On the maximal L p -L q regularity of the Stokes problem with first order boundary condition; model problems

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1 c 212 The Mathematical Society of Japan J. Math. Soc. Japan Vol. 64, No ) pp doi: /jmsj/ On the maximal L p -L q regularity of the Stokes problem with first order boundary condition; model problems By Yoshihiro Shibata and Senjo Shimizu Received May 25, 21) Revised Nov. 2, 21) Abstract. In this paper, we proved the generalized resolvent estimate and the maximal L p-l q regularity of the Stokes equation with first order boundary condition in the half-space, which arises in the mathematical study of the motion of a viscous incompressible one phase fluid flow with free surface. The core of our approach is to prove the R boundedness of solution operators defined in a sector Σ ɛ,γ = {λ C \ {} arg λ π ɛ, λ γ } with < ɛ < π/2 and γ. This R boundedness implies the resolvent estimate of the Stokes operator and the combination of this R boundedness with the operator valued Fourier multiplier theorem of L. Weis implies the maximal L p-l q regularity of the non-stationary Stokes. For a densely defined closed operator A, we know that what A has maximal L p regularity implies that the resolvent estimate of A in λ Σ ɛ,γ, but the opposite direction is not true in general cf. Kalton and Lancien [19). However, in this paper using the R boundedness of the operator family in the sector Σ ɛ,λ, we derive a systematic way to prove the resolvent estimate and the maximal L p regularity at the same time. 1. Introduction. This paper is concerned with the generalized resolvent estimate and the maximal L p -L q regularity of the Stokes problem with first order boundary condition in the half-space, which arises in the study of the free boundary problem of viscous incompressible one phase fluid flow. This problem is mathematically to find a time dependent domain Ω t, t being time variable, in the n-dimensional Euclidean space R n, an n-vector of functions vx, t) = v 1 x, t),..., v n x, t)) and a scalar function px, t) satisfying the following Navier-Stokes equations: 21 Mathematics Subject Classification. Primary 35Q3; Secondary 76D7. Key Words and Phrases. Stokes equation, half space problem, maximal regularity, resolvent estimate, surface tension, gravity force. The second author was partially supported by Challenging Exploratory Research , MEXT, Japan.

2 562 Y. Shibata and S. Shimizu v t + v )v Div Sv, p) = f, div v = in Ω t, t >, Sv, p)n t c σ Hn t =, V n = v n t on Γ t, t >, 1.1) v t= = v, in Ω, where V n is the velocity of the evolution of Γ t in a normal direction, n t = n t,1,..., n t,n ) is the unit outer normal to Γ t, Sv, p) = pi + µdv) is the Stokes stress tensor, I = δ ij ) is the n n identity matrix, Dv) is the Cauchy stress tensor with elements D jk v) = D j v k + D k v j D j = / x j ), i th component of Div Sv, p) n = D j µd i v j + D j v i ) δ ij p) = µ v i + D i div v) D i p, j=1 i th component of Sv, p)n t n = {µd i v j + D j v i ) δ ij p}n t,j = j=1 n µd i v j + D j v i )n t,j pn t,i, div v = n j=1 D jv j, v )v = n j=1 v jd j v, H is the doubled mean curvature of Γ t, µ is a positive constant describing viscosity, and c σ is a positive constant describing the coefficient of surface tension. The following two problems have been studied by many mathematicians: 1) the motion of an isolated liquid mass and 2) the motion of a viscous incompressible fluid contained in an ocean of infinite extent. In case 1), the initial domain Ω is bounded. A local in time unique existence theorem was proved by Solonnikov [39, [42, [44, [45 in the L 2 Sobolev-Slobodetskii space, by Schweizer [29 in the semigroup setting, by Moglilevskiĭ and Solonnikov [22, [45 in the Hölder spaces when c σ > ; and by Solonnikov [41 and Mucha and Zaj aczkowski [23 in the L p Sobolev-Slobodetskii space and by Shibata and Shimizu [33, [34 in the L p in time and L q in space setting when c σ =. A global in time unique existence theorem for small initial velocity was proved by Solonnikov [41 in the L p Sobolev-Slobodetskii space and by Shibata and Shimizu [33, [34 in the L p in time and L q in space setting when c σ = ; and by Solonnikov [4 in the L 2 Sobolev-Slobodetskii space and by Padula and Solonnikov [25 in the Hölder spaces under the additional assumption that the initial domain Ω is sufficiently close to a ball when c σ >. In case 2), the initial domain Ω is a perturbed layer like: Ω = {x R 3 b < x 3 < ηx ), x = x 1, x 2, x 3 ) R 2 }. A local in time unique existence theorem was proved by Beale [7, Allain [3 and Tani [51 in the L 2 Sobolev-Slobodetskii space when c σ > and by Abels [1 in the j=1

