and t being the time variable. The motion is described by the following equations:

 ∂ ρ + div (ρ u) = 0   t ρ(∂t u + u · ∇ u) − Div S(u) + ∇ P (ρ) = 0  D(u)n − ⟨D(u)n, n⟩n = 0, u · n = 0 (ρ, u)|t =0 = (ρ∗ + θ0 , u0 )

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

(1.1)

(cf. [1,2]), where ∂t = ∂/∂ t, ρ∗ is a positive constant describing the mass density of the reference body Ω , Γ the boundary of Ω and n the unit outward normal to Γ . Moreover, P (ρ) is a C ∞ function defined on ρ > 0 satisfying the condition: P ′ (ρ) > 0 for ρ > 0 and S(u) the stress tensor of the form: S(u) = α D(u) + (β − α)div uI, where α and β are positive constants describing the first and second viscosity coefficients, respectively, D(v) denotes the deformation tensor whose (j, k) components are Djk (v) = ∂j vk + ∂k vj with ∂j = ∂/∂ xj , and I the N × N identity matrix.

∗

Tel.: +81 3 5286 3000. E-mail address: [email protected].

http://dx.doi.org/10.1016/j.na.2014.04.012 0362-546X/© 2014 Elsevier Ltd. All rights reserved.

M. Murata / Nonlinear Analysis 106 (2014) 86–109

87

Finally, for any matrix field K with components Kjk , j, k = 1, . . . , N, the quantity Div K is an N-vector with jth component

∂k Kjk , and also for any vector of functions u = (u1 , . . . , uN ) we set div u = ∇ = (∂1 , . . . , ∂N ). N

k=1

N

j =1

and u · ∇ =

N

j =1

uj ∂j with

A local in time unique existence theorem was proved by Burnat and Zaja¸czkowski [3] in a bounded domain of 3-dimensional Euclidean space R3 , where their velocity field u and density of the fluid ρ belong to Sobolev–Slobodeckij 2+α,1+α/2 1+α,1/2+α/2 spaces W2 and W2 with α ∈ (1/2, 1), respectively. Moreover, a global in time unique existence theorem was proved by Kobayashi and Zaja¸czkowski [4] in the same class as in the local in time unique existence theorem by the 3−1/q 1 energy method. The purpose of this paper is to prove a local in time unique existence theorem in a uniform Wq and 2 ,1 1 ,1 our velocity field u and density of the fluid ρ belong to Wq,p (Ω × (0, T )) and Wq,p (Ω × (0, T )) with 2 < p < ∞ and N < q < ∞, where we have set Wqℓ,,pm (Ω × (0, T )) = Lp ((0, T ), Wqℓ (Ω )) ∩ Wpm ((0, T ), Lq (Ω )).

(1.2)

One of the merits of our approach is less compatibility condition than [3]. To obtain such a local in time unique existence theorem, it is key to prove the Lp –Lq maximal regularity for the linearized problem of the following form:

 ∂ ρ + γ div u = f  γt ∂ u −2Div S(u) + ∇(γ ρ) = g 0 t 1 α[ D ( u ) u − ⟨ D ( u ) n , n ⟩ n ] = h − ⟨h, n⟩n, u · n = h˜   (ρ, u)|t =0 = (ρ0 , u0 )

in Ω × (0, ∞), in Ω × (0, ∞), on Γ × (0, ∞), in Ω .

(1.3)

¯ satisfying the assumptions: γi = γi (x) (i = 0, 1, 2) are uniformly continuous functions defined on Ω ρ∗ /2 ≤ γ0 (x) ≤ 2ρ∗ ,

0 ≤ γk (x) ≤ ρ1

(x ∈ Ω , k = 1, 2),

∥∇γℓ ∥Lr (Ω ) ≤ ρ1 (ℓ = 0, 1, 2)

(1.4)

with some positive constant ρ1 . The maximal regularity result was first proved by Solonnikov [5] for the general parabolic equations satisfying the uniform Lopatinski–Shapiro conditions. In his paper, the problem in a domain is transformed locally to the model problems in a neighbourhood of either an interior point or a boundary point by using the localization technique and the partition of unity associated with the domain Ω . The boundary neighbourhood problem (1.3) is transformed to a problem in the half-space xN > 0. By applying the Fourier transform with respect to time and tangential directions, his problem becomes a system of ordinary differential equations. Solonnikov calculated explicitly the inverse Fourier transform of solutions of such ordinary differential equations and expressed them in the form of potentials in the half-space. Then, he estimated them in suitable norms. Burnat and Zaja¸czkowski [3] also used the same procedure as in Solonnikov [5] to transform problem (1.3) to the half-space problem and they estimated the inverse Fourier transform of solutions of the ordinary differential equations by the Plancherel theorem because they worked in the L2 framework. Kakizawa [6] proved R-boundedness of solutions to the ordinary differential equations corresponding to (1.3) and used the Weis operator valued Fourier multiplier theorem [7] to prove the maximal Lp –Lq regularity in the half-space. The idea in [6] is similar to that in Shibata and Shimizu [8]. Our approach is completely different from these papers [5,3,6], that is we prove the existence of an R bounded solution operator to the following generalized resolvent problem corresponding to time dependent problem (1.3):

 λθ + γ2 div v = f γ0 λv − Div S(v) + ∇(γ1 θ ) = g α[D(v)n − ⟨D(v)n, n⟩n] = h − ⟨h, n⟩n, v · n = h˜

in Ω , in Ω , on Γ .

(1.5)

In fact, we prove that for any ϵ ∈ (0, π /2), there exist a constant λ0 ≥ 1 and an operator family R(λ) ∈ Hol (Σϵ,λ0 ,

L(Xq (Ω ), Wq2 (Ω )N )) such that for any f ∈ Wq1 (Ω ), g ∈ Lq (Ω )N , h ∈ Wq1 (Ω )N and h˜ ∈ Wq2 (Ω ), problem (1.5)

admits a unique solution (ρ, v) = R(λ)(f, g, λ1/2 h, ∇ h, λh˜ , λ1/2 ∇ h˜ , ∇ 2 h˜ ) and (λ, λ1/2 ∇ Pv , ∇ 2 Pv )R(λ) is R-bounded for ˜ λ ∈ Σϵ,λ0 ∩Kϵ with value in L(Xq (Ω ), Wq1 (Ω )×Lq (Ω )N ). Here, Pv is the projection such that Pv (ρ, u) = u, N˜ = N +N 2 +N 3 ,

Σϵ,λ0 = {λ ∈ C | |λ| ≥ λ0 , | arg λ| ≤ π − ϵ},   2  2  γ γ 2 + ϵ + (Im λ) ≥ +ϵ , Kϵ = λ ∈ C | λ + α+β α+β

(1.6) 2

Xq (Ω ) = {F = (F1 , . . . , F7 ) | F1 ∈ Wq1 (Ω ), F5 ∈ Lq (Ω ), F2 , F3 , F6 ∈ Lq (Ω )N , F4 , F7 ∈ Lq (Ω )N }, with γ = supx∈Ω γ1 (x)γ2 (x), and F1 , F2 , F3 , F4 , F5 , F6 and F7 are independent variables corresponding to f , g, λ1/2 h∇ h, ˜ λ1/2 ∇ h˜ and ∇ 2 h, ˜ respectively. Moreover, Hol (U , L(X , Y )) denotes the set of all L(X , Y ) valued holomorphic functions λh, 1 The definition of W 3−1/q domain is given in Definition 1.1, below. q

88

M. Murata / Nonlinear Analysis 106 (2014) 86–109

defined on a complex domain U and L(X , Y ) the set of all bounded linear operators from a Banach space X into another Banach space Y . Since the solution (ρ, u) to problem (1.3) is represented by the inverse Laplace transform of the solution (θ, v) to problem (1.5), the maximal Lp –Lq result for problem (1.3) is obtained with the help of the Weis operator valued Fourier multiplier theorem [7]. The R sectoriality of the resolvent operator was introduced by Clément and Prüß [9], and Denk, Hieber and Prüß [10] proved the R-sectoriality for the boundary value problem of parameter elliptic equations satisfying the uniform Lopatinski–Shapiro condition in a domain with compact boundary. As a result, they proved the maximal Lp –Lq regularity for the corresponding time dependent problem. They [10] considered only the zero boundary condition case, while the non-homogeneous boundary condition case is treated in this paper. This is a different point than [9,10]. The approach to the maximal regularity theorem through R bounded solution operators in a general unbounded domain is also found in Enomoto and Shibata [11] and Shibata [12]. Once showing the existence of R bounded solution operators to problem (1.5), the maximal Lp –Lq results is rather easy consequence with the help of the Weis operator valued Fourier multiplier theorem. And, a local in time existence theorem follows by the standard argument of the contraction mapping principle based on the maximal Lp –Lq result. Thus, the main part of this paper is devoted to the proof of the existence of R bounded solution operators to problem (1.5) and the spirit of the proof is the same as in Shibata [12]. Before ending the introduction, we summarize several symbols and functional spaces used throughout the paper. For the differentiations of scalar f and N-vector g, we use the following symbols:

∇ f = (∂1 f , . . . , ∂N f ),

∇ 2 f = (∂i ∂j f | i, j = 1, . . . , N ),

∇ g = (∂i gj | i, j = 1, . . . , N ),

∇ 2 g = (∂i ∂j gk | i, j, k = 1, . . . , N )

with ∂j = ∂/∂ xj . For any Banach space X with norm ∥ · ∥X , X d denotes the d-product space of X , while its norm is denoted by ∥ · ∥X instead of ∥ · ∥X d for the sake of simplicity. For any domain D, Lq (D) and Wqm (D) denote the usual Lebesgue space and Sobolev space, while ∥ · ∥Lq (D) and ∥ · ∥Wqm (D) denote their norms, respectively. We set Wqm,ℓ (D) = {(f , g) | f ∈ Wqm (D), g ∈ Wqℓ (D)N } with Wq0 (D) = Lq (D). For 1 ≤ p ≤ ∞, Lp ((a, b), X ) and Wpm ((a, b), X ) denote the usual Lebesgue space and Sobolev space of X -valued functions defined on the interval (a, b), while ∥ · ∥Lp ((a,b),X ) and ∥ · ∥Wpm ((a,b),X ) denote their norms, respectively. Set Lp,γ1 (R, X ) = {f (t ) ∈ Lp,loc (R, X ) | e−γ1 t f (t ) ∈ Lp (R, X )}, Lp,γ1 ,0 (R, X ) = {f (t ) ∈ Lp,γ1 (R, X ) | f (t ) = 0 (t < 0)}, j Wpm,γ1 (R, X ) = {f (t ) ∈ Lp,γ1 (R, X ) | e−γ1 t ∂t f (t ) ∈ Lp (R, X ) (j = 1, . . . , m)},

Wpm,γ1 ,0 (R, X ) = Wpm,γ1 (R, X ) ∩ Lp,γ1 ,0 (R, X ),

∥e−γ t f ∥Wpm (I ,X ) =

m   j =0

(e−γ t ∥∂tj f (t )∥X )p dt

1/p

.

I

Let Fx = F and Fξ−1 = F −1 denote the Fourier transform and the Fourier inverse transform, respectively, which are defined by

Fx [f ](ξ ) =

 RN

e−ix·ξ f (x) dx,

Fξ−1 [g ](x) =



1

(2π )N

RN

eix·ξ g (ξ ) dξ .

We also write fˆ (ξ ) = Fx [f ](ξ ). Let L and L−1 denote the Laplace transform and the Laplace inverse transform, respectively, which are defined by



∞

e−λt f (t ) dt ,

L[f ](λ) = −∞

L−1 [g ](λ) =

1 2π

∞



eλt g (τ ) dτ

−∞

with λ = γ + iτ ∈ C. Given s ∈ R and X -valued function f (t ), we set 1 s Λsγ f (t ) = L− λ [λ L[f ](λ)](t ).

Moreover, the Bessel potential space of X -valued functions of order s is defined by the following: Hps,γ1 (R, X ) = {f ∈ Lp (R, X ) | e−γ t Λsγ [f ](t ) ∈ Lp (R, X ) for any γ ≥ γ1 }, Hps,γ1 ,0 (R, X ) = {f ∈ Hps,γ1 (R, X ) | f (t ) = 0 (t < 0)}. 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. N and C denote the set of all natural numbers and complex numbers, respectively, and we set N0 = N ∪ {0}. 3−1/r Finally, we introduce the definition of a uniform Wr domain, R boundedness of operator families and the Weis operator valued Fourier multiplier theorem.

M. Murata / Nonlinear Analysis 106 (2014) 86–109

89 3−1/r

Definition 1.1. Let 1 < r < ∞ and let Ω be a domain in RN with boundary Γ . We say that Ω is a uniform Wr domain if there exist positive constants α , β and K such that for any x0 = (x01 , . . . , x0N ) ∈ Γ there exist a coordinate 3−1/r number j and a Wr function h(x′ )(x′ = (x1 , . . . , xˆ j , . . . , xN )) defined on B′α (x′0 ) with x′0 = (x01 , . . . , xˆ 0j , . . . x0N ) and ∥h∥W 3−1/r (B′ (x′ )) ≤ K such that α

r

0

Ω ∩ Bβ (x0 ) = {x ∈ RN | xj > h(x′ ) (x′ ∈ B′α (x′0 ))} ∩ Bβ (x0 ), Γ ∩ Bβ (x0 ) = {x ∈ RN | xj = h(x′ ) (x′ ∈ B′α (x′0 ))} ∩ Bβ (x0 ). 3−1/r

Here, B′α (x′0 ) = {x′ ∈ RN −1 | |x′ − x′0 | < α}, Bβ (x0 ) = {x ∈ RN | |x − x0 | < β} and Wr functions h ∈ Wr2 (B′α (x′0 )) such that

  B′α (x0 )×B′α (x0 )

|∂k ∂l h(x′ ) − ∂k ∂l h(y′ )|r ′ ′ dx dy |x′ − y′ |N −2+r

1/r

(B′α (x′0 )) denotes the set of all

<∞

for k, l ̸= j with ∂k ∂l h = ∂ 2 h/∂ xk ∂ xl . Definition 1.2. 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 sequences {rj (u)}nj=1 of independent, symmetric, {−1, 1}-valued random variables on [0, 1] there holds the inequality:



1 0

p  1p p  1p   n n         rj (u)xj  du rj (u)Tj xj  du ≤C  .     j =1  j =1 X

Y

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 supports 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 [M F [φ]],

(F [φ] ∈ D (R, X )).