3 Maximal L p-l q regularity of the Stokes problem 563 L p Sobolev-Slobodetskii space when c σ =. A global in time unique existence theorem for small initial velocity was proved in the L 2 Sobolev-Slobodetskii space by Beale [7 and Tani and Tanaka [52 when c σ > and by Sylvester [47 when c σ =. The decay rate was studied by Beale and Nishida [9, Sylvestre [48 and Hataya [17. We remark that in the two phase fluid flow case, the free boundary problem also has been studied by Abels [2, Denisova and Solonnikov [12, [13, [14, [15, Giga and Takahashi [18, Takahashi [49, Nouri and Poupaud [24, Prüß and Simonett [26, [27, [28, Shibata and Shimizu [32, Shimizu [36, [37, [38, Tanaka [5 and references therein. Our purpose is to prove a local in time unique existence theorem of 1.1) in the L p in time and L q in space setting and in the case where the initial domain Ω satisfies more general assumptions including the above physical situation. In fact, the L p in time and L q in space approach relaxes the regularity assumption and compatibility condition on initial data and the general domain setting allows us to treat several different physical situations at the same time. The core of our approach is to prove the maximal L p -L q regularity of the Stokes problems with first order boundary condition in a general domain. Since to achieve our approach is a rather long journey, we decide to divide it into three parts. In this paper, we prove the maximal L p -L q regularity of Stokes equations with first order boundary condition in the half-space. And in the forthcoming papers, we shall discuss the same problem in a general domain and the local in time unique existence theorem of free boundary problems of the Navier-Stokes equations in a general domain in the L p in time and L q in space setting. Another issue of this paper is to drive a systematic way to prove the resolvent estimate and the maximal L p regularity at the same time in the model problem case. On the one hand, we know that the maximal regularity implies the resolvent estimate, but that the opposite direction is not true in general cf. Kalton and Lancien [19), but on the other hand, using the R boundedness of the operator family in the sector Σ ɛ,λ, we can prove the resolvent estimate and the maximal L p regularity at the same time at least in the model problem case. Now, we formulate our problem in this paper. Let R n + and R n be a half-space and its boundary and let Q + and Q be their cylindrical domains. Namely, R n + = {x = x 1,..., x n ) R n x n > }, R n = {x = x 1,..., x n ) R n x n = }, Q + = {x, t) x R n +, t > } Q = {x, t) x R n, t > }. Let n =,...,, 1) be the unit outer normal to R n. We consider the following four problems:

4 564 Y. Shibata and S. Shimizu λu Div Su, θ) = f, div u = g in R n +, Su, θ)n = h on R n ; 1.2) U t Div SU, Θ) = F, div U = G in Q +, SU, Θ)n = H on Q, U t= = ; 1.3) λu Div Su, θ) = f, div u = g in R n +, λη + u n = d on R n, Su, θ)n + c g c σ )ηn = h on R n ; 1.4) U t Div SU, Θ) = F, div U = G in Q +, Y t + U n = D in Q, SU, Θ)n + c g c σ )Y n = H on Q, U t= =. 1.5) Here, c g and c σ are positive constants; U = U 1,..., U n ), u = u 1,..., u n ), Θ, θ, Y and η are unknown functions while F = F 1,..., F n ), H = H 1,..., H n ), f = f 1,..., f n ), h = h 1,..., h n ), G, g, D and d are given functions; η = n 1 j=1 D2 j η, and 1.2) and 1.4) are the corresponding generalized resolvent problems to the evolution equations 1.3) and 1.5), respectively. To state our main results exactly, we introduce several symbols. Given ɛ < ɛ < π/2) and γ, we set Σ ɛ,γ = {λ C \ {} arg λ π ɛ, λ γ }, where C stands for the set of all complex numbers. Given domain G, L q G) and Wq m G) denote the usual Lebesgue space and Sobolev space while LqG) and W m q G) denote their norms, respectively. For the differentiations of scalar θ and n-vector u = u 1,..., u n ), we use the following symbol: θ = D 1 θ,..., D n θ), u = D i u j i, j = 1,..., n), 2 θ = D i D j θ i, j = 1,..., n), 2 u = D i D j u k i, j, k = 1,..., n). Given Banach space X with norm X, X n denotes the n-product space of X, that is X n = {f = f 1,..., f n ) f i X}. The norm of X n is also denoted by X for simplicity and