(1.7)

The following theorem is obtained by Weis [7]. Theorem 1.3. 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 dτ

ℓ

 M (τ ) | τ ∈ R \ {0}

≤ κ < ∞ (ℓ = 0, 1)

with some constant κ . Then, the operator TM defined in (1.7) 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 )) ≤ C κ for some positive constant C depending on p, X and Y . Remark 1.4. For the definition of UMD space, we refer to a book due to Amann [13]. For 1 < q < ∞, Lebesgue space Lq (Ω ) and Sobolev space Wqm (Ω ) are both UMD spaces. 2. Main results In this section, we summarize our main results. First, we are concerned with generalized resolvent problem (1.5). Theorem 2.1. Let 1 < q < ∞, N < r < ∞, max(q, q′ ) ≤ r (q′ = q/(q − 1)), and 0 < ϵ < π /2. Assume that Ω is a uniform 3−1/r domain. Let Σϵ,λ0 , Kϵ and Xq (Ω ) be the sets defined in (1.6) and set Λϵ,λ0 = Σϵ,λ0 ∩ Kϵ . Moreover, we define the space Wr Xq (Ω ) by Xq (Ω ) = {(f , g, h, h˜ ) | (f , g) ∈ Wq1,0 (Ω ), h ∈ Wq1 (Ω )N , h˜ ∈ Wq2 (Ω )}. Then, there exist a positive constant λ0 and an operator family R(λ) ∈ Hol(Λϵ,λ0 , L(Xq (Ω ), Wq1,2 (Ω ))) such that for any (f , g, h, h˜ ) ∈ Xq (Ω ) and λ ∈ Λϵ,λ0 , (θ , v) = R(λ)(f , g, λ1/2 h, ∇ h, λh˜ , λ1/2 ∇ h˜ , ∇ 2 h˜ ) is a unique solution to problem (1.5).

90

M. Murata / Nonlinear Analysis 106 (2014) 86–109

Moreover, there exists a constant C depending on ϵ , λ0 , q and N such that

RL(X

({(τ ∂τ )ℓ (λR(λ)) | λ ∈ Λϵ,λ0 }) ≤ C ,

RL(X

({(τ ∂τ )ℓ (γ R(λ)) | λ ∈ Λϵ,λ0 }) ≤ C ,

1,0 q (Ω ),Wq (Ω ))

1,0 q (Ω ),Wq (Ω ))

RL(X

2

N ) q (Ω ),Lq (Ω )

RL(X

N3 ) q (Ω ),Lq (Ω )

(2.1)

({(τ ∂τ )ℓ (λ1/2 ∇ Pv R(λ)) | λ ∈ Λϵ,λ0 }) ≤ C , ({(τ ∂τ )ℓ (∇ 2 Pv R(λ)) | λ ∈ Λϵ,λ0 }) ≤ C

with λ = γ + iτ and ℓ = 0, 1. To prove Theorem 2.1 in the case λ ̸= 0, inserting the formula: θ = λ−1 (f − γ1 (x)div v) into the second equation in (1.5), we have

λv − γ0−1 Div S(v) − λ−1 γ0−1 ∇(γ1 γ2 div v) = g − λ−1 γ0−1 ∇(γ1 f ). Thus, instead of (1.5), we mainly consider the equations:

 γ0 λv − Div S(v) − δ∇(γ3 div v) = f α[D(v)n − ⟨D(v)n, n⟩n] = h − ⟨h, n⟩n,

v · n = h˜

in Ω on Γ ,

(2.2)

with γ3 = γ1 γ2 . Let 0 < ϵ < π /2 and λ0 > 0. As δ , we consider the following three cases: (C1) δ = λ−1 and λ ∈ Λϵ,λ0 ;

Re δ (C2) δ ∈ Σϵ with Re δ < 0, λ ∈ C with Re λ ≥ | Im ||Im λ| and |λ| ≥ λ0 ; δ (C3) Re δ ≥ 0, and λ ∈ C with Re λ ≥ λ0 |Im λ| and |λ| ≥ λ0 . 1 In the following, for δ0 > 0 we assume that |δ| ≤ δ0 in any cases of (C1), (C2) and (C3). In particular, in (C1), δ0 = λ− 0 . In (C2), we note that Im δ ̸= 0, because δ ∈ Σϵ and Re δ < 0. We may include the case where δ = 0 in (C1), which is corresponding to the Lamé system. The case (C1) is used to prove the existence of an R-bounded solution operator pertaining to (1.5) and the cases (C2) and (C3) enable us the application of a homotopic argument in proving the exponential stability of the analytic semigroup associated with (1.3) in a bounded domain. Such homotopic argument already appeared in [14] and [11] in the non-slip condition case. For the sake of simplicity, we introduce the set Γϵ,λ0 ,δ0 defined by

Γϵ,λ0 ,δ0

 Λϵ,λ0        Re δ   |Im λ|, |λ| ≥ λ0 = λ ∈ C | Re λ ≥   Im δ    {λ ∈ C | Re λ ≥ λ0 |Im λ|, |λ| ≥ λ0 }

for (C1) for (C2)

(2.3)

for (C3)

with |δ| ≤ δ0 . We have Theorem 2.1 immediately with the help of Theorem 2.2 in case (C1). Theorem 2.2. Let 1 < q < ∞, N < r < ∞, max(q, q′ ) ≤ r (q′ = q/(q − 1)), 0 < ϵ < π /2, δ0 > 0 and λ0 > 0. Assume that 3−1/r Ω is a uniform Wr domain and that |δ| ≤ δ0 . Let Γϵ,λ0 ,δ0 be the set defined in (2.3). Set Yq (Ω ) = {(f, h, h˜ ) | f ∈ Lq (Ω )N , h ∈ Wq1 (Ω )N , h˜ ∈ Wq2 (Ω )}, 2

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

(2.4)

Then, there exist a positive constant λ0 and an operator family A(λ) ∈ Hol(Γϵ,λ0 ,δ0 , L(Yq (Ω ), Wq2 (Ω )N )) such that for any

(f, h, h˜ ) ∈ Yq (Ω ) and λ ∈ Λϵ,λ0 , v = A(λ)Fλ (f, h, h˜ ) is a unique solution to problem (2.2), and A(λ) satisfies the following estimates:

 RL(Y

N˜ q (Ω ),Lq (Ω ) )

τ

d dτ

ℓ

 (Gλ A(λ)) | λ ∈ Λϵ,λ0

≤ C (ℓ = 0, 1)

with some constant C depending on ϵ , λ0 , δ0 , a, b, q and N. Here and in the following, we set N˜ = N 3 + N 2 + 2N, Gλ v = (λv, γ v, λ1/2 ∇ v, ∇ 2 v), and Fλ (f, h, h˜ ) = (f, λ1/2 h, ∇ h, λh˜ , λ1/2 ∇ h˜ , ∇ 2 h˜ ). Secondly, we are concerned with time dependent problem (1.3). Let B be a linear operator defined by

B (θ , v) = (−γ2 div v, γ0−1 Div S(v) − γ0−1 ∇(γ1 θ ))

for (θ , v) ∈ Wq1 (Ω ) × Dq (Ω )

M. Murata / Nonlinear Analysis 106 (2014) 86–109

91

with Dq (Ω ) = {v ∈ Wq2 (Ω ) | [D(v)n − ⟨D(v)n, n⟩n]|Γ = 0, v · n = 0}. Since Definition 1.2 with n = 1 implies the boundedness of the operator family T , it follows from Theorem 2.1 that Λϵ,λ0 is contained in the resolvent set of B and for any λ ∈ Λϵ,λ0 and (f , g, h, h˜ ) ∈ Xq (Ω ),

|λ|∥θ ∥Wq1 (Ω ) + ∥(λv, λ1/2 ∇ v, ∇ 2 v)∥Lq (Ω ) ≤ C (∥(f , g)∥W 1,0 (Ω ) + ∥(λ1/2 h, ∇ h)∥Lq (Ω ) + ∥(λh˜ , λ1/2 ∇ h˜ , ∇ 2 h˜ )∥Lq (Ω ) ).

(2.5)

q

By (2.5), we have the following theorem. Theorem 2.3. Let 1 < q < ∞, N < r < ∞, max(q, q′ ) ≤ r (q′ = q/(q − 1)), 0 < ϵ < π /2, δ0 > 0 and λ0 > 0. 3−1/r Assume that Ω is a uniform Wr domain and that |δ| ≤ δ0 . Then, the operator B generates an analytic semigroup {T (t )}t ≥0 1 ,0 on Wq (Ω ). Moreover, there exist constants γ0 > 0 and M > 0 such that for any (f , g) ∈ Wq1,0 (Ω ), (ρ(t ), u(t )) = T (t )(f , g) solves (1.3) with f = 0, g = 0, h = 0 and h˜ = 0 and satisfies the following estimate:

∥T (t )(f , g)∥W 1,0 (Ω ) + t 1/2 ∥∇ Pv T (t )(f , g)∥Lq (Ω ) + t ∥∇ 2 Pv T (t )(f , g)∥Lq (Ω ) ≤ Meγ0 t ∥(f , g)∥W 1,0 (Ω ) . q

q

(2.6)

Combining Theorem 2.3 with a real interpolation method (cf. Shibata and Shimizu [8, Proof of Theorem 3.9]), we have the following result for equation (1.3) with f = 0, g = 0, h = 0 and h˜ = 0. Theorem 2.4. Let 1 < p, q < ∞, N < r < ∞, max(q, q′ ) ≤ r (q′ = q/(q − 1)), 0 < ϵ < π /2, δ0 > 0 and λ0 > 0. Assume 3−1/r that Ω is a uniform Wr domain and that |δ| ≤ δ0 . Set Bq2,(p1−1/p) (Ω ) = (Wq1,0 (Ω ), Dq (Ω ))1−1/p,p

(2.7) 2(1−1/p)

with real interpolation functor (·, ·)θ,p . Then, for any (ρ0 , u0 ) ∈ Wq1 (Ω ) × Bq,p

(Ω ), problem (1.3) with f = 0, g = 0, h = 0

and h˜ = 0 admits a unique solution (ρ, u) with

ρ ∈ Wp1,γ1 ((0, ∞), Wq1 (Ω )),

u ∈ Lp,γ1 ((0, ∞), Wq2 (Ω )) ∩ Wp1,γ1 ((0, ∞), Lq (Ω ))

possessing the estimate:

∥e−γ t ρ∥Wp1 ((0,∞),Wq1 (Ω )) + ∥e−γ t ∂t u∥Lp ((0,∞),Lq (Ω )) + ∥e−γ t u∥Lp ((0,∞),Wq2 (Ω )) ≤ C (∥ρ0 ∥Wq1 (Ω ) + ∥u0 ∥B2(1−1/p) (Ω ) ) q,p

for any γ ≥ γ1 with some constant C depending on p, q, γ1 and N. Combining Theorems 1.3 and 2.1, we have the maximal Lp –Lq regularity result for equation (1.3) with ρ0 = 0 and u0 = 0. Theorem 2.5. Let 1 < p, q < ∞, N < r < ∞, max(q, q′ ) ≤ r (q′ = q/(q − 1)), 0 < ϵ < π /2, δ0 > 0 and λ0 > 0. 3−1/r Assume that Ω is a uniform Wr domain and that |δ| ≤ δ0 . Then, there exists a positive constant γ2 such that for any 1/2 1,0 (f , g) ∈ Lp,γ2 ,0 (R, Wq (Ω )), h ∈ Lp,γ2 ,0 (R, Wq1 (Ω )N ) ∩ Hp,γ2 ,0 (R, Lq (Ω )N ), and h˜ ∈ Lp,γ2 ,0 (R, Wq2 (Ω )) ∩ Wp1,γ2 ,0 (R, Lq (Ω )), problem (1.3) with ρ = 0 and u = 0 admits a unique solution (ρ, u) with

ρ ∈ Wp1,γ2 ,0 (R, Wq1 (Ω )),

u ∈ Lp,γ2 ,0 (R, Wq2 (Ω )N ) ∩ Wp1,γ2 ,0 (R, Lq (Ω )N ),

possessing the estimate

∥e−γ t (γ ρ, ∂t ρ)∥Lp (R,Wq1 (Ω )) + ∥e−γ t (γ u, ∂t u)∥Lp (R,Lq (Ω )) + ∥e−γ t u∥Lp (R,Wq2 (Ω )) ≤ C (∥e−γ t (f , g)∥Lp (R,W 1,0 (Ω )) + ∥e−γ t (∇ h, Λ1γ/2 h)∥Lp (R,Lq (Ω )) q

h∥W 1 (R,Lq (Ω )) + ∥e−γ t h˜ ∥Lp (R,W 2 (Ω )) )

−γ t ˜

+ ∥e

p

q

for any γ ≥ γ2 with some constant C depending on N, p and q. Remark 2.6. As was seen in Shibata and Shimizu [8], we know that 1/2

Hp,γ2 ,0 (R, Wq1 (Ω )) ⊂ Lp,γ2 ,0 (R, Wq2 (Ω )N ) ∩ Wp1,γ2 ,0 (R, Lq (Ω )N ),

∥e−γ t Λ1γ/2 ∇ f ∥Lp (R,Lq (Ω )) ≤ C {∥e−γ t f ∥Wp1 (R,Lq (Ω )) + ∥e−γ t f ∥Lp (R,Wq2 (Ω )) }. Finally, we are concerned with a local in time unique existence theorem for (1.1).

(2.8)

92

M. Murata / Nonlinear Analysis 106 (2014) 86–109 3−1/q

Theorem 2.7. Let 2 < p < ∞, N < q < ∞, R > 0 and assume that Ω be a uniform Wq domain in RN . Let ρ∗ be a positive constant and let P (ρ) be a C ∞ function defined on ρ > 0 such that ρ1 < P ′ (ρ) < ρ2 with some positive constants ρ1 and ρ2 for 2(1−1/p) any ρ ∈ (ρ∗ /4, 4ρ∗ ). Let Bq,p (Ω ) be the space defined in (2.7). Then, there exists a time T depending on R such that for any 2(1−1/p)

initial data (θ0 , u0 ) ∈ Wq1 (Ω ) × Bq,p

(Ω ) with ∥θ0 ∥Wq1 (Ω ) + ∥u0 ∥B2(1−1/p) (Ω ) ≤ R satisfying the range condition: q,p

ρ∗ /2 < ρ∗ + θ0 (x) < 2ρ∗ (x∈Ω )

(2.9)

problem (1.1) admits a unique solution (ρ, u) with

ρ ∈ Wq1,,p1 (Ω × (0, T )),

u ∈ Wq2,,p1 (Ω × (0, T )).

Employing a similar argumentation to that in Shibata and Shimizu [8], we can show the existence part of Theorem 2.5 by using Theorems 1.3 and 2.1. Moreover, the uniqueness of solutions to (1.3) can be proved by using the existence of solutions to the dual problem as was seen also in Shibata and Shimizu [8]. Thus, we may omit the proof of Theorem 2.5. In the following, we give detailed proofs of Theorem 2.2 from Section 3 through Section 6. And, in Section 7, we prove Theorem 2.7 applying the usual contraction mapping principle to the Lagrangian coordinate description of problem (1.1) based on a time local maximal Lp –Lq regularity result derive from Theorems 2.4 and 2.5. 3. On the R boundedness of solution operators in RN In this section, following Enomoto and Shibata [11, Section 3], we discuss the R boundedness of solution operators to the equation:

γ0 λv − α ∆v − β∇ div v − δ∇(γ3 div v) = f in RN .

(3.1)

Here, we note that Div S(v) = α ∆v + β∇ div v.

(3.2)

In this section, γ0 and γ3 are positive constants. First, we derive the solution operator of (3.1). To obtain it, applying div to (3.1), we have

(γ0 λ − (α + β + γ3 δ)∆)div v = f,

(3.3)

from which it follows that (γ0 λ − α ∆)v = f + (β + γ3 δ)(γ0 λ − (α + β + γ3 δ)∆)

−1

v = (γ0 λ − α ∆)

−1

f + (β + γ3 δ)(γ0 λ − α ∆)

−1

∇ div f. Thus,

(γ0 λ − (α + β + γ3 δ)∆) ∇ div f −1

= (γ0 λ − α ∆) f − (γ0 λ − α ∆) ∆ ∇ div f + (γ0 λ − (α + β + γ3 δ)∆)−1 ∆−1 ∇ div f. −1

−1

−1

By the Fourier transform and the inverse Fourier transform for f = (f1 , . . . , fN ) we define the solution operator S0 (λ) to (3.1) by S0 (λ)f = (v1 , . . . , vN ) with

vj (x) =

    N N   1 −1 δjk − ξj ξk |ξ |−2 1 ξj ξk |ξ |−2 F [fk ](ξ ) −1 Fξ F [ f ](ξ ) ( x ) − F (x). (3.4) k α α −1 γ0 λ + |ξ |2 α + β + γ3 δ ξ (α + β + γ3 δ)−1 γ0 λ + |ξ |2 k=1 k=1

Concerning the spectrum, we know the following result. Lemma 3.1. Let 0 < ϵ < π /2, λ0 > 0, and s ≥ 1. (1) For any λ ∈ Σϵ , ξ ∈ RN and a > 0 we have |aλ + |ξ |2 | ≥ (sin 2ϵ )(a|λ| + |ξ |2 ). (2) For any λ ∈ Γϵ,λ0 ,δ0 we have (sα + β + γ3 δ)−1 γ0 λ ∈ Σσ with some constant σ ∈ (0, π /2) depending on s, α , β , γ0 , γ3 , λ0 , δ0 and ϵ . (3) For any λ ∈ Γϵ,λ0 ,δ0 and ξ ∈ RN we have

δ1 (|λ| + |ξ |2 ) ≤ |(sα + β + γ3 δ)−1 γ0 λ + |ξ |2 | ≤ δ2 (|λ| + |ξ |2 ) with some constants δ1 and δ2 depending on s, α , β , γ0 , γ3 , λ0 , δ0 and ϵ . Proof. The lemma was proved by Götz and Shibata [15] (cf. also Shibata and Tanaka [14]), but to make the paper self-contained as much as we can, we quote their proof here. (1) is well-known. To prove (2), it suffices to observe that

| arg(sα + β + γ3 δ)−1 λ| < π .

(3.5)

We prove (3.5) by the contradiction argument, so that we suppose that | arg(sα + β + γ3 δ) (sα + β + γ3 δ)−1 λ = −p2 for some p ∈ R \ {0}, from which it follows that Re λ + (sα + β)p2 + p2 γ3 Re δ = 0,

Im λ = −p2 γ3 Im δ.

−1

λ| = π . Thus we have (3.6)

M. Murata / Nonlinear Analysis 106 (2014) 86–109

93

First, we consider the (C1) case. In this case, Re δ = Re λ|λ|−2 and Im δ = −Im λ|λ|−2 , because δ = 1/λ. If Im λ = 0, we have Re λ > 0, because λ ∈ Σϵ , which furnishes that

(1 + p2 γ3 |λ|−2 )Re λ + (sα + β)p2 > 0.

(3.7)

This contradicts to the first relation of (3.6). If Im λ ̸= 0, by the second relation of (3.6) we have 1 = p γ3 |λ| combined with the first relation of (3.6) furnishes that 2



Re λ +

γ3 sα + β

2

+ (Im λ)2 =



γ3 sα + β

2

−2

, which

.

This contradicts to λ ∈ Kϵ , because sα + β ≥ α + β and γ3 ≤ γ in (1.6). Thus, we have (3.5) in the case of (C1). Secondly, we consider the (C2) case. By the second relation of (3.6), p2 = |Im λ|/(γ3 |Im δ|), so that by the first relation of (3.6) and Re δ < 0 we have

|Re δ| |Im λ| ≤ 0, |Im δ|

(sα + β)p2 = −Re λ + |Re δ|

because Re λ ≥ |Im δ| |Im λ| in the case of (C2). This contradicts to (sα + β)p2 > 0, so that we have (3.5). Thirdly, we consider the (C3) case. In this case, Re λ ≥ 0 and Re δ ≥ 0, which contradicts to the first relation of (3.6). Thus, we have (3.5), which completes the proof of the assertion (2). Finally, we prove the assertion (3). Since (sα + β + γ3 δ)−1 γ0 λ ∈ Σσ with some σ ∈ (0, π /2) as follows from (2), by (1) we have |(sα + β + γ3 δ)−1 γ0 λ + |ξ |2 | ≥ (sin σ /2)(|(sα + β + γ3 δ)−1 γ0 λ| + |ξ |2 ). Since |sα + β + γ3 δ| ≤ sα + β + γ3 δ0 , we have |(sα + β + γ3 δ)−1 γ0 λ| ≥ (sα + β + γ3 δ0 )−1 |γ0 λ|, so that

|(sα + β + γ3 δ)−1 γ0 λ + |ξ |2 | ≥ (sin σ /2) min(1, γ0 /(sα + β + γ3 δ0 ))(|λ| + |ξ |2 ). On the other hand, we have |(sα + β + γ3 δ)−1 γ0 λ + |ξ |2 | ≤ (|sα + β + γ3 δ|−1 γ0 λ| + |ξ |2 ). Since δ ∈ Σϵ in any cases of (C1), (C2) and (C3), by (1) we have |sα + β + γ3 δ| ≥ (sin ϵ/2)(sα + β + γ3 |δ|) ≥ (sin ϵ/2)(sα + β), so that

|(sα + β + γ3 δ)−1 γ0 λ + |ξ |2 | ≤ max(1, (γ0 /(sin ϵ/2)(sα + β))|λ| + |ξ |2 ), which completes the proof of Lemma 3.1.



The following theorem is the main result of this section. Theorem 3.2. Let 1 < q < ∞, 0 < ϵ < π /2, δ0 > 0 and λ0 > 0. Let S0 (λ) be the operator defined in (3.4). Then, S0 (λ) ∈ Hol (Γϵ,λ0 ,δ0 , L(Lq (RN )N , Wq2 (RN )N )), for any f ∈ Lq (RN )N and λ ∈ Γϵ,λ0 ,δ0 , v = S0 (λ)f is a unique solution to the problem (3.1), and we have

 RL(L

q (R

N )N ,L (RN )N˜ ) q

τ

d dτ

ℓ

 (Gλ S0 (λ)) | λ ∈ Γϵ,λ0 ,δ0

≤ C (ℓ = 0, 1)

with some constant C depending on ϵ , λ0 , δ0 , α , β , γ0 , γ3 , q and N. Theorem 3.2 follows immediately from Lemma 3.1 and the following theorem due to Enomoto and Shibata [11]. Theorem 3.3. Let 1 < q < ∞ and let Λ be a set in C. Let m(λ, ξ ) be a function defined on Λ × (RN \ {0}) such that for any multi-index α ∈ NN0 (N0 = N ∪ {0}) there exists a constant Cα depending on α and Λ such that

|∂ξα m(λ, ξ )| ≤ Cα |ξ |−|α|

(3.8)

for any (λ, ξ ) ∈ Λ × (RN \ {0}). Let Kλ be an operator defined by Kλ f = Fξ−1 [m(λ, ξ )F [f ](ξ )]. Then, the set {Kλ | λ ∈ Λ} is R-bounded on L(Lq (Rn ))2 and

RL(Lq (RN )) ({Kλ | λ ∈ Λ}) ≤ Cq,N max Cα

(3.9)

|α|≤N +1

with some constant Cq,N that depends solely on q and N. 4. On the R boundedness of solution operators in RN + Let RN+ and RN0 be the half-space and its boundary defined by

RN+ = {x = (x1 , . . . , xN ) ∈ RN | xN > 0}, 2 We write L(X , X ) = L(X ) for short.

RN0 = {x = (x1 , . . . , xN ) | xN = 0},

94

M. Murata / Nonlinear Analysis 106 (2014) 86–109

respectively. When Ω = RN+ , n = n0 = (0, . . . , 0, −1), we consider the following equation in this section:

 γ0 λv − α ∆v − (β + γ3 δ)∇ div v = f α(∂N vj + ∂j vN ) = −hj , vN = −h˜

in RN+ ,

(4.1)

on RN0 ,

where j = 1, . . . , N − 1. In this section, we assume that γ0 and γ3 are positive numbers and we prove the following theorem. Theorem 4.1. Let 1 < q < ∞, 0 < ϵ < π /2, δ0 > 0 and λ0 > 0. Let Γϵ,λ0 ,δ0 be the set defined in (2.3). Set Yq′ (RN+ ) = {(f, h′ , h˜ ) | f ∈ Lq (RN+ )N ,

h′ = (h1 , . . . , hN −1 ) ∈ Wq1 (RN+ )N −1 ,

h˜ ∈ Wq2 (Ω )},

Yq′ (RN+ ) ={F = (F2 , F3′ , F4′ , F5 , F6 , F7 ) | F2 , F6 ∈ Lq (RN+ )N , F3′ ∈ Lq (RN+ )N −1 , F4′ ∈ Lq (RN+ )N (N −1) , F5 ∈ Lq (RN+ ), F7 ∈ Lq (RN+ )N }. 2

Then, there exists an operator family T (λ) ∈ Hol(Γϵ,λ0 ,δ0 , L(Yq′ (RN+ ), Wq2 (RN+ )N )) such that for any (f, h′ , h˜ ) ∈ Yq′ (RN+ ) and

λ ∈ Λϵ,λ0 , v = T (λ)Fλ′ (f, h′ , h˜ ) is a unique solution to the problem (4.1), and A(λ) satisfies the following estimates:   ℓ d RL(Y (RN ),L (RN )N˜ ) τ (Gλ T (λ)) | λ ∈ Λϵ,λ0 ≤ κ0 (ℓ = 0, 1) q + q + dτ with some constant κ0 depending on ϵ , λ0 , δ0 , α , β , γ0 , γ3 , q and N. Here N˜ and Gλ are the same constant and operator in Theorem 2.2 and we have set Fλ′ (f, h′ , h˜ ) = (f, λ1/2 h′ , ∇ h′ , λh˜ , λ1/2 ∇ h˜ , ∇ 2 h˜ ). Remark 4.2. F3′ and F4′ are corresponding variables to λ1/2 h′ and ∇ h′ , respectively. To prove the R-boundedness of the sum of operators and the composition of operators, we use the following two lemmas whose proofs are found in [16,10]. Lemma 4.3. (1) Let X and Y be Banach spaces, and let T and S be R-bounded families in L(X , Y ). Then, T + S = {T + S | T ∈ T , S ∈ S } is also an R-bounded family in L(X , Y ) and

RL(X ,Y ) (T + S ) ≤ RL(X ,Y ) (T ) + RL(X ,Y ) (S ).

(2) Let X , Y and Z be Banach spaces, and let T and S be R-bounded families in L(X , Y ) and L(Y , Z ), respectively. Then, ST = {ST | T ∈ T , S ∈ S } is also an R-bounded family in L(X , Z ) and RL(X ,Z ) (ST ) ≤ RL(X ,Y ) (T )RL(Y ,Z ) (S ). Lemma 4.4. Let 1 < p, q < ∞ and let D be the domain in RN . (1) Let m(λ) be a bounded function defined on a subset Λ in a complex plane C and let Mm (λ) be a multiplication operator with m(λ) defined by Mm (λ)f = m(λ)f for any f ∈ Lq (D). Then,

RL(Lq (D)) ({Mm (λ) | λ ∈ Λ}) ≤ CN ,q,D ∥m∥L∞ (Σ ) .

(2) Let n(τ ) be a C 1 function defined on R \ {0} that satisfies the conditions: |n(τ )| ≤ γ and |τ ′ (τ )| ≤ γ with some constants γ > 0 for any τ ∈ R \ {0}. Let Tn be an operator valued Fourier multiplier defined by Tn f = F −1 [nF [f ]] for any f with F [φ] ∈ D (R, X ). Then, Tn is extended to the bounded linear operator from Lq (R, Lq (D)) into itself. Moreover, denoting this extension also by Tn , we have

∥Tn ∥L(Lp (R,Lq (D))) ≤ Cp,q,D,γ . To prove Theorem 4.1, first we reduce the problem (4.1) to the following problem:

 γ0 λu − α ∆u − (β + γ3 δ)∇ div u = 0 α(∂N uj + ∂j uN ) = −hj ,

uN = 0

in RN+ , on RN0 ,

(4.2)

where j = 1, . . . , N − 1. For this purpose, first setting vj = vj′ (j = 1, . . . , N − 1) and vN = −h˜ + vN′ in (4.1), we see that the equations (4.1) are transformed to the following equations:

 γ0 λv′ − α ∆v′ − (β + γ3 δ)∇ div v′ = f − f′ α(∂N vj′ + ∂j vN′ ) = −hj + α∂j h˜ , vN′ = 0

in RN+ , on RN0 ,

with v′ = (v1′ , . . . , vN′ ) and f′ = (β + γ3 δ)∇∂N h˜ + n0 (γ0 λh˜ − α ∆h˜ ), where j = 1, . . . , N − 1.

(4.3)

M. Murata / Nonlinear Analysis 106 (2014) 86–109

95

Second, given function f defined on RN+ , f e and f o denote its even extension and odd extension to RN , respectively, that is f e ( x) =



f (x) f (x′ , −xN )

xN > 0, xN < 0,

f o (x) =



f (x) −f (x′ , −xN )

xN > 0, xN < 0,

with x′ = (x1 , . . . , xN −1 ). Let f − f′ = (g1 , . . . , gN ) ∈ Lq (RN+ )N and set F = (g1e , . . . , gNe −1 , gNo ). Let S0 (λ) be the operator given in Theorem 3.2 and let S1 (λ) be an operator in Hol (Γϵ,λ0 ,δ0 , L(Yq′ (RN+ ), Wq2 (RN+ )N )) such that S1 (λ)Fλ′ (f, h′ , h˜ ) = S0 (λ)F.