5 Maximal L p-l q regularity of the Stokes problem 565 n f X = f j X for f = f 1,..., f n ) X n. j=1 Set Ŵ 1 q G) = { θ L q,loc G) θ L q G) n}, Ŵ 1 q,g) = { θ Ŵ 1 q G) θ G = }, where G denotes the boundary of G. Let Ŵ q 1 G) denote the dual space of Ŵq 1, G), where 1/q + 1/q = 1. For θ Ŵ q 1 G) L q G), we have { θ Ŵ 1 q G) = sup G } θϕ dx ϕ Ŵ q 1,G), ϕ Lq G) = 1. For 1 p, L p R, X) and Wp m R, X) denote the usual Lebesgue space and Sobolev space of X-valued functions defined on the whole line R, and LpR,X) and W m p R,X) denote their norms, respectively. Set L p,,γ R, X) = {f : R X e γt ft) L p R, X), ft) = for t < }, W m p,,γ R, X) = { f L p,,γ R, X) e γt D j t ft) L p R, X), j = 1,..., m }, L p, R, X) = L p,, R, X), W m p,r, X) = W m p,,r, X). Let L and L 1 λ denote the Laplace transform and its inverse, that is L [fλ) = e λt ft) dt, L 1 λ [gt) = 1 e λt gλ) dτ, 2π where λ = γ + iτ. Given s R and X-valued function ft), we set Λ s γft) = L 1 λ [ λ s L [fλ)t). We introduce the Bessel potential space of X valued functions of order s as follows: H s p,,γ R, X) = { f : R X e γt Λ s γft) L p R, X) H s p,r, X) = H s p,,r, X). for any γ γ, ft) = for t < }, The following four theorems are our main results of the paper.

6 566 Y. Shibata and S. Shimizu Theorem 1.1. Let < ɛ < π/2 and 1 < q <. Then, for any λ Σ ɛ,, f L q R n + ) n, g Ŵq 1 ) R n + W 1 q R+), n h Wq 1 ) R n +, problem 1.2) admits a unique solution u, θ) W 2 q R n +) n Ŵ 1 q R n +) that satisfies the estimate: λ u, λ 1/2 u, 2 u, θ) LqR n + ) C { f, λ 1/2 g, g, λ 1/2 h, h) LqR n + ) + λ g Ŵ 1 q R n + ) } for some constant C = C n,q,ɛ,µ depending only on n, q, ɛ and µ. Theorem 1.2. Let 1 < p, q < and γ. Then, for any F L p,,γ R, Lq R n +) n), G L p,,γ R, W 1 q R n +) ) W 1 p,,γ R, Ŵ 1 q R n +) ), H L p,,γ R, W 1 q R n +) n) H 1/2 p,,γ R, Lq R n +) n), problem 1.3) admits a unique solution U, Θ) such that U L p,,γ R, W 2 q R n +) n) W 1 p,,γ R, Lq R n +) n)), Θ L p,,γ R, Ŵ 1 q R n +) ) that satisfy the estimate: e γt U t, γu, Λ 1/2 γ U, 2 U, Θ) LpR,L qr n + )) C { e γt F, Λ 1/2 γ G, G, Λ 1/2 γ H, H ) LpR,L qr+ n )) + e γt } G t, γg) LpR,Ŵ q 1 R+ n )) for any γ γ with some constant C = C n,p,q,µ depending only on n, p, q and µ. Theorem 1.3. Let < ɛ < π/2 and 1 < q <. Then, there exists a constant γ 1 depending on ɛ such that for any λ Σ ɛ,γ, f L q R n +) n, g Ŵ 1 q R n +) W 1 q R n +), h W 1 q R n +), d W 2 q R n +), problem 1.4) admits a unique solution u, θ, η) W 2 q R n +) n Ŵ 1 q R n +) W 3 q R n +) that satisfies the estimate:

7 Maximal L p-l q regularity of the Stokes problem 567 λ u, λ 1/2 u, 2 u, θ, λ 1/2 θ, θ) LqR n + ) + λ η W 2 q Rn + ) + η W 3 q R n + ) C { f, λ 1/2 g, g, λ 1/2 h, h) LqR n + ) + λ g Ŵ 1 q R n + ) + d W 2 q Rn + ) }, λ 3/2 η W 1 q R n + ) C { f, λ 1/2 g, g, λ 1/2 h, h ) LqR n + ) + λ g Ŵ 1 q R n + ) + d W 2 q Rn + ) + λ 1/2 d W 1 q R n + ) }, for some constant C = C n,q,ɛ,µ depending only on n, q, ɛ and µ. Theorem 1.4. that for any Let 1 < p, q <. Then, there exists a constant γ 1 such F L p,,γ R, Lq R n +) n), G L p,,γ R, W 1 q R n +) ) W 1 p,,γ R, Ŵ 1 q R n +) ), H L p,,γ R, W 1 q R n +) n) H 1/2 p,,γ R, Lq R n +) n), D L p,,γ R, W 2 q R n +) ), problem 1.5) admits a unique solution U, Θ, Y ) such that U L p,,γ R, W 2 q R n +) n) W 1 p,,γ R, Lq R n +) n), Θ L p,,γ R, Ŵ 1 q R n +) ), Y L p,,γ R, W 3 q R n +) ) W 1 p,,γ R, W 2 q R n +) ) that satisfy the estimate: e γt U t, γu, Λ 1/2 γ U, 2 U, Θ) LpR,L qr n + )) + e γt Y t, γy ) LpR,W 2 q Rn + )) + e γt Y LpR,W 3 q Rn + )) C { e γt F, Λ 1/2 γ G, G, Λ 1/2 γ H, H) LpR,L qr+ n )) + e γt G t, γg) LpR,Ŵ q 1 R+ n )) + } e γt D LpR,Wq 2Rn + )) for any γ γ with some constant C = C n,p,q,µ depending only on n, p, q and µ. If we assume that D H 1/2 p,,γ R, Wq 1 R+)) n in addition, then Y H 3/2 p,,γ R, Wq 1 R+)) n and