Then, w = S1 (λ)Fλ′ (f, h′ , h˜ ) satisfies the equation:

 γ0 λw − α ∆w − (β + γ3 δ)∇ div w = f − f′ wN = 0 on RN0 ,

in RN+ ,

(4.4)

with w = (w1 , . . . , wN ), and using Lemmas 4.3 and 4.4, we see that

 RL(Y′ (RN ),L q

+

N N˜ q (R+ ) )

τ

d dτ

ℓ

 Gλ S1 (λ) | λ ∈ Γϵ,λ0 ,δ0

≤ C (ℓ = 0, 1)

(4.5)

with some constant C . Here and in the following, C denotes a generic constant depending at most on α , β , γ0 , γ3 , ϵ , λ0 , δ0 , q ˜ 0 + w + u in equation (4.1), and then u satisfies equation (4.2) replacing −hj by and N for the sake of simplicity. Set v = hn

−hj + α∂j h˜ − α∂N wj , because ∂j wN = 0 (j = 1, . . . , N − 1). Thus, in the following we consider the problem (4.2). First, we derive a solution formula of (4.2). For this purpose, applying the partial Fourier transform to (4.2) with respect to x′ = (x1 , . . . , xN −1 ), we have

 −1 ′ 2 2 ′ ′ (xN > 0),  α(α γ0 λ + |ξ | )ˆuj − α∂N uˆ j − (β + γ3 δ)iξj (iξ · uˆ + ∂N uˆ N ) = 0 −1 ′ 2 2 ′ ′ α(α γ0 λ + |ξ | )ˆuN − α∂N uˆ N − (β + γ3 δ)∂N (iξ · uˆ + ∂N uˆ N ) = 0 (xN > 0), (4.6)   ′ ˆ α∂N uˆ j + iαξj uˆ N |XN =0 = −hj (ξ , 0), uˆ N |xN =0 = 0.   N −1 ′ ′ ˆ k , ξ ′ = (ξ1 , . . . , ξN −1 ) and fˆ = fˆ (ξ ′ , xN ) = RN −1 e−ix ·ξ f (x′ , xN ) dx′ for f = uj and hj . Here and in with iξ ′ · uˆ ′ = k=1 iξk u the following, j runs from 1 through N − 1. Since (γ0 λ − α ∆)(γ0 λ − (α + β + γ3 δ)∆)v = 0 as was seen in (3.3), we have (∂N2 − A2 )(∂N2 − B2 )ˆv = 0 with   A = (α + β + γ3 δ)−1 γ0 λ + |ξ ′ |2 , B = α −1 γ0 λ + |ξ ′ |2 . ˆ = (ˆu1 , . . . , uˆ N ) of the form: We look for a solution u uˆ ℓ = Pℓ (e−BxN − e−AxN ) + Qℓ e−BxN for ℓ = 1, . . . , N. Substituting vˆ ℓ into (4.6) and equating the coefficients of e−AxN and e−BxN , we have

 2 α(B − A2 )Pj − (β + γ3 δ)iξj (iξ ′ · P ′ − APN ) = 0,    α(B2 − A2 )PN + (β + γ3 δ)A(iξ ′ · P ′ − APN ) = 0, iξ ′ · P ′ + iξ ′ · Q ′ − BPN − BQN = 0,  ′   α(−B(Pj + Qj ) + APj + iξj QN ) = −hˆ j (ξ , 0), QN = 0, N −1 with iξ ′ · R′ = i k=1 iξk Rk for R = P and Q . Solving the linear equations (4.7), we have  β + γ3 δ iξj  Pj = − iξ ′ · hˆ ′ (ξ ′ , 0),   2 − A2 )  α(α + β + γ δ) A ( B 3     β + γ3 δ 1 P = iξ ′ · hˆ ′ (ξ ′ , 0), N α(α + β + γ3 δ) B2 − A2   iξj 1 β + γ3 δ    Qj = hˆ j (ξ ′ , 0) + iξ ′ · hˆ ′ (ξ ′ , 0),   α B α(α + β + γ 3 δ) AB(A + B)   QN = 0. Setting M (xN ) =

e−BxN − e−AxN B−A

,

(4.7)

(4.8)

96

M. Murata / Nonlinear Analysis 106 (2014) 86–109

we have N −1  1 −BxN β + γ3 δ hˆ j (ξ ′ , 0) − e αB α(α + β + γ3 δ) k=1

uˆ j (ξ ′ , xN ) = uˆ N (ξ ′ , xN ) =



 (iξj )(iξk ) (iξj )(iξk ) −BxN ˆ ′ hk (ξ , 0) , (4.9) M (xN )hˆ k (ξ ′ , 0) − e A(A + B) AB(A + B)

N −1  β + γ3 δ iξk M (xN )hˆ k (ξ ′ , 0). α(α + β + γ3 δ) k=1 A + B

Using the Volevich trick: a(ξ ′ , xN )b(ξ ′ , 0) = −

∞

 0

∂ (a(ξ ′ , xN + yN )b(ξ ′ , yN )) dyN ∂ yN

∞



(∂N a)(ξ ′ , xN + yN )b(ξ ′ , yN ) dyN −

=− 1/2

and the identity: 1 = λα B2 λ1/2 −



1

α2

∞

0 N −1

Fξ−′ 1



a(ξ ′ , xN + yN )(∂N b)(ξ ′ , yN ) dyN , 0

0

uj (x) =

∞



iξk k=1 B2 i k ,

ξ we have

N −1

λ1/2 B3



Be−B(xN +yN ) Fx′ [λ1/2 hj ](ξ ′ , yN ) (x′ ) dyN

  1 −1 iξk ′ −B(xN +yN ) − Fξ ′ Be Fx′ [∂k hj ](ξ , yN ) (x′ ) dyN α 0 B3 k=1    ∞ 1 1 − Fξ−′ 1 2 Be−B(xN +yN ) Fx′ [∂N hj ](ξ ′ , yN ) (x′ ) dyN α 0 B   N −1  ∞  β (iξj )(iξk ) 2 ′ ′ ′ [∂N hk ](ξ , yN ) (x ) dyN + Fξ−′ 1 B M ( x + y ) F N N x α(α + β + δ) k=1 0 A(A + B)B2    ∞ (iξj )(iξk ) −1 −B(xN +yN ) ′ − Fξ ′ Be Fx′ [∂N hk ](ξ , yN ) (x′ ) dyN 2  A(A + B)B   0 ∞ iξk −1 ′ ′ 2 − Fξ ′ B M (xN + yN )Fx′ [∂j hk ](ξ , yN ) (x ) dyN , (A + B)B2 0   N −1  ∞  β +δ iξk ′ ′ 2 ′ [∂N hk ](ξ , yN ) (x ) dyN uN (x) = − B M ( x + y ) F Fξ−′ 1 N N x α(α + β + δ) k=1 0 (A + B)B2    ∞ 1 −1 ′ −B(xN +yN ) − Fξ ′ Be Fx′ [∂k hk ](ξ , yN ) (x′ ) dyN    (A + B)B 0 ∞ A −1 2 ′ ′ − Fξ ′ B M (xN + yN )Fx′ [∂k hk ](ξ , yN ) (x ) dyN . (A + B)B2 0 

∞

(4.10)

Here, we have denoted the partial Fourier transform and the partial inverse Fourier transform by Fx′ and Fξ−′ 1 , respectively. Let F3′ = (F31 , . . . , F3N −1 ) and F4′ = (F4kℓ | k = 1, . . . N , ℓ = 1, . . . , N − 1) be variables corresponding to λ1/2 h′ = (λ1/2 h1 , . . . , λ1/2 hN −1 ) and ∇ h′ = (∂ℓ hk | k = 1, . . . , N − 1, ℓ = 1, . . . , N ), respectively. In view of (4.10) we define operators T1j (λ) by

T1j (λ)(F3 , F4 ) = ′

′

1

α2

∞

 0

Fξ−′ 1



λ1/2 B3

Be

−B(xN +yN )



Fx′ [F3j ](ξ , yN ) (x′ ) dyN ′

   ∞ N −1  1 iξk −B(xN +yN ) ′ ′ Fξ−′ 1 Be F [ F ](ξ , y ) (x′ ) dyN 4kj N x 3 α B 0 k=1    ∞ 1 1 −B(xN +yN ) −1 ′ ′ − Fξ ′ Be Fx [F4Nj ](ξ , yN ) (x′ ) dyN α 0 B2   N −1  ∞  β (iξj )(iξk ) 2 ′ ′ ′ [F4Nk ](ξ , yN ) (x ) dyN + Fξ−′ 1 B M ( x + y ) F N N x α(α + β + δ) k=1 0 A(A + B)B2    ∞ (iξj )(iξk ) −B(xN +yN ) ′ ′ − Fξ−′ 1 Be F [ F ](ξ , y ) (x′ ) dyN 4Nk N x A(A + B)B2 0     ∞ iξk ′ ′ 2 ′ [F4jk ](ξ , yN ) (x ) dyN , B M ( x + y ) F − Fξ−′ 1 N N x (A + B)B2 0 −

M. Murata / Nonlinear Analysis 106 (2014) 86–109

T1N (λ)(F3′ , F4′ ) = −

β +δ α(α + β + δ)   ∞ Fξ−′ 1

− 0

∞



Fξ−′ 1

− 0

k=1

1

(A + B)B 

97

iξk B2 M (xN + yN )Fx′ [F4Nk ](ξ ′ , yN ) (x′ ) dyN ( A + B)B2 0  Be−B(xN +yN ) Fx′ [F4kk ](ξ ′ , yN ) (x′ ) dyN

N −1  

∞

Fξ−′ 1







A

B2 M (xN + yN )Fx′ [F4kk ](ξ ′ , yN ) (x′ ) dyN (A + B)B2



.

(4.11)

Combining (4.10) and (4.11), we have u(x) = T1 (λ)(λ1/2 h′ , ∇ h′ )

(4.12)

with u = (u1 , . . . , uN ) and T1 (λ)(F3′ , F4′ ) = (T11 (λ)(F3′ , F4′ ), . . . , T1N (λ)(F3′ , F4′ )). Set

Zq (RN+ ) = {(F3′ , F4′ ) | F3′ ∈ Lq (RN+ )N −1 , F4 ∈ Lq (RN+ )N (N −1) }. To obtain

 RL(Z

N N N˜ q (R+ ),Lq (R+ ) )

τ

ℓ

d dτ

 (Gλ T1 (λ)) | λ ∈ Γϵ,λ0 ,δ0

≤ C (ℓ = 0, 1),

(4.13)

we use the following lemma proved in Götz and Shibata [15]. Lemma 4.5. Let 1 < q < ∞, 0 < ϵ < π /2, λ0 > 0 and δ0 > 0. Let m(λ, ξ ′ ) be a function defined on Γϵ,λ0 ,δ0 × RN −1 \ {0} such that for any multi-index α ′ ∈ NN0 −1 there exists a constant Cα ′ such that

   ℓ   ′ d ′   α ′ τ m(λ, ξ )  ≤ Cα ′ (|λ|1/2 + |ξ ′ |)−2−|α | (ℓ = 0, 1) ∂ξ ′   dτ

(4.14)

for any (λ, ξ ′ ) ∈ Γϵ,λ0 ,δ0 × (RN −1 \ {0}). Let Φj (λ) (j = 1, 2) be operators defined by

Φ1 (λ)f =

∞

 0

Φ2 (λ)f =

Fξ−′ 1 [m(λ, ξ ′ )Be−B(xN +yN ) Fx′ [f ](ξ ′ , yN )](x′ ) dyN ,

∞

 0

Fξ−′ 1 [m(λ, ξ ′ )B2 M (xN + yN )Fx′ [f ](ξ ′ , yN )](x′ ) dyN .

Then, we have

 RL(L

N N N˜ q (R+ ),Lq (R+ ) )

τ

d

ℓ

dτ

 (Gλ Φi (λ)) | λ ∈ Γϵ,λ0 ,δ0

≤ C (ℓ = 0, 1, i = 1, 2)

with some constant C . Here and in the following, Cα ′ denotes a generic constant depending on α ′ , ϵ , λ0 , δ0 , α , β and γ . We check the multiplier condition (4.14) in the present case. As was seen in Enomoto and Shibata [11, Lemma 4.3], using Lemma 3.1 and the following Bell formula for the derivatives of the composite functions: ′

′ ∂ξα′ f (g (ξ ′ ))

=

|α | 

f (ℓ) (g (ξ ′ ))

ℓ=1

α1′ +···+αℓ′ |α ′ |≥1

α′

α′

Γαα′ ,...,α′ (∂ξ ′1 g (ξ ′ )) · · · (∂ξ ′ℓ g (ξ ′ )) ′



1

ℓ

(4.15)

i

with f

(ℓ)

(t ) = d f (t )/dt and suitable coefficients Γαα′ ,...,α′ , we see easily that ℓ

′

ℓ

1

′ |∂ξα′ As | ′ |∂ξα′ Bs |

1/2

≤ Cα′ (|λ|

1/2

≤ Cα′ (|λ|

′

s−|α ′ |

,

′

s−|α ′ |

,

+ |ξ |) + |ξ |)

ℓ

(4.16)

for any multi-index α ′ ∈ NN0 −1 and (λ, ξ ′ ) ∈ Γϵ,λ0 ,δ0 × (RN −1 \ {0}). Here, s is an arbitrary real number, f = f (t ) is a C ∞ function defined on the real line and g (ξ ′ ) is also a C ∞ function defined on RN −1 \ {0}. By Lemma 3.1 we see that | arg((α + β + γ3 δ)−1 γ0 λ + |ξ ′ |2 )| ≤ π − σ with σ ∈ (0, π /2), and we have

|A + B| ≥ Re A + Re B ≥ c3 (|λ|1/2 + |ξ ′ |)

(4.17)

with some positive constant c3 depending solely on ϵ , λ0 , δ0 , α , β , γ0 and γ3 . Thus, by (4.15) and (4.16)

|∂ξα′ (A + B)s | ≤ Cα′ (|λ|1/2 + |ξ ′ |)s−|α | ′

′

(4.18)

98

M. Murata / Nonlinear Analysis 106 (2014) 86–109

for any multi-index α ′ ∈ NN0 −1 and (λ, ξ ′ ) ∈ Γϵ,λ0 ,δ0 × RN −1 \ {0}. Since |∂ξα′ ξj | ≤ (|λ|1/2 + |ξ ′ |)1−|α | for any α ′ ∈ N0N −1 (j = 1, . . . , N − 1), by the Leibniz rule, (4.16) and (4.18), we see that all the multipliers: ′

λ1/2 B3

,

iξk B3

,

1

,

(iξj )(iξk )

B2 A(A + B)B2

′

iξk 1 A , , (A + B)B2 (A + B)B (A + B)B2

,

appearing in (4.11) satisfy the multiplier condition (4.14), so that (4.13) follows immediately from Lemma 4.5. 5. On the R-boundedness of solution operators in a bent half-space Ω+ Let Φ : RN → RN be a bijection of C 1 class and let Φ −1 be its inverse map of Φ . Writing ∇ Φ = R + R(x) and ∇ Φ −1 = R−1 + R−1 (x), we assume that R and R−1 are orthonormal matrices with constant coefficients and R(x) and R−1 (x) are matrices of functions in Wr2 (RN ) with N < r < ∞ such that

∥(R, R−1 )∥L∞ (RN ) ≤ M1 ,

∥∇(R, R−1 )∥Wr1 (RN ) ≤ M2 .

(5.1)

We will choose M1 small enough eventually, so that we may assume that 0 < M1 ≤ 1 ≤ M2 . Note that RR−1 = I. Let Ω+ be a bent half-space with boundary Γ+ defined by Ω+ = Φ (RN+ ) and Γ+ = Φ (RN0 ), and let n+ be the unit outer normal to Γ+ . We see that Γ+ is represented by Φ−1,N (y) = 0 with Φ −1 = (Φ−1,1 , . . . , Φ−1,N ), which furnishes that n+ = (RN1 + RN1 (x), . . . , RNN + RNN (x))R(x)−1 , where we have set R(x) = {

N 

(5.2)

(RNj + RNj (x))2 }1/2 with R−1 = (Rij ) and R−1 (x) = (Rij (x)). By (5.1) and the fact that

j =1

N

j =1

2 RNj = 1, we have

˜+ n+ = (RN1 , . . . , RNN ) + n

(5.3)

˜ + ∥L∞ (RN ) ≤ CM1 and ∥∇ n˜ + ∥Wr1 (RN ) ≤ CM2 provided that M1 is chosen small enough. Let γ0 (x) and γ3 (x) be with ∥n real-value functions defined on RN that satisfy the following conditions: ρ0 /2 ≤ γ0 (x) ≤ 2ρ0 ,

0 ≤ γ3 (x) ≤ ρ12

(x ∈ Ω+ ),

∥γℓ − γˆℓ ∥L∞ (Ω+ ) ≤ M1 ,

∥∇γℓ ∥Lr (Ω+ ) ≤ CM2

(5.4)

for ℓ = 0, 3, where γˆℓ (ℓ = 0, 3) are some constants with ρ0 /2 < γˆ0 < 2ρ0 and 0 ≤ γˆ3 < ρ In this section, we consider the following problem:

2 1.



γ0 λu − Div S(u) − δ∇(γ3 div u) = f α[D(u)n+ − ⟨D(u)n+ , n+ ⟩n+ ] = h − ⟨h, n+ ⟩n+ ,

u · n+ = h˜

in Ω+ , on Γ+ .

(5.5)

We choose λ0 large enough, so that we assume that λ0 ≥ 1. Thus, we may assume that given δ0 > 0|δ| ≤ δ0 in the (Case 1). We have the following theorem. Theorem 5.1. Let 1 < q < ∞, N < r < ∞, 0 < ϵ < π /2 and δ0 > 0. Let Γϵ,λ0 ,δ0 be the set defined in (2.3) and let Yq (Ω+ ) and Yq (Ω+ ) be the sets defined by replacing Ω by Ω+ in (2.4). Assume that γ0 and γ3 satisfy the condition in (5.4). Then, there exist constants M1 ∈ (0, 1] and λ0 ≥ 1, and an operator family U(λ) ∈ Hol(Γϵ,λ0 ,δ0 , L(Yq (Ω+ )N , Wq2 (Ω+ )N )) such that for any (f, h, h˜ ) ∈ Yq (Ω+ ) and λ ∈ Λϵ,λ0 , u = U(λ)Fλ (f, h, h˜ ) solves equation (5.5) uniquely, and U(λ) satisfies the following estimates:

 RL(Y

N˜ q (Ω ),Lq (Ω ) )

τ

d dτ

ℓ

 (Gλ U(λ)) | λ ∈ Λϵ,λ0

≤ C (ℓ = 0, 1)

with some constant C depending on ϵ , λ0 , ρ0 , ρ1 , α , β , M2 , q and N. Proof. In order to prove the theorem, we transform problem (5.5) to a problem in RN+ by the change of variable: x = Φ −1 (y) with y ∈ Ω+ and x ∈ RN+ and the change of unknown: w(x) = R−1 u(y). Since ∂/∂ yj = k=1 (Rkj + Rkj (x))∂/∂ xk , employing the same argument due to Enomoto, von Below and Shibata [17,11] and using (5.1)–(5.4), we have the following equations in RN+ :

N

 γˆ0 λw − α ∆w − (β + γˆ3 δ)∇ div w + F (w) = f+

in RN+ ,

(5.6)

α(∂N ws + ∂s wN ) + Fbs (w) = h+s , wN + FbN (w) = h+N on RN0 , N N −1 −1 ˜ where f+ = R−1 f ◦ Φ , h+s = j=1 Rsj (hj ◦ Φ − ℓ=1 ⟨hℓ ◦ Φ , (RN ℓ + RN ℓ )R ⟩(RNj + RNj )R ), h+N = h ◦ Φ and s = 1, . . . , N − 1. Moreover, F (w), Fbs (w) and FbN (w) have the following forms: F (w) = λM1 w + M2 ∇ 2 w + N1 ∇ w,

Fbs (w) = M3 ∇ w,

FbN = M4 w

M. Murata / Nonlinear Analysis 106 (2014) 86–109

99

satisfying

∥(M1 , M2 , M3 , M4 )∥L∞ (RN ) ≤ CN M1 , +

∥(N1 , ∇ M1 , ∇ M2 , ∇ M3 , ∇ M4 )∥Lr (RN ) ≤ CN M2 ,

(5.7)

∥∇ 2 M4 ∥Lr (RN ) ≤ CN M2 .

+

+

Here, we have used (5.1) and Sobolev’s embedding theorem. By Theorem 4.1, there exists an operator family T (λ) such that for any (f+ , h′+ , h+N ) ∈ Yq′ (RN+ ) with h′+ = (h+1 , . . . , h+N −1 ) and λ ∈ Λϵ,λ0 , w = T (λ)Fλ′ (f+ , h′+ , h+N ) satisfies the equations:

 ′ ′  γˆ0 λw − Div S (w) − δ γˆ3 ∇ div w + F (w) = f+ + R1 (λ)Fλ (f+ , h+ , h+N ) ′ ′ α(∂N ws + ∂s wN ) + Fbs (w) = h+s + R2s (λ)Fλ (f+ , h+ , h+N )   wN + FbN (w) = h+N + R2N (λ)Fλ′ (f+ , h′+ , h+N )

in RN+ , on RN0 , on

(5.8)

RN0

with R1 (λ)F ′ = F (T (λ)F ′ ) and R2j (λ)F ′ = Fbj (T (λ)F ′ ) for F ′ = (F2 , F3′ , F4′ , F5 , F6 , F7 ) ∈ Yq′ (RN+ ), where s = 1, . . . , N − 1

and j = 1, . . . , N. To estimate N1 ∇ w, (∇ M3 )∇ w, (∇ M4 )w and (∇ 2 M4 )w, we use the following lemma in Shibata [18, Lemma 2.4]. Lemma 5.2. Let 1 < q < and N < r < ∞. Then, there exists a constant CN ,r ,q such that for any σ > 0, a ∈ Lr (RN+ ) and b ∈ Wq1 (RN+ ) there holds the estimate: r

N

∥ab∥Lq (RN ) ≤ σ ∥∇ b∥Lq (RN ) + CN ,r ,q σ − r −N ∥a∥Lr −(NRN ) ∥b∥Lq (RN ) . +

+

r

+

+

Setting R(λ)F ′ = (R1 (λ)F ′ , R21 (λ)F ′ , . . . , R2N (λ)F ′ ) ∈ Yq′ (RN+ ) for F ′ = (F2 , F3′ , F4′ , F5 , F6 , F7 ) ∈ Yq′ (RN+ ), by Theorem 4.1, Lemmas 5.2, 4.3 and 4.4, we have −1/2

RL(Yq′ (RN )) ({(τ ∂τ )ℓ Fλ′ R(λ) | λ ∈ Γϵ,λ0 }) ≤ {CN (σ + M1 ) + Cσ λ0 +

}κ0 (ℓ = 0, 1)

for any σ > 0 with some constant CN depending solely on N and Cσ depending solely on M2 , ϵ , λ0 , α , β , ρ0 , ρ1 , r, q and N, where Fλ′ is the same operator as in Theorem 4.1, that is Fλ′ (f+ , h′+ , h+N ) = (f+ , λ1/2 h′+ , ∇ h′+ , λh+N , λ1/2 ∇ h+N , ∇ 2 h+N ) for −1/2

(f+ , h′+ , h+N ) ∈ Yq′ (RN+ ). We choose σ and M1 so small that CN (σ + M1 )κ0 ≤ 1/4 and λ0 ≥ 1 so large that Cσ λ0

κ0 ≤ 1/4.

Thus, we have

RL(Yq′ (RN )) ({(τ ∂τ )ℓ Fλ′ R(λ) | λ ∈ Γϵ,λ0 }) ≤ 1/2 (ℓ = 0, 1).

(5.9)

+

Since R-boundedness implies the uniform boundedness for the operator, by (5.9) we have

∥Fλ′ R(λ)Fλ′ (f+ , h′+ , h+N )∥Lq (RN ) ≤ (1/2)∥Fλ′ (f+ , h′+ , h+N )∥Lq (RN ) +

(5.10)

+

for λ ∈ Γϵ,λ0 . Since ∥Fλ′ (f+ , h′+ , h+N )∥Lq (RN ) = ∥(f+ , λ1/2 h′+ , ∇ h′+ , λh+N , λ1/2 ∇ h+N , ∇ 2 h+N )∥Lq (RN ) gives us equivalent +

+

norms of the space Yq′ (RN+ ) for λ ̸= 0, by (5.10), R(λ)Fλ′ is a contraction map from Yq′ (RN+ ) into itself, so that for each

λ ∈ Γϵ,λ0 ,δ0 , (I + R(λ)Fλ′ )−1 exists and ∥(I + R(λ)Fλ′ )−1 ∥L(Yq′ (RN )) ≤ 2. +

Here, I is the identity operator and ∥ · ∥L(X ) denotes the operator norm of L(X ). If we define w by w = T (λ)Fλ′ (I + R(λ)Fλ′ )−1 (f+ , h′+ , h+N ), then by (7.17) w is a solution to (5.6). In addition, by (5.9) and Lemma 4.3 we have

RL(Yq′ (RN )) ({(τ ∂τ )ℓ (I + Fλ′ R(λ))−1 | λ ∈ Γϵ,λ0 )} ≤ 4 (ℓ = 0, 1). +

(5.11)

Since Fλ′ (I + R(λ)Fλ′ )−1

  ∞ ∞   = Fλ′ (−1)j (R(λ)Fλ′ )j = (−1)j (Fλ′ R(λ))j Fλ′ = (I + Fλ′ R(λ))−1 Fλ′ , j =0

(5.12)

j =0

if we define an operator family B+ (λ) by B+ (λ) = T (λ)(I + Fλ′ R(λ))−1 , then by (5.12), Theorem 4.1 and Lemma 4.3, we have

RL(Y′ (RN ),L q

+

N N˜ q (R+ ) )

({(τ ∂τ )ℓ Gλ B+ (λ) | λ ∈ Γϵ,λ0 ,δ }) ≤ 4κ0 (ℓ = 0, 1).

100

M. Murata / Nonlinear Analysis 106 (2014) 86–109

Moreover, by (5.12) w = B+ (λ)Fλ′ (f+ , h′+ , h+N ) is a solution to (5.6). Recalling that the change of unknown functions: T −13 w = (R−1 v) ◦ Φ and the change variable : y = Φ (x) transfer (5.6) to (7.17), we see that v = (R− is a solution to 1 w) ◦ Φ (5.6), and also we can construct an R-bounded solution operator A+ (λ) for the problem (5.6) from B+ (λ). The uniqueness of solutions follows from the existence of solutions to the dual problem. This completes the proof of Theorem 4.1.  Finally, we consider the following generalized Lamé equations in RN :

γ0 λu − Div S(u) − δ∇(γ3 div u) = f in RN .

(5.13)

Employing a similar argumentation to that in the proof of Theorem 5.1 and using Theorem 3.2, we have the following theorem. Theorem 5.3. Let 1 < q < ∞, δ0 > 0 and 0 < ϵ < π /2. Let Γϵ,λ0 ,δ0 be the set defined in (2.3). Assume that γ0 and γ3 satisfy the condition (5.4) replacing Ω+ by RN . Then, there exist constants M1 ∈ (0, 1) and λ0 ≥ 1, and an operator family A0 (λ) with A0 (λ) ∈ Hol (Γϵ,λ0 , L(Lq (RN )N , Wq2 (RN )N )) such that for any g ∈ Lq (RN )N and λ ∈ Γϵ,λ0 , u = A0 (λ)f is a unique solution to problem (5.13). Moreover, A0 (λ) possesses the estimate:

RL(L

N N N˜ q (R ) ,Lq (Ω+ ) )

({(τ ∂τ )ℓ Gλ A0 (λ) | λ ∈ Γϵ,λ0 }) ≤ C (ℓ = 0, 1).

Here, M1 depends solely on ϵ , α , β , δ0 , M2 , q and N, and λ0 and C depend solely on ϵ , α , β , δ0 , ρ0 , ρ1 , M2 , q and N. 6. A proof of Theorem 2.2 6.1. Some preparation for the proof of Theorem 2.2 Employing the same argument Appendix in [11], we can obtain the following proposition concerning some important 3−1/r properties of the uniform Wr domain that will be used to construct a solution operator in Ω . 3−1/r

Proposition 6.1. Let N < r < ∞ and let Ω be a uniform Wr domain in RN . Let M1 be the number given in Section 5. Then, 0 1 there exist constants M2 > 0, 0 < d , d < 1, an open set U, at most countably many N-vector of functions Φj ∈ Wr3 (RN )N and points x0j ∈ Γ and x1j ∈ Ω such that the following assertions hold: (i) The maps: RN ∋ x → Φj (x) ∈ RN are bijective. ∞ ∞ (ii) Ω = { j=1 (Φj (RN+ ) ∩ B0j )} ∪ ( j=1 B1j ), Φj (RN+ ) ∩ B0j = Ω ∩ B0j and Φj (RN0 ) ∩ B0j = ∂ Ω ∩ B0j , where Bij = Bdi (xij ). (iii) There exist C ∞ functions ζji , ζ˜ji (i = 0, 1, j = 1, 2, . . .) such that 0 ≤ ζji , ζ˜ji ≤ 1,

supp ζji , supp ζ˜ji ⊂ Bij , 1  ∞ 

ζ˜ji = 1 on supp ζji ,

∥ζji ∥W∞ 3 (RN ) ,

ζji = 1 in Ω ,

i=0 j=0

∞ 

∥ζ˜ji ∥W∞ 3 (RN ) ≤ c0 (i = 0, 1)

ζj0 = 1 on Γ .

j=1

Here, c0 is a constant which depends on M2 , N, q, and r, but independent of j = 1, 2, . . . . (iv) ∇ Φj = Rj + Rj , ∇(Φj )−1 = R−j + R−j , where Rj , R−j are N × N constant orthonormal matrices, and Rj , R−j are N × N matrices of Wr2 (RN ) functions defined on RN which satisfy the conditions: ∥Rj ∥L∞ (RN ) , ∥R−j ∥L∞ (RN ) ≤ M1 and ∥∇ Rj ∥L∞ (RN ) , ∥∇ R−j ∥L∞ (RN ) ≤ M2 for any j = 1, 2, . . . . (v) There exists a natural number L ≥ 2 such that any L + 1 distinct sets of {Bij }∞ j=1 have an empty intersection. After choosing M2 and di according to M1 in Proposition 6.1, we choose M2 again so large that ∥∇γℓ ∥Lr (Bi ) ≤ M2 (ℓ = 0,

1, 2). Moreover, in view of the assumption (1.4) we assume that

ρ∗ /2 ≤ γ0 (x) ≤ 2ρ∗ ,

0 ≤ γ3 (x) ≤ ρ12 (x ∈ Ω ),

∥γℓ − γℓ (xij )∥L∞ (Ω ∩Bi ) ≤ M1 , j

∥∇γℓ ∥Lr (Ω ∩Bi ) ≤ M2 (ℓ = 0, 3)

j

(6.1)

j

Let n0j be the unit outer normal to Γj0 = Φj (RN0 ). In view of (5.1) and (5.2), we can extend the unit outward normal n to Γ to the whole space RN such that n = n0j

on B0j ,

2 ∥ n ∥ W∞ 1 (Bi ) + ∥∇ n∥L (Bi ) ≤ CM2 r j

j

for any i = 0, 1 and j ∈ N with some positive constant C independent of j ∈ N. 3 M T denotes the transposed M.