8 568 Y. Shibata and S. Shimizu e γt Λ 3/2 γ Y LpR,W 1 q Rn + )) C { e γt F, Λ 1/2 γ G, G, Λ 1/2 γ H, H) LpR,L qr n + )) + e γt G t, γg) LpR,Ŵ 1 q R n + )) + e γt D LpR,W 2 q Rn + ) + e γt Λ 1/2 γ D LpR,W 1 q Rn + )) }. Remark 1.5. The case of non-zero initial values in 1.3) and 1.5) can be treated by the semigroup, whose generation are guaranteed by Theorem 1.1 and Theorem 1.3 with g = and h =, respectively. Remark 1.6. We use the Fourier multiplier theorem with respect to time variable, so that it is natural to use Bessel potential spaces with respect to time variable. Especially, Λγ 1/2 plays an essential role to treat the original nonlinear problem. Theorem 1.1 and Theorem 1.3 were proved by Shibata and Shimizu [31 and [35, respectively. Theorem 1.2 was essentially proved by Shibata and Shimizu [34. Theorem 1.4 is only new. But it is the purpose of this paper that we investigate a systematic approach by means of the R boundedness of the operator family in the sector Σ ɛ,γ to obtain the both of the generalized resolvent estimate and the maximal L p -L q regularity at the same time, and therefore we reprove the results obtained in [31, [35 and [34. The paper is organized as follows. In Section 2, we introduce R boundedness, operator valued Fourier multiplier theorem and several results used in later sections. In Section 3, we prove the generalized resolvent estimate and the maximal L p -L q regularity in the whole space. In Section 4, we derive solution formulas to problems 1.2) and 1.3). In Section 5, we prepare several technical lemmas used to prove our main results. In Section 6, applying technical lemmas to the solution formulas, we prove Theorems 1.1 and 1.2 at the same time. In Section 7, we derive solution formulas of 1.4) and 1.5), and applying technical lemmas to these formulas, we prove Theorems 1.3 and 1.4 at the same time. 2. R-boundedness and operator valued Fourier multiplier theorem. Let X and Y be two Banach spaces, and X and Y denote their norms, respectively. Let L X, Y ) denote the set of all bounded linear operators from X into Y and L X) = L X, X). Definition 2.1. A family of operators T L X, Y ) is called R-bounded,

9 Maximal L p-l q regularity of the Stokes problem 569 if there exist constants C > and p [1, ) such that for each m N, N being the set of all natural numbers, T j T, x j X j = 1,..., N) and for all sequences {r j u)} N j=1 of independent, symmetric, { 1, 1}-valued random variables on [, 1, there holds the inequality: 1 N r j u)t j x j ) j=1 p Y du C 1 N p r j u)x j du. 2.1) The smallest such C is called R-bound of T, which is denoted by RT ). Let DR, 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, while D R, X) = L DR), X) and S R, X) = L S R), X), respectively. Here, DR) = DR, C) and S R) = S R, C). Definition 2.2. A Banach space X is said to be a UMD Banach space, if the Hilbert transform is bounded on L p R, X) for some and then all) p with 1 < p <. Here, the Hilbert transform H of a function f S R, X) is defined by Hf = 1 π lim fs) ɛ + t s >ɛ t s ds j=1 t R). Given M L 1,loc R, L X, Y )), let us define the operator T M : DR, X) S R, Y ) by the formula: T M φ = [MF [φ, F [φ DR, X)). 2.2) Here and hereafter, F and denote the Fourier transform and its inversion formula, that is F [f) = ˆf) = e ix fx) dx, R n [gx) = [gx) = 1 2π) n X R n e ix g) d. Theorem 2.3 Weis [55). Let X and Y be two UMD Banach spaces and 1 < p <. Let M be a function in C 1 R \ {}, L X, Y )) such that R{Mρ) ρ R \ {}}) = κ <, R{ρM ρ) ρ R \ {}}) = κ 1 <.

10 57 Y. Shibata and S. Shimizu Then, the operator T M defined in 2.2) is extended to a bounded linear operator from L p R, X) into L p R, Y ). Moreover, denoting this extension by T M, we have T M L LpR,X),L pr,y )) Cκ + κ 1 ) for some positive constant C depending on p, X and Y. Theorem 2.4 Bourgain [1). Let X be a UMD Banach space and 1 < p <. Let mρ) be a scalar function in C 1 R \ {}) such that mρ) M, ρm ρ) M r R \ {}) for some positive constant M. Let T m be a Fourier multiplier defined by the formula: T m f = [mf [f F [f DR, X)). Then, T m is extended to a bounded linear operator on L p R, X). Moreover, denoting this extension by T m, we have T m L LpR,X)) CM for some positive constant C depending on p and X. Remark 2.5. Zimmermann [56. Theorem 2.4 was extended to the several variables case by In order to prove the R-boundedness, we use the following two propositions whose proofs were given in Denk-Hieber-Prüß [11. Proposition 2.6. Let 1 < q < and Λ be an index set. Let {k λ x) λ Λ} be a family of functions in L 1,loc R n ) and let us define the operator K λ of a function f by the formula: K λ fx) = k λ x y)fy) dy R n λ Λ). Assume that there exists a constant M independent of λ Λ such that K λ f L2R n ) M f L2R n ) f L 2 R n ), λ Λ), 2.3)