(6.2)

M. Murata / Nonlinear Analysis 106 (2014) 86–109

101

Next, we prepare some proposition used to construct a parametrix. By Proposition 6.1 (v), for any r ∈ (1, ∞) there exists a constant Cr ,L such that

 ∞ 

1/r ∥f ∥Lr (Ω ∩Bi )

for any f ∈ Lr (Ω ).

≤ Cr ,L ∥f ∥Lr (Ω )

j

j =1

(6.3)

Using (6.3), we can show the following proposition in a similar manner to the proof of Lemma 4.3 in Shibata [18]. Proposition 6.2. Let 1 < q < ∞, q′ = q/(q − 1) and i = 0, 1. Then, the following assertions hold. (ℓ) ∞ m (i) Let m be a non-negative integer. Let {fj }∞ j=1 be a sequence in Wq (Ω ) and let {gj }j=1 (ℓ = 0, 1, . . . , m) be sequences of positive real numbers. Assume that ∞  (gj(ℓ) )q < ∞,

|(∇ ℓ fj , φ)Ω | ≤ M3 gj(ℓ) ∥ϕ∥Lq′ (Ω ∩Bi ) for any ϕ ∈ Lq (Ω ) and ℓ = 0, 1, . . . , m j

j=1

with some constant M3 independent of j = 1, 2, . . . . Here and in the following, for any domain D we set (u, v)D = ∞ m Then, f = j=1 fj exists in the strong topology of Wq (Ω ) and

 ℓ

∥∇ f ∥Lq (Ω ) ≤ Cq′ ,L M3

∞  (gj(ℓ) )q

 D

u(x)v(x)dx.

1/q .

j =1

(i)

(i)

∞ (ii) Let n be a natural number. Let {fj }∞ j=1 (i = 1, . . . , n) be sequences in Lq (Ω ) and {gj }j=1 (i = 1, . . . , n) be sequences of positive numbers. Let ai (i = 1, . . . , n) be any complex numbers. Assume that ∞  (gj(i) )q < ∞,

|(fj(i) , ϕ)Ω | ≤ M3 gj(i) ∥ϕ∥L ′ (Ω ∩Bi ) q

j=1

j

for any ϕ ∈ L1 (Ω ) and i = 1, . . . , n

with some constant M3 independent of j = 1, 2, . . . . In addition, we assume that there exists a sequence of positive numbers { hj } ∞ j=1 such that

  n    (i) a f ,φ   i =1 i j

∞  (hj )q < ∞, j =1

Ω

    ≤ M3 hj ∥ϕ∥Lq′ (Ω ∩Bi ) . j 

Then,

  n    (i)  ai f    i=1 

 ≤ Cq′ ,L M3

Lq (Ω )

where f (i) =

∞

j=1 fj

(i)

∞  (hj )q

1/q ,

j =1

.

6.2. Local solution In the following, we write Hj0 = Φj (RN+ ), Γj0 = Φj (RN0 ) and Hj1 = RN for the sake of simplicity. In view of (6.1), we

define the functions γjiℓ by

γjiℓ (x) = (γℓ (x) − γℓ (xij ))ζ˜ji (x) + γℓ (xij ) (i = 0, 1, ℓ = 0, 3). Since 0 ≤ ζ˜ji ≤ 1 and ∥∇ ζ˜ji ∥L∞ (RN ) ≤ c0 , by (6.1) we have

ρ0 /2 ≤ γj0i (x) ≤ 2ρ0 ,

0 ≤ γj3i (x) ≤ ρ12

∥γjiℓ − γjiℓ(xi ) ∥L∞ (H i ) ≤ M1 , j

j

∥∇γjiℓ ∥Lr (H i ) ≤ CM2 (ℓ = 0, 3). j

(6.4)

Moreover, we have

γjiℓ (x) = γℓ (x) (x ∈ suppζji , ℓ = 0, 3), because ζ˜ = 1 on supp ζ For f ∈ Lq (Ω ) and h ∈

(6.5)

(Ω ) , we consider the equations:  0 0 γj0 λvj − Div S(v0j ) − δ∇(γj30 div v0j ) = ζ˜j0 f in Hj0 , α[D(v0j )n0j − ⟨D(v0j )n0j , n0j ⟩n0j ] = ζ˜j0 h − ⟨ζ˜j0 h, n0j ⟩n0j , v0j · n0j = ζ˜j0 h˜ on Γj0 , i j

i j.

N

Wq1

γj01 λv1j − Div S(v1j ) − δ∇(γj31 div v1j ) = ζ˜j1 f in Hj1 .

N

(6.6) (6.7)

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M. Murata / Nonlinear Analysis 106 (2014) 86–109

Note that n0j is defined on RN and ∥∇ n0j ∥W 1 (RN ) ≤ CN M2 . Since γjiℓ (ℓ = 0, 3) satisfy the condition (5.4), by Theorems 5.1 r

and 5.3 there exist operator families Tj i (λ) with

Tj 0 (λ) ∈ Hol(Γϵ,λ0 , L(Yq (Hj0 ), Wq2 (Hj0 )N )), Tj 1 (λ) ∈ Hol(Γϵ,λ0 , L(Lq (Hj1 )N , Wq2 (Hj1 )N ))

(6.8)

such that v0j = Tj 0 (λ)Fλ (ζ˜j0 f, ζ˜j0 h, ζ˜j0 h˜ ),

v1j = Tj 1 (λ)ζ˜j1 f

(6.9)

uniquely solve the problems (6.6) and (6.7), respectively. Moreover, we have

RL(Y

0 0 N˜ q (Hj ),Lq (Hj ) )

RL(L

1 N 1 N˜ q (Hj ) ,Lq (Hj ) )

({(τ ∂τ )ℓ Gλ Tj 0 (λ) | λ ∈ Γϵ,λ0 }) ≤ κ2 (ℓ = 0, 1), (6.10)

({(τ ∂τ )ℓ Gλ Tj 1 (λ) | λ ∈ Γϵ,λ0 }) ≤ κ2 (ℓ = 0, 1)

with some constant κ2 independent of j ∈ N. By (6.10), we have

∥(λv0j , λ1/2 ∇ v0j , ∇ 2 v0j )∥Lq (H 0 ) ≤ κ2 ∥(ζ˜j0 f, λ1/2 ζ˜j0 h, ∇(ζ˜j0 h))∥Lq (H 0 ) , j

(6.11)

j

∥(λv1j , λ1/2 ∇ v1j , ∇ 2 v1j )∥Lq (H 1 ) ≤ κ2 ∥ζ˜j1 f∥Lq (H 1 ) j

j

for any j ∈ N. 6.3. Construction of the parametrix For (f, h, h˜ ) ∈ Yq (Ω ), we consider Eq. (2.2). Let us define the parametrix W (λ)(f, h, h˜ ) by W (λ)(f, h, h˜ ) =

1  ∞ 

ζji vij =

∞ 

ζj0 Tj 0 (λ)Fλ (ζ˜j0 f, ζ˜j0 h, ζ˜j0 h˜ ) +

j =1

i =0 j =1

∞ 

ζj1 Tj 1 (λ)ζ˜j1 f.

(6.12)

j=1

By Proposition 6.1 and (6.11), we have W (λ)(f, h, h˜ ) ∈ Wq2 (Ω )N . Inserting W (λ)(f, h, h˜ ) into (2.2) and noting the facts that n = n0j on supp ζj0 ∩ Γ and (6.5), we have



γ0 λv − Div S(v) − δ∇(γ3 div v) = f − V1 (λ)(f, h, h˜ ) α[D(v)n − ⟨D(v)n, n⟩n] = h − ⟨h, n⟩n − V2′ (λ)(f, h, h˜ ),

v = h˜

in Ω , on Γ

with v = W (λ)(f, h, h˜ ), V1 (λ)(f, h, h˜ ) =

1  ∞  [{Div S(ζji vij ) − ζji Div S(vij )} − δ{∇(γj3i div (ζji vij )) − ζji ∇(γj3i div vij )}]

(6.13)

i=0 j=1

V2′ (λ)(f, h, h˜ ) =

∞ 

α{D(ζj0 v0j ) − ζj0 D(v0j ) − ⟨D(ζj0 v0j ) − ζj0 D(v0j ), n0j ⟩n0j }

j =1

Set V (λ)(f, h, h˜ ) = (V1 (λ)(f, h, h˜ ), V2′ (λ)(f, h, h˜ ), 0). By Proposition 6.2, (6.2) and (6.11) we have V (λ)(f, h, h˜ ) ∈ Yq (Ω ) and −1/2

∥Fλ V (λ)(f, h, h˜ )∥Lq (Ω ) ≤ C λ0

∥Fλ (f, h)∥Lq (Ω )

(6.14)

for any λ ∈ Γϵ,λ0 , where Fλ is the same operator as in Theorem 2.2. Here and in the following, C denotes a generic constant

depending solely on ϵ , q, r, α , β , ρ0 , ρ2 and N, but independent of j ∈ N. Since ∥Fλ (f, h, h˜ )∥Lq (Ω ) is an equivalent norm of −1/2

Yq (Ω ) for λ ̸= 0, choosing λ0 ≥ 1 so large that C λ0

≤ 1/2 in (6.14), we see that (I − V (λ))−1 ∈ L(Yq (Ω )) exists ˜ and u = W (λ)(I − V (λ))(f, h, h) is a solution to problem (2.2). The uniqueness of solutions follows from the existence of solutions to the dual problem. 6.4. Construction of R-bounded solution operators Observing that Fλ (ζ˜j0 f, ζ˜j0 h, ζ˜j0 h˜ ) = ζ˜j0 Fλ (f, h, h˜ ) + (∇ ζ˜j0 )(0, 0, h, 0, λ1/2 h˜ , 2∇ h˜ ) + (0, 0, 0, 0, 0, (∇ 2 ζ˜j0 )h˜ ), in view of (6.12) we define operators Sji (λ)F by Sj1 (λ)F = Tj 1 (λ)ζ˜j1 F2 and Sj0 (λ)F = Tj 0 (λ)ζ˜j0 F + λ−1/2 Tj 0 (λ)(∇ ζ˜j0 )(0, 0, F3 , 0, F5 , 2F6 ) + λ−1 Tj 0 (λ)(0, 0, 0, 0, 0, (∇ 2 ζ˜j0 )F5 ),

(6.15)

M. Murata / Nonlinear Analysis 106 (2014) 86–109

103

and then Sj0 (λ)Fλ (f, h, h˜ ) = Tj 0 (λ)Fλ (ζ˜j0 f, ζ˜j0 h, ζ˜j0 h˜ ) = v0j and Sj1 (λ)Fλ (f, h, h˜ ) = Tj 1 (λ)ζ˜j1 f = v1j . In view of (6.12) and (6.13), we set 1  ∞ 

W (λ)F =

ζji Sji (λ)F ,

i=0 j=1 1  ∞  V1 (λ)F = [{Div S(ζji Sji (λ)F ) − ζji Div S(Sji (λ)F )} − δ{∇(γj3i div (ζji Sji (λ)F )) − ζji ∇(γj3i div Sji (λ)F )}] i=0 j=1

∞ 

V2′ (λ)F =

α{D(ζj0 Sj0 (λ)F ) − ζj0 D(Sj0 (λ)F ) − ⟨D(ζj0 Sj0 (λ)F ) − ζj0 D(Sj0 (λ)F ), n0j ⟩n0j }.

j =1

Set V (λ)F = (V1 (λ)F , V2′ (λ)F , 0). By (6.2), (6.8), (6.10), (6.12), (6.13) and Proposition 6.2, we see that

W (λ) ∈ Hol Γϵ,λ0 ,δ0 , L(Lq (Yq (Ω ), Wq2 (Ω )N )), W (λ)Fλ (f, h, h˜ ) = W (λ)(f, h, h˜ ),

V (λ) ∈ HolΓϵ,λ0 ,δ0 , LLq (Yq (Ω )),

V (λ)Fλ (f, h, h˜ ) = V (λ)(f, h, h˜ ).

(6.16)

Moreover, as was seen in Enomoto, Below and Shibata [17], using Lemmas 4.3 and 4.4, (6.2), (6.10) and Proposition 6.2, we have

RL(Y

N˜ q (Ω ),Lq (Ω ) )

({(τ ∂τ )ℓ Gλ W (λ) | λ ∈ Γϵ,λ0 ,δ0 }) ≤ C1 κ2 ,

(6.17)

RL(Yq (Ω )) ({(τ ∂τ )ℓ Fλ V (λ) | λ ∈ Γϵ,λ0 ,δ0 }) ≤ C2 κ2 |λ0 |−1/2 −1/2

for ℓ = 0, 1 with some constants C1 and C2 independent of j ∈ N. Thus, choosing λ0 ≥ 1 so large that C2 κ2 λ0 Lemma 4.3 we have

RL(Y

2N +N 2 ) q (Ω ),Lq (Ω )

≤ 1/2, by

({(τ ∂τ )ℓ (I − Fλ V (λ))−1 | λ ∈ Γϵ,λ0 }) ≤ 4 (ℓ = 0, 1).

By (6.16), W (λ)(I − V (λ))−1 (f, h, h˜ ) = W (λ)Fλ (I − V (λ))−1 (f, h, h˜ ) = W (λ)(I − Fλ V (λ))−1 Fλ (f, h, h˜ ).

(6.18)

Thus, setting A(λ) = W (λ)(I − Fλ V (λ)) , by (6.17), (6.16), (6.18) and Lemma 4.3, we see that A(λ) is the required R-bounded solution operator to problem (2.2). The uniqueness follows from the existence of solutions to the dual problem, which completes the proof of Theorem 2.2. −1

7. Local in time unique existence theorem for nonlinear problem (1.1) As was done in Burnat and Zajaczkowski [3] and Kobayashi and Zajaczkowski [4], in order to prove Theorem 2.7, we formulate (1.1) in Lagrangian coordinates. 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: x=ξ+

t



v(ξ , s)ds ≡ Xv (ξ , t ),

(7.1)

0

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

 

max  

i,j=1...,N

0

t

  (∂vi /∂ξj )(·, s) ds 

L∞ (Ω )

< σ (0 < t < T ).