11 α =1 Maximal L p-l q regularity of the Stokes problem 571 α x k λ x) M x n+1) x R n \ {}, λ Λ). 2.4) Then, the set {K λ λ Λ} is an R-bounded family in L L q R n )) and R{K λ λ Λ}) C n,q M for some constant C n,q depending on n and q. Proposition 2.7. Let G be a domain in R n, Λ an index set and 1 < q <. Let {k λ x, y) λ Λ} be a family of functions in L 1,loc G G) and let us define the operator K λ of a function f by the formula: T λ fx) = G kx, y)fy) dy x G). Assume that there exists a function k x, y) L 1,loc G G) such that k λ x, y) k x, y) x, y) G G, λ Λ). 2.5) Let us define the operator K of a function f by the formula: T fx) = G kx, y)fy) dy x G, λ Λ). If T L L q G)), then the set {T λ λ Λ} is an R-bounded family in L L q G)) and R{T λ λ Λ}) C q,g T L LqG)) for some positive constant C q depending on q. Finally, we give a theorem used in proving the resolvent estimate and the maximal L p -L q regularity. Theorem 2.8. Let 1 < p, q <, < ɛ < π/2 and γ. Let G be a domain in R n and Φ λ be a C 1 function of τ R\{} when λ = γ +iτ Σ ɛ,γ with its value in L L q G)). Assume that the sets {Φ λ λ Σ ɛ,γ } and {τd/dτ)φ λ λ = γ + iτ Σ ɛ,γ } are R-bounded families in L L q G)). In addition, we assume that there exists a constant M such that R{Φ λ λ Σ ɛ,γ }) M, Then, we have { R τ d }) dτ Φ λ λ = γ + iτ Σ ɛ,γ M. Φ λ f LqG) M f LqG) f L q G), λ Σ ɛ,γ )

12 572 Y. Shibata and S. Shimizu for some constant C q depending on q. Moreover, if we define the operator Ψ of a function f L p R, L q G)) by the formula: Ψfx, t) = L 1 λ [Φ λl [fλ)x, t) = e γt Fτ 1 [Φ λ F [e γt fτ)t) λ = γ + iτ) where F [e γt fτ) = e γ+iτ)t fx, t) dt, then there exists a constant C p,q depending on p and q such that for any γ γ. e γt Ψf LpR,L qg)) C p,q M e γt f LpR,L qg)) Proof. Since the set {Φ λ λ Σ ɛ,γ } is an R-bounded family in L L q G)), it is easy to see from the definition of the R boundedness that the set {Φ λ λ Σ ɛ,γ } is a bounded family in L L q G)). Moreover, we have Φ λ f LqG) R{Φ λ λ Σ ɛ,γ }) f LqG) M f LqG). On the other hand, by the assumption we have R { Φ γ+iτ τ R \ {} }) M, { R τ d }) dτ Φ γ+iτ τ R \ {} M for any γ γ, and therefore applying Theorem 2.3 with X = L q G) to the formula: we have e γt Ψfx, t) = Fτ 1 [Φ γ+iτ F [e γt fτ)t), e γt Ψf LpR,L qg)) C p,q M e γt f LpR,L qg)).

13 Maximal L p-l q regularity of the Stokes problem Problems in the whole space. In this section we consider the generalized resolvent problem and nonstationary Stokes equations in R n as follows: λu Div Su, θ) = f, div u = g in R n, 3.1) U t Div SU, Θ) = F, div U = G in R n, ), 3.2) subject to the initial condition: Ux, ) =. We prove the following theorem. Theorem 3.1. Let 1 < p, q <, < ɛ < π/2 and γ. 1) For any λ Σ ɛ,, f L q R n ) n and g Ŵ q 1 R n ) Wq 1 R n ), problem 3.1) admits a unique solution u, θ) Wq 2 R n ) n Ŵ q 1 R n ) that satisfies the following estimate: λ u, λ 1/2 u, 2 u, θ) LqR n ) { C n,q,ɛ,µ f, λ 1/2 g, g) + λ g } LqR n ) Ŵq 1 R n ). 2) For any F L p,,γ R, L q R n )) and G L p,,γ R, Wq 1 R n )) Wp,,γ 1 R, Ŵ q 1 R n )), problem 3.2) admits a unique solution U, Θ) L p,,γ R, W 2 q R n ) n) W 1 p,,γ R, L q R n ) n )) L p,,γ R, Ŵ 1 q R n ) ) that satisfies the estimate: e γt U t, γu, Λ 1/2 γ U, 2 U, Θ) LpR,L qr n )) { C n,p,q,µ e γt F, Λ 1/2 γ G, G) + LpR,L qr n )) e γt G t, γg) LpR,Ŵ q 1 R n )) for any γ γ. First, we reduce the problems 3.1) and 3.2) to the case where g = and G =. To do this, we use the following lemma. Lemma 3.2. Let 1 < p, q <. 1) For any g Ŵ q 1 R n ) Wq 1 R n ), there exists a v Wq 2 R n ) n such that div v = g in R n and there hold the estimates: }

14 574 Y. Shibata and S. Shimizu v LqR n ) C n,q g Ŵ 1 q R n ), v LqR n ) C n,q g LqR n ), 3.3) 2 v LqR n ) C n,q g LqR n ). 2) For any G Wp,R, 1 Ŵ q 1 R n )) L p, R, Wq 1 R n )), there exists a V L p, R, Wq 2 R n ) n ) such that div V = G in R n R and e γt V t, γv ) LpR,L qr n )) C n,p,q e γt G t, γg) LpR,Ŵ q 1 R n )), e γt Λ 1/2 γ V C LpR,L qr n )) n,p,q e γt Λ 1/2 γ G, 3.4) LpR,L qr n )) for any γ. e γt 2 V LpR,L qr n )) C n,p,q e γt G LpR,L qr n )) Proof. 1) Defining v j x) by the formula: v j x) = [ ij ĝ) 2 x), and setting v = v 1,..., v n ), obviously we have div v = g in R n. Moreover, by the Fourier multiplier theorem of S. G. Mihlin, we have v LqR n ) C n,q g LqR n ), 2 v LqR n ) C n,q g LqR n ). In order to estimate v LqR n ), we take any ϕ C R n ) and we consider a vector of function Φx) = Φ 1 x),..., Φ n x)) defined by the formula: [ ij [ϕ) Φ j x) = F 2 x). Setting η, ζ) R n = ηx)ζx) dx, by the definition we have v R n j, ϕ) R n = g, Φ j ) R n. Since Φ j Lq R n ) C n,q ϕ L q R n ) as follows from the Fourier multiplier theorem of S. G. Mihlin, we have

15 Maximal L p-l q regularity of the Stokes problem 575 v j, ϕ) R n g Ŵ 1 q R n ) Φ j Lq R n ) C n,q g Ŵ 1 q R n ) ϕ L q R n ), which implies that v LqR n ) C n,q g Ŵ 1 q R n ). Therefore, we have proved 3.3). 2) Regarding t as a parameter, defining V j x, t) by the formula: V j x, t) = [ ij Ĝ, t) 2 x), and setting V = V 1,..., V n ), obviously we have div V = G in R n for all t >. Moreover, V j x, t) = for t < because Gx, t) = for t < by the assumption. Since Dt V j x, t), γv j x, t), Λ 1/2 γ V j x, t) ) [ = ij Ĝt, t), γĝ, t), F [Λ1/2 γ 2 G, t)) x), applying the same argument as in the proof of 1), we have 3.4). This completes the proof of the lemma. In view of Lemma 3.2, we set u = v + w and U = V + W in 3.1) and 3.2), respectively, and then setting f = f λv µ Div Dv)) and F = F V t µ Div DV )), we see that 3.1) and 3.2) are converted to the following equations: λw Div Sw, θ) = f, div w = in R n, 3.5) W t Div SW, Θ) = F, div W = in R n, ), 3.6) subject to the initial condition: W x, ) =, respectively. By Lemma 3.2 we have f { LqR n ) f LqR n ) + C n,q λ g Ŵ 1 q R n ) + µ g } L qr n ), 3.7) e γt F LpR,L qr n )) e γt F LpR,L qr n )) { + C n,q e γt G t LpR,Ŵ q 1 R n )) + } µ e γt G LpR,L qr n )). 3.8) Since Div Du) = u when div u =, in what follows instead of 3.5) and 3.6) we consider the equations:

16 576 Y. Shibata and S. Shimizu λu µ u + θ = f, div u = in R n, 3.9) U t µ U + Θ = F, div U = in R n, ), 3.1) subject to the initial condition: Ux, ) =, respectively. By using the Fourier transform, we have the following solution formulas: ux) = Ux, t) = L 1 λ [ P ) ˆf) λ + µ 2 x), [ P )L F [F, λ) λ + µ 2 x, t), θx) = Θx, t) = [ i ˆf) 2 [ i ˆF, t) 2 x), x), where P ) is an n n matrix whose j, k) component P jk ) is given by the formula: P jk ) = δ jk j k 2 and δ jk denote the Kronecker delta symbols defined by the formula: δ jk = 1 when j = k and δ jk = when j k. Note that L F [F, λ) = = e λt+i x) F x, t) dxdt R n R n e iτt+ x) e γt F x, t) dxdt = F [e γt F, τ) λ = γ + iτ), L 1 λ 1 [G, λ)x, t) = 2π) n+1 e λt+i x G, λ) ddτ R n = eγt 2π) n+1 = e γt [Gx, t). By a technical reason, instead of u and U we consider u ɛ x) = R n e iτt+ x) G, λ) ddτ [ e ɛ 2 P ) ˆf) [ ie λ + µ 2 x), θ ɛ x) = ɛ 2 ˆf) 2 [ e U ɛ x, t) = L 1 λ ɛ 2 P )L F [F, λ) λ + µ 2 x, t), Θ ɛ x, t) = [ ie ɛ 2 ˆF, t) 2 x), x),