1 In this case, we have ∇x = A− v ∇ξ = (I + V0 (

t 0

(∂vi /∂ξj )

(7.2)

t

∇ v(ξ , s) ds))∇ξ , where V0 (K) is an N × N matrix of C ∞ functions with t respect to K = (kij ) defined on |K| < 2σ and V0 (0) = 0, where kij are corresponding variables to 0 (∂vi /∂ξj )(·, s) ds. Assume that ρ(x, t ) and u(x, t ) are solutions of (1.1) in the Euler coordinate. Setting ρ(Xv (ξ , t ), t ) = ρ∗ + θ0 (ξ ) + θ (ξ , t ), 0

we write (1.1) in the Lagrangian coordinate introduced by (7.1) as follows:

 ∂t θ + (ρ∗ + θ0 )div v = F (θ , v)    (ρ∗ + θ0 )∂t v − Div S(v) + ∇(P ′ (ρ∗ + θ0 )θ ) = g + G(θ , v) α[D(v)n − ⟨D(v)n, n⟩n] = H (v)    v · n = −v · (nˆ v − n) (θ , v)|t =0 = (0, u0 )

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

(7.3)

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M. Murata / Nonlinear Analysis 106 (2014) 86–109

ˆ v = n(Xv (ξ , t )), g = −P ′ (ρ∗ + θ0 )∇θ0 and F (θ , v), G(θ , v) and H (v) are nonlinear functions of the following forms: where n F (θ , v) = −θ div v − (ρ∗ + θ0 + θ )V1

t



 ∇ vds ∇ v,

0

    t   t ∇ vds ∇ vI ∇ vds ∇ v + (β − α)V1 G(θ , v) = −θ ∂t v + Div α V2 0    t  t   0  t ∇ vds ∇ v ∇ vds ∇ v + (β − α)div vI + (β − α)V1 ∇ vds ∇ α D(v) + α V2 + V3 0 0   t   01 2 ′ ′′ ∇ vds ∇(θ0 + θ ), P (ρ∗ + θ0 + τ θ )(1 − τ ) dτ θ − P (ρ∗ + θ0 + θ )V0 −∇ 0

0

ˆ v − n) − α{⟨D(v)nˆ v , nˆ v ⟩nˆ v − ⟨D(v)n, n⟩n} H (v) = −α D(v)(n −α



t







∇ v ds ∇ v nˆ v −

V2

t













∇ v ds ∇ v nˆ v , nˆ v nˆ v .

V2

(7.4)

0

0

Here, V0 (K), V1 (K), V2 (K) and V3 (K) are some matrices of C ∞ functions with respect to K defined on |K| ≤ σ , which satisfy conditions: V0 (0) = 0,

V1 (0) = 0,

V2 (0) = 0, V3 (0) = 0 (7.5)    t t t 1 ˆ ds)∇ w ˆ and ˆ )∇ w, Dx (w) = Dξ (w ˆ ) + V2 ( 0 ∇ w ˆ + V1 ( 0 ∇ wds and relations: A− = I + V0 ( 0 ∇ v ds), div x w = div ξ w v  ˆ + V3 ( t ∇ wds ˆ with w ˆ = K(Xv (ξ , t ), t ). ˆ ˆ Div x K(w) = Div ξ K )∇ K = w ( X (ξ , t ), t ) and K v 0 Since we can show eventually that the correspondence x = Xv (ξ , t ) is invertible by using the argument due to Ströhmer [19], our main task is to prove the following theorem. Theorem 7.1. Let N < q < ∞, 2 < p < ∞ and R > 0. If the initial data (θ0 , u0 ) for (1.1) satisfy the condition (2.9) and ∥θ0 ∥W 1 (Ω ) + ∥u0 ∥B2(1−1/p) (Ω ) ≤ R, then there exists a time T > 0 depending on R such that the problem (7.3) admits a unique q

solution (θ , v) with

q,p

θ ∈ Wp1 ((0, T ), Wq1 (Ω )),

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

To prove Theorem 7.1, first we show the maximal Lp –Lq regularity for the following time local linear problem:

 ∂t ρ + γ2 div u = f , γ0 ∂t u − Div S(u) + ∇(γ1 ρ) = g α[D(u)n − ⟨D(u)n, n⟩n] = h − ⟨h, n⟩n, u · n = h˜ (ρ, u)| = (ρ , u ) t =0 0 0

in Ω × (0, T ), on Γ × (0, T ), in Ω .

(7.6)

1/2

For this purpose, we have to replace the nonlocal operator Λγ with value in Lq (Ω ) by the local operator ∂t with value in Wq−1 (Ω ). For this purpose, according to Shibata [20], we introduce the extension map ι : L1,loc (Ω ) → L1,loc (RN ) having the following properties: (e-1) For any 1 < q < ∞ and f ∈ Wq1 (Ω ), ιf ∈ Wq1 (RN ), ιf = f in Ω and ∥ιf ∥W ℓ (RN ) ≤ Cq ∥f ∥W ℓ (Ω ) for ℓ = 0, 1 with some q

constant Cq depending on q, r and Ω .

q

(e-2) For any 1 < q < ∞ and f ∈ Wq1 (Ω ), ∥(1 − ∆)−1/2 ι(∇ f )∥Lq (RN ) ≤ Cq ∥f ∥Lq (Ω ) with some constant Cq depending on q, r and Ω . Here, (1 − ∆)−1/2 is the operator defined by (1 − ∆)−1/2 f = Fξ−1 [(1 + |ξ |2 )1/4 F [f ](ξ )]. In the following, such extension map ι is fixed. We define Wq−1 (Ω ) by

Wq−1 (Ω ) = {f ∈ L1,loc (Ω ) | (1 − ∆)−1/2 ιf ∈ Lq (Ω )}. Employing a similar argumentation to the proof of Proposition 2.8 in Shibata and Shimizu [8] (cf. also the appendix in Shibata [20]), we have 1/2

Wp1,γ0 ,0 (R, Wq−1 (Ω )) ∩ Lp,γ0 ,0 (R, Wq1 (Ω )) ⊂ Hp,γ0 ,0 (R, Lq (Ω )),

(7.7)

∥e−γ t Λ1γ/2 f ∥Lp (R,Lq (Ω )) ≤ C {∥e−γ t ∂t [(1 − ∆)−1/2 (ιf )]∥Lp (R,Lq (Ω )) + ∥e−γ t f ∥Lp (R,Wq1 (Ω )) }.

(7.8)

Combining Theorems 2.4 and 2.5 and (7.7), we have the following theorem.

M. Murata / Nonlinear Analysis 106 (2014) 86–109

105

Theorem 7.2. Let 1 < p, q < ∞, N < r < ∞ and max(q, q′ ) ≤ r (q′ = q/(q − 1)). Let T and R be any positive numbers and 2(1−1/p) Bq,p (Ω ) be the same set as in (2.7). Assume that θ0 ∈ Wq1 (Ω ) satisfies the range condition (2.9) and ∥∇θ0 ∥Lq (Ω ) ≤ R. Then, 2(1−1/p)

there exists a positive number γ0 = γ0 (R) depending on R and ρ∗ such that for any initial data (ρ0 , u0 ) ∈ Wq1 (Ω )× Bq,p

(Ω )

and right members f , g, h and h˜ with f ∈ Wp1 ((0, T ), Wq1 (Ω )N ),

g ∈ Lp ((0, T ), Lq (Ω )),

h ∈ Lp ((0, T ), Wq1 (Ω )) ∩ Wp1 ((0, T ), Wq−1 (Ω )),

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

(7.9)

satisfying the conditions: h|t =0 = 0 and h˜ |t =0 = 0, problem (7.6) admits unique solutions ρ and u with

ρ ∈ Wp1 ((0, T ), Wq1 (Ω )),

u ∈ Lp ((0, T ), Wq2 (Ω )) ∩ Wp1 ((0, T ), Lq (Ω ))

possessing the estimate:

∥ρ∥Wp1 ((0,t ),Wq1 (Ω )) + ∥u∥Lp ((0,t ),Wq2 (Ω )) + ∥u∥Wp1 ((0,t ),Lq (Ω )) ≤ C (R)eγ t {∥(ρ0 , u0 )∥Eq,p (Ω ) + ∥(f , g)∥Lp ((0,t ),W 1,0 (Ω )) + ∥h∥Lp ((0,t ),Wq1 (Ω )) q

+ ∥∂s [(1 − ∆)−1/2 ιh]∥Lp ((0,t ),Lq (Ω )) + ∥h˜ ∥Lp ((0,t ),Wq2 (Ω )) + ∥h˜ ∥Wp1 ((0,t ),Lq (Ω )) } for any t ∈ (0, T ] and γ ≥ γ0 with some constant C (R) depending on R but independent of γ ≥ γ0 and t ∈ (0, T ]. Proof. Let t be any number with 0 < t ≤ T . Given f (·, s) defined for s ≥ 0, f0 (·, s) denotes the zero extension of f to s < 0, that is f0 (·, s) = f (·, s) for s ≥ 0 and f0 (·, s) = 0 for s < 0. Let Et f be the extension of f defined by

 Et f =

f0 (·, s) f0 (·, 2t − s)

for s ≤ t , for s ≥ t .

(7.10)

Note that Et f vanishes for s ̸∈ [0, 2t ]. Moreover, if f |s=0 , then

 ∂s f (·, s) ∂s Et f = −(∂s f )(·, 2t − s) 0

for s ≤ t , for s ≥ t , for s ̸∈ [0, 2t ].

(7.11)

Let ρ t = θ (·, s) and ut = v(·, s) be solutions to the equations:

 ∂t θ + γ2 div v = Et f , γ0 ∂t v − Div S(v) + ∇(γ1 θ ) = Et g α[D(v)u − ⟨D(v)n, n⟩n] = Et h − ⟨Et h, n⟩n, v · n = Et h˜ (ρ, u)| = (ρ , u ) t =0 0 0

in Ω × (0, ∞), on Γ × (0, ∞), in Ω .

(7.12)

Since Et1 f = Et2 f for 0 < t1 < t2 ≤ T , by the uniqueness of solutions yields that ρ t1 (·, s) = ρ t2 (·, s) and ut1 (·, s) = ut2 (·, s) for s ∈ [0, t1 ] with 0 < t1 < t2 ≤ T . By Theorems 2.4 and 2.5, we see that there exist positive constants γ0 (R) and C (R) depending on R such that

∥e−γ s (θ , v)∥W 1 (R,W 1,0 (Ω )) + ∥e−γ s v∥Lp (R,Wq2 (Ω )) p

q

≤ C (R){∥(ρ0 , u0 )∥Eq,p (Ω ) + ∥e−γ s (Et f, Et g)∥Lp (R,W 1,0 (Ω )) q

+ ∥e

−γ s

1/2

(∇ Et h, Λγ Et h)∥Lp (R,Lq (Ω )) + ∥e

−γ s

∂s Et h˜ ∥Lp (R,Lq (Ω )) + ∥e−γ s Et h˜ ∥Lp (R,Wq2 (Ω )) }

(7.13)

for any γ ≥ γ0 (R). By (7.8), (7.10) and (7.11), we have

∥e−γ s (Et f , Et g)∥Lp (R,W 1,0 (Ω )) ≤ 2∥e−γ s (f , g)∥Lp ((0,t ),W 1,0 (Ω )) , q

q

∥e−γ s (∇ Et h, Λ1γ/2 Et h)∥Lp (R,Lq (Ω )) ≤ C {∥e−γ s ∂s [(1 − ∆)−1/2 ιh]∥Lp ((0,t ),Lq (Ω )) + ∥e−γ s h∥Lp ((0,t ),Wq1 (Ω )) }, −γ s

∥e

∂s Et h˜ ∥Lp (R,Lq (Ω )) + ∥e

−γ s

−γ s ˜

Et h˜ ∥Lp (R,W 2 (Ω )) ≤ 2{∥e q

(7.14) −γ s

h∥Lp ((0,t ),W 2 (Ω )) + ∥e q

∂s h˜ ∥Lp ((0,t ),Lq (Ω )) }.

Setting u = uT and ρ = ρ T , noting that u(·, s) = ut (·, s) for 0 < s < t and combining (7.13) and (7.14), we complete the proof of Theorem 7.2. 

106

M. Murata / Nonlinear Analysis 106 (2014) 86–109 3−1/q

Proof of Theorem 7.1. In the following, we assume that 2 < p < ∞ and N < q < ∞, and that Ω is a uniform Wq domain in RN (N ≥ 2). By Sobolev’s embedding theorem we have

  m      fj   j =1 

Wq1 (Ω ) ⊂ L∞ (Ω ),

≤C

m 

∥fj ∥Wq1 (Ω ) .

(7.15)

j =1

Wq1 (Ω )

Let T and L be any positive numbers and we define a space IL,T by

IL,T = {(θ , v) | θ ∈ Wp1 ((0, T ), Wq1 (Ω )), v ∈ Wp1 ((0, T ), Lq (Ω )) ∩ Lp ((0, T ), Wq2 (Ω ))

(θ , v)|t =0 = (0, u0 ) in Ω , ∥θ ∥Wp1 ((0,T ),Wq1 (Ω )) + Iv (0, T ) ≤ L},

(7.16)

where we have set Iv (0, T ) = ∥v∥Lp ((0,T ),W 2 (Ω )) + ∥∂t v∥Lp ((0,T ),Lq (Ω )) . Since we choose T > 0 small enough eventually, we q

may assume that 0 < T ≤ 1 in the following. Given (κ, w) ∈ IL,T , let θ and v be solutions to problem:

 ∂t θ + (ρ∗ + θ0 )div v = F (κ, w)    (ρ∗ + θ0 )∂t v − Div S(v) + ∇(P ′ (ρ∗ + θ0 )θ ) = g + G(κ, w) α[D(v)n − ⟨D(v)n, n⟩n] = H (w)    v · n = −w · (nˆ w − n) (θ , v)|t =0 = (0, u0 )

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

(7.17)

First, we estimate the right-hand sides of (7.17). By (7.15), Hölder’s inequality and the identity: κ(·, t ) = have

 t     sup  ∇ w(·, s) ds  t ∈(0,T )

≤ M1 T

1/p′

L∞ (Ω )

0

 t     L, sup  ∇ w(·, s) ds  t ∈(0,T )

sup ∥κ(·, t )∥L∞ (Ω ) ≤ M1 T

t ∈(0,T )

′

(7.18)

q

t ∈(0,T )

∂s κ(·, s) ds, we

≤ CT 1/p L,

sup ∥κ(·, t )∥W 1 (Ω ) ≤ CT 1/p L,

L,

0

′

Wq1 (Ω )

0

1/p′

t

with p′ = p/(p − 1). Here and in the following, C denotes a generic constant independent of T and R and we use the letter M1 to denote a special constant independent of T and L. The value of C may change from line to line. To treat polynomials t ′ of functions with respect to κ and 0 ∇ w(·, s) ds, in view of (2.9) and (7.2) we choose T so small that M1 T 1/p L ≤ ρ∗ /4 and M1 T 1/p L ≤ σ , so that ′

 t     sup  ∇ w(·, s) ds  t ∈(0,T )

ρ∗ /4 ≤ ρ∗ + θ0 + τ κ ≤ 4ρ∗ (τ ∈ [0, 1]),

L∞ (Ω )

0

< σ.