17 Maximal L p-l q regularity of the Stokes problem 577 for ɛ >. We see that where f ɛ x) = λu ɛ µ u ɛ + θ ɛ = f ɛ, div u ɛ = in R n, U ɛ ) t µ U ɛ + Θ ɛ = F ɛ, div U ɛ = in R n, ), [e ɛ 2 ˆf)x) and Fɛ x, t) = [e ɛ 2 ˆF, t)x). In order to estimate λu ɛ and U ɛ ) t, we consider a family of kernel functions k ɛ,λ x) and operators K ɛ,λ defined by the formulas: We have [ λe k ɛ,λ x) = ɛ 2 P ) λ + µ 2 x), [ λe K ɛ,λ [fx) = k ɛ,λ x y)fy) dy = ɛ 2 P ) ˆf) R λ + µ 2 x). n U ɛ ) t x, t) = L 1 λ λu ɛ x) = K ɛ,λ [fx), [K ɛ,λ[l [F λ)t) = e γt Fτ 1 [K ɛ,γ+iτ [F [e γt F τ)t). Now, we prove that for any < σ < π/2 sets {K ɛ,λ λ Σ σ, } and {τd/dτ)k ɛ,λ λ Σ σ, } are R bounded families in L L q R n )), whose R bounds do not exceed some constant C which depends only on σ, µ and n. We start with the following well-known fact. Lemma 3.3. Let < σ < π/2 and λ Σ σ,. Then, λ + µ 2 sinσ/2) λ + µ 2 ). Lemma 3.4. Let < σ < π/2 and s R. Set N = N {}. Then, for any λ Σ σ, and multi-index α = α 1,..., α n ) N n, we have D α λ + µ 2 ) s Cα,s,µ,σ λ 1/2 + ) 2s α. Proof. D α λ + µ 2 ) s Setting ft) = t s, by the Bell formula and Lemma 3.3 we have α C α f l) λ + µ 2 ) l=1 α 1 + +α l =α α i 1 D α1 λ + µ 2 ) D α l λ + µ 2 )

18 578 Y. Shibata and S. Shimizu α C α,s,µ,σ λ 1/2 + ) 2s l) l=1 C α,s,µ,σ λ 1/2 + ) 2s α k+2l k)= α where we have used the facts that 2l k = α and k λ 1/2 + ) k = λ 1/2 + ) 2l α. k Now, we check the conditions stated in Proposition 2.6. formula, we have By the Parseval K ɛ,λ [f L2R n ) = R n λe ɛ 2 P ) ˆf) λ + µ 2 2 ) 1/2 d C σ f L2R n ) λ Σ σ, ). 3.11) Noting that τ τ λ λ + µ 2 = iµτ 2 λ + µ 2 ) 2, we have also τ τ K ɛ,λ[f = L2R n ) where we have used the fact that R n iµτ 2 e ɛ 2 P ) ˆf) λ + µ 2 ) 2 2 ) 1/2 d C σ f L2R n ) λ Σ σ, ), 3.12) iµτ 2 λ + µ 2 ) 2 C µ λ 2 σ λ + µ 2 ) 2 C σ. To continue the estimate, we use the following lemma. Lemma 3.5. For any ɛ > and multi-index α N n, there exists a constant C α independent of ɛ and such that D α e ɛ 2 C α e ɛ/2) 2 α.

19 Maximal L p-l q regularity of the Stokes problem 579 Proof. Setting ft) = e ɛt, by the Bell formula we have α D α e ɛ 2 Cα f l) 2 ) l=1 α 1 + +α l =α α i 1 D α 1 2 D α l 2 α C α e ɛ 2 ɛ l 2l α C α e ɛ/2) 2 α, l=1 where we have used the facts that D α α and e ɛ 2 ɛ 2 ) l C l e ɛ/2) 2 for some constant C l independent of ɛ and. This completes the proof of the lemma. By Lemma 3.5 and the Leibniz formula, we have Dα ) λe ɛ 2 P ) Dα i j λ + µ 2 C α,σ,µ 1 α, { τ )} λe ɛ 2 P ) i j τ λ + µ 2 C α,σ,µ 1 α for j = 1,..., n and λ, ) Σ σ, R n \ {}), which combined with Lemma 3.6 below due to Shibata and Shimizu [3, Theorem 2.3 implies that D j k ɛ,λ x) C σ,µ x n+1), { D j τ } τ k ɛ,λx) C σ,µ x n+1) 3.13) for any λ Σ σ, and x R n \ {}. Lemma 3.6. Let N and n be a non-negative integer and positive integer, respectively. Let < σ 1 and set s = N + σ n. Let lσ) be a number defined in such a way that lσ) = when < σ < 1 and lσ) = 1 when σ = 1. Let f) be a function in C N+lσ)+1 R n \ {}) which satisfies the following two conditions: 1) D γ f L 1R n ) for any multi-index γ N n with γ N. 2) For any multi-index γ N n with γ N lσ) there exists a number C σ such that D γ f) Cσ s γ R n \ {}).