(7.19)

Recall that ∥θ0 ∥W 1 (Ω ) + ∥u0 ∥B2(1−1/p) (Ω ) ≤ R. By (7.5), (7.18) and (7.19), we have q,p

q

  t    ′  ≤ CT 1/p L, sup  V ∇ w (·, t ) ds i   t ∈(0,T ) 0 Wq1 (Ω )   1    ′′  P (ρ∗ + θ0 + τ κ)(1 − τ ) dτ  sup ∇  t ∈(0,T )

Lq (Ω )

0

 

t ∈(0,T )

≤ C (R + T

t



sup  ∇ W 1/p′

0

  ∇ w(·, t ) ds  

≤ CT 1/p L ′

Lq (Ω )

(7.20)

L),

where i = 1, 2, 4, 5 and 6, and W = W(K) is any matrix of polynomials with respect to K. By (7.4), (7.15) and (7.18)–(7.20), we have

∥F(κ, w)∥Wp1 ((0,T ),Wq1 (Ω )) ≤ C (L2 T 1/p + L3 (T 1/p )2 ), ∥g∥Lp ((0,T ),Lq (Ω )) ≤ CRT 1/p ,   ′ ′ ′ ′ ′ ′ ∥G(κ, w)∥Lp ((0,T ),Lq (Ω )) ≤ C L2 T 1/p + L3 (T 1/p )2 + (1 + R + LT 1/p )(LT 1/p )2 T 1/p + (R + LT 1/p )(LT 1/p )T 1/p . (7.21) ′

′

To obtain sup ∥w(·, t )∥B2(1−1/p) (Ω ) ≤ C (Iw (0, T ) + eγ T ∥u0 ∥B2(1−1/p) (Ω ) ),

t ∈(0,T )

q,p

(7.22)

q,p

we use the embedding relation: Lp ((0, ∞), X1 ) ∩ Wp1 ((0, ∞), X0 ) ⊂ BUC (J , [X0 , X1 ]1−1/p,p )

(7.23)

for any two Banach spaces X0 and X1 such that X1 is dense in X0 and 1 < p < ∞ (cf. [13]). In fact, let Et be the extension operator defined in (7.10) and let Z be a solution to problem:

∂t Z − ∆Z = 0 in Ω × (0, ∞),

Z|Γ = 0,

Z|t =0 = u0

in Ω .

(7.24)

M. Murata / Nonlinear Analysis 106 (2014) 86–109

107

Employing similar argumentations to those from Section 3 through Section 6 we can prove the existence of such a Z ∈ Wp1,γ ((0, ∞), Lq (Ω )N ) ∩ Lp,γ ((0, ∞), Wq2 (Ω )N ) possessing the estimate:

∥e−γ t ∂t Z∥Lp ((0,∞),Lq (Ω )) + ∥e−γ t Z∥Lp ((0,∞),Wq2 (Ω )) ≤ C ∥u0 ∥B2(1−1/p) (Ω ) .

(7.25)

q,p

We choose γ so large that (7.25) hold and fix it in the following. If we set y = w − z, then y ∈ Wp1 ((0, T ), Lq (Ω )) ∩

Lp ((0, T ), Wq2 (Ω )) and y|t =0 = 0. Let Et be the operator defined in (7.10). We have w = z + ET y for 0 ≤ t ≤ T , so that by (7.23) sup ∥w(·, t )∥B2(1−1/p) (Ω ) ≤ sup ∥ET y(·, t )∥B2(1−1/p) (Ω ) + eγ T sup ∥e−γ t z∥B2(1−1/p) (Ω ) q,p

t ∈(0,T )

q,p

t ∈(0,∞)

≤ C {IET y (0, ∞) + e

γT

q,p

t ∈(0,∞)

sup Ie−γ t z (0, ∞)}

t ∈(0,∞)

with some constant C independent of T ∈ (0, ∞), which combined with (7.10), (7.11) and (7.25) furnishes (7.22). 2(1−1/p) Since Bq,p (Ω ) ⊂ Wq1 (Ω ) as follows from the assumption: 2 < p < ∞, by (7.19) and (7.22) sup ∥w(·, t )∥W 1 (Ω ) ≤ C (L + eγ T ∥u0 ∥B2(1−1/p) (Ω ) ), q,p

q

t ∈(0,T )

  t     ∂ W ∇ w (·, s ) ds sup  t   t ∈(0,T ) 0

Lq (Ω )

(7.26)

≤ C (L + eγ T ∥u0 ∥B2(1−1/p) (Ω ) ). q,p

In the following, we assume that LT 1/p ≤ 1 and 0 < σ ≤ 1 to obtain ′

ˆ w − n∥W 1 (Ω ) ≤ C (R + L)T sup ∥n

(7.27)

q

t ∈(0,T )

ˆw − n = we write n

1 0

(∇ n)(ξ + τ

∥ n ∥ W∞ 1 (RN ) ≤ C ,

t 0

t

w(ξ , s) ds) dτ (

w(ξ , s) ds). By (6.2), we have 0     t   (∇ n) · + τ ≤ C (τ ∈ [0, 1]), w(·, s) ds   

∥nˆ w ∥W∞ 1 (Ω ) ≤ C ,

(7.28)

L∞ (Ω )

0

ˆ w − n∥Lq (Ω ) ≤ C (R + L)T . To estimate ∇(nˆ w − n), we write ∇(nˆ w − n) = A1 + A2 with so that by (7.15) and (7.26) we have ∥n 1





(∇ n) ξ + τ

A1 = 0



(∇ n) ξ + τ 2

A2 =

w(ξ , s) ds



dτ

0 1



t





∇ w(ξ , s) ds , 0

t



0

t





w(ξ , s) ds

t



I+τ

0

∇ w(ξ , s) ds



dτ

t



0



w(ξ , s) ds . 0

By (7.26) and (7.28), we have ∥A1 ∥Lq (Ω ) ≤ C (R + L)T . On the other hand, by Proposition 6.1 and (7.18) we observe that q A2 Lq (Ω )

∥ ∥

≤C

1  ∞   i=0 j=1

Ω

1



ζ

i j

0

 q     t  t   2  (∇ n) ξ + τ ∥w(ξ , s)∥L∞ (Bi ∩Ω ) ds dξ . w(ξ , s) ds  dτ 

Choosing σ > 0 smaller if necessary, we have ξ + variable x = ξ + τ q A2 Lq (Ω )

∥ ∥

t

≤C

0

t 0

j

0

0

w(ξ , s) ds ∈ Bij = Bdi (xij ) for ξ ∈ supp ζji , so that by the change of

w(ξ , s) ds, (7.19), (6.2) and Hölder’s inequality, we have

1  ∞  

t 0

i=0 j=1

∥w(·, s)∥L∞ (Bi ∩Ω ) ds

q

j

≤C

1  ∞ 

T

i =0 j =1

q/q′

t

 0

∥w(·, s)∥qL

∞ (Bj ∩Ω ) i

ds.

By Proposition 6.1 and Sobolev’s embedding theorem, we see that there exists a constant C independent of j ∈ N such that

∥a∥L∞ (Bi ∩Ω ) ≤ C ∥a∥Wq1 (Bi ∩Ω ) , j

(7.29)

j

so that by (6.3) we have ∥A2 ∥Lq (Ω ) ≤ supt ∈(0,T ) CT ∥w(·, t )∥W 1 (Ω ) , which combined with (7.26) furnishes (7.27). Using Propoq

sition 6.1, (6.2), (6.3), (7.19), (7.28) and (7.29), we also have

∥f (∇ 2 n)∥Lq (Ω ) ≤ C ∥f ∥Lq (Ω ) ,

∥f (∇ 2 nˆ w )∥Lq (Ω ) ≤ C ∥f ∥Wq1 (Ω ) .

(7.30)

By (7.15), (7.20), (7.26), (7.27), (7.28), and (7.30) we have

∥H (w)∥Lp ((0,T ),Wq1 (Ω )) ≤ C ((R + L)LT + L2 T 1/p ), ′

∥w · (nˆ w − n)∥Lp ((0,T ),Wq2 (Ω )) ≤ C ((R + L)LT + (R + L)T 1/p ).

(7.31)

108

M. Murata / Nonlinear Analysis 106 (2014) 86–109

ˆ w = (∇ n)(Xw )w, by (7.28) we have Since ∂t n ∥∂t nˆ w ∥Lq (Ω ) ≤ C ∥w(·, t )∥Lq (Ω ) ,

(7.32)

so that by (7.15), (7.26), (7.27) and (7.32), we have

∥∂t (w · (nˆ w − n))∥Lp ((0,T ),Lq (Ω )) ≤ C ((R + L)LT + (R + L)2 T 1/p ).

(7.33)

To estimate ∥∂t [(1 − ∆)−1/2 ιH (w)]∥Lp (0,T ),Lq (RN ) , we prepare the following lemma. 3−1/r

Lemma 7.3. Let 1 < p < ∞, N < q, r < ∞ and let Ω be a uniform Wr properties (e-1) and (e-2). Then,

domain. Let ι be the extension map satisfying the

∥∂t [(1 − ∆)−1/2 ι((∇ f )g )]∥Lp ((0,T ),Lq (RN ))  1/p  1/p  T T (∥∇ f (·, t )∥Lq (Ω ) ∥∂t g (·, t )∥Lq (Ω ) )p dt . (∥∂t f (·, t )∥Lq (Ω ) ∥g (·, t )∥Wq1 (Ω ) )p dt + ≤C 0

0

Proof. To prove the lemma, we use an inequality:

∥(1 − ∆)−1/2 ι(fg )∥Lq (RN ) ≤ C ∥f ∥Lq (Ω ) ∥g ∥Lq (Ω )

(7.34)

provided that N < q < ∞, which follows from the following observation: For any ϕ ∈ C0∞ (RN ) by Hölder’s inequality and (e-1) we have

|((1 − ∆)−1/2 ι(fg ), ϕ)RN | = |(ι(fg ), (1 − ∆)−1/2 ϕ)RN | ≤ C ∥f ∥Lq (Ω ) ∥g ∥Lq (Ω ) ∥(1 − ∆)−1/2 ϕ∥Ls (RN ) , where s is an index such that 2/q + 1/s = 1. Since N (1/q′ − 1/s) = N /q < 1, by Sobolev’s embedding theorem we have ∥(1 − ∆)−1/2 ϕ∥Ls (RN ) ≤ C ∥ϕ∥L ′ (RN ) , which furnishes (7.34). Since q

∂t [(1 − ∆)−1/2 ι((∇ f )g )] = (1 − ∆)−1/2 ι[∇{(∂t f )g }] − (1 − ∆)−1/2 ι[(∂t f )(∇ g )] + (1 − ∆)−1/2 ι[(∇ f )∂t g ], by (7.34), (7.15) and (e-2) we have Lemma 7.3.



Applying Lemma 7.3 and using (7.28) and (7.32), we have

 t ∥∂t [(1 − ∆)−1/2 H (w)]∥pL ((0,T ),L (RN )) ≤ C ∥∂s w(·, s)∥pLq (Ω ) ∥nˆ w − n∥pW 1 (Ω ) ds p q q 0   s p  t  t    ∥∇ w(·, s)∥pLq (Ω ) ∥w(·, s)∥pLq (Ω ) ds + + V ∇ w (·, r ) dr ∥∂s w(·, s)∥pLq (Ω )  2   0

 t +

∥∇ w(·, s)∥

0

2 Lq (Ω )

  s     V ∇ w (·, r ) dr + 2   0

ds

Wq1 (Ω )

0

0

p L∞ (Ω )

∥∇ w(·, s)∥Lq (Ω ) ∥w(·, s)∥Lq (Ω )

ds.

(7.35)

Applying (7.19), (7.20), (7.26) and (7.27) to the right-hand side of (7.35), we have

∥∂t [(1 − ∆)−1/2 H (w)]∥Lp ((0,T ),Lq (RN )) ≤ C ((R + L)LT + (R + L)2 T + L2 T 1/p ). ′

(7.36)

Noting that θ0 satisfies the condition (2.9) and ∥∇θ0 ∥Lq (Ω ) ≤ R, by Theorem 7.2, (7.21), (7.31), (7.33) and (7.36), we have

∥θ ∥Wp1 ((0,T ),Wq1 (Ω )) + Iv (0, T ) ≤ C (R){∥u0 ∥B2(1−1/p) (Ω ) + C (R, L, T )}, q,p

(7.37)

where we have set

C (R, L, T ) = M2 {L2 T 1/p + L3 (T 1/p )2 + (1 + R + LT 1/p )(LT 1/p )2 T 1/p ′

′

′

′

+ (R + LT 1/p )(LT 1/p )T 1/p + (R + L)LT + (R + L)2 T + (R + L)T 1/p } with some special constant M2 independent of R and T . Recalling that ∥u0 ∥B2(1−1/p) (Ω ) ≤ R, setting L = 2C (R)R and choosing q,p T ∈ (0, 1) so small that C (R, L, T ) ≤ R, by (7.37) we have ∥θ ∥Wp1 ((0,T ),Wq1 (Ω )) + Iv (0, T ) ≤ L. (7.38) ′

′

If we define a map Φ by Φ (κ, w) = (θ , v), then by (7.38) Φ is the map from IL,T into itself. Considering the difference Φ (κ1 , w1 ) − Φ (κ2 , w2 ) for (κi , wi ) ∈ IL,T (i = 1, 2), employing the same argument and choosing T ∈ (0, 1) smaller if necessary, we see that Φ is a contraction map on IL,T , so that by the Banach fixed point theorem there exists a unique fixed point (θ , v) ∈ IL,T such that (θ , v) = Φ (θ , v) which solves the nonlinear equation (7.3). Since the existence of solutions to (7.3) is proved by the contraction mapping principle, the uniqueness of solutions automatically follows, which completes the proof of Theorem 7.1.

M. Murata / Nonlinear Analysis 106 (2014) 86–109

109

Proof of Theorem 2.7. Following Ströhmer [19], we see that the correspondence x = Xv (ξ , t ) is injective for any t ∈ [0, T ] ˆ v is parallel provided that (7.2) holds with some small constant σ > 0. Let Γ be defined by F (x) = 0 locally. Since n ˆ v = 0 on Γ implies that ∂ F (Xv (ξ , t ))/∂ t = 0, which furnishes that F (Xv (ξ , t )) = to (∇ F )(Xv (ξ , t )), the fact that v · n F (Xv (ξ , 0)) = F (ξ ) = 0 for ξ ∈ Γ . Therefore, we have {Xv (ξ , t ) | ξ ∈ Γ } ⊂ Γ for each t ∈ (0, T ). Note that Xv ∈ C 0 [0, T ], Wq2 (Ω ) ∩ Wp1 ((0, T ), Wq2 (Ω )). Let us fix t ∈ (0, T ). Employing the same argument as in proving the inverse

mapping theorem for a non-degenerate C 1 map, we see that Xv gives us a local diffeomorphism. We see that {Xv (ξ , t ) | ξ ∈ Γ } is non-empty, open and closed subset of Γ , so that the connectedness of Γ implies that Γ = {Xv (ξ , t ) | ξ ∈ Γ }. Thus, we also see that {Xv (ξ , t ) | ξ ∈ Ω } ⊂ Ω , because of the injectivity of Xv . Since Ω is a connected open set, we also 1 1 2 see that Ω = {Xv (ξ , t ) | ξ ∈ Ω }. Moreover, the inverse map: ξ = X− v (x, t ) ∈ Wp ((0, T ), Wq (Ω )), and therefore setting 1 −1 ρ = θ(X− v (x, t ), t ) and u = v(Xv (x, t ), t ), we see that (ρ, u) satisfies the nonlinear equations (1.1) and

ρ ∈ Wq1 ((0, T ), Lq (Ω )) ∩ Lp ((0, T ), Wq1 (Ω )),

u ∈ Wp1 ((0, T ), Lq (Ω )) ∩ Lp ((0, T ), Wq2 (Ω )).

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