20 58 Y. Shibata and S. Shimizu Then, there exists a constant C n,s depending essentially only on n and α such that F 1 [fx) Cn,α max γ N+1+lσ) C γ ) x n+s) x R n \ {}). In view of 3.12), 3.11) and 3.13) we can apply Proposition 2.6 and therefore, there exists a constant M n,q,σ,µ > depending essentially only on n, q, σ and µ such that R{K ɛ,λ λ Σ σ, }) M n,q,σ,µ, { R τ }) τ K ɛ,λ λ Σ σ, M n,q,σ,µ, 3.14) which combined with Theorem 2.8 implies that λu ɛ LqR n ) C n,q,σ,µ f LqR n ), 3.15) e γt U ɛ ) t LpR,L qr n )) C n,p,q,µ e γt F LpR,L qr n )) 3.16) for any γ. Now, we discuss the limit process. We have Since [ e u ɛ x) u ɛ x) = ɛ 2 e ɛ 2 )P ) ˆf) λ + µ 2 x). 1 e ɛ 2 e ɛ 2 = ɛ ɛ ) 2 e θɛ+1 θ)ɛ ) 2 dθ, by Lemma 3.5 and the Leibniz formula we have D α e ɛ 2 e ɛ 2 ) Cα ɛ ɛ 2 α e minɛ,ɛ )/2) 2, which implies that Dα { )} e τ τ ) l ɛ 2 e ɛ 2 )P ) λ + µ 2 C α,σ,µ ɛ ɛ α e minɛ,ɛ )/2) 2 for l =, 1. Setting

21 Maximal L p-l q regularity of the Stokes problem 581 [ e k ɛ,ɛ,λx) = ɛ 2 e ɛ 2 )P ) λ + µ 2 K ɛ,ɛ,λ[fx) = k ɛ,ɛ,λx y)fy) dy R n x), for λ Σ σ,, by the same argument as in the proof of 3.14) we have Since R{K ɛ,ɛ,λ λ Σ σ, }) M n,q,σ,µ ɛ ɛ, { R τ }) τ K ɛ,ɛ,λ λ Σ σ, M n,q,σ,µ ɛ ɛ. 3.17) u ɛ u ɛ = K ɛ,ɛ,λ[f, U ɛ U ɛ = e γt Fτ 1 [K ɛ,ɛ,λ[f [e γt F τ), combining 3.14) and Theorem 2.8 implies that u ɛ u ɛ LqR n ) C n,q,σ,µ ɛ ɛ f LqR n ), e γt U ɛ U ɛ ) LpR,L qr n )) C n,p,q,µ ɛ ɛ e γt F LpR,L qr n )). Therefore, {u ɛ } ɛ> and {U ɛ x, t)} ɛ> are Cauchy sequences in L q R n ) and L p R, L q R n )), respectively, which implies that there exist u L q R n ) and U L p R, L q R n )) such that lim u ɛ u LqR ɛ + n ) =, lim ɛ + e γt U ɛ U) LpR,L qr n )) = 3.18) for any γ, respectively. Combining 3.15) and 3.18) implies that λu LqR n ) C n,q,σ,µ f LqR n ). On the other hand, to estimate U t we use the following fact. Theorem 3.7 cf. [16). Let 1 < p <, let Ω be a domain in R n and let X be a reflexive Banach space. Let p be a conjugate exponent of p, that is 1/p + 1/p = 1. Then, L p Ω, X) = L p Ω, X ), L p Ω, X) = L p Ω, X), where Y stands for the dual space of Y.

22 582 Y. Shibata and S. Shimizu Note that C R n+1 ) is dense in L p R, L q R n )) when 1 p < and 1 q <. Given any ϕ C R n+1 ), we have e γt U t, ϕ) R n+1 = U, e γt ϕ) t ) R n+1 = U U ɛ + U ɛ, e γt ϕ) t ) R n+1, and therefore e γt U t, ϕ) R n+1 e γt ) U U ɛ ) LpR,L qr )) n γ ϕ Lp R,L q Ω)) + ϕ t Lp R,L q R n )) + e γt U ɛ ) t LpR,L qr n )) ϕ Lp R,L q R n )). Letting ɛ + and using 3.16) and 3.18), we have e γt U t, ϕ) R n+1 C n,p,q,µ e γt F LpR,L qr n )) ϕ Lp R,L q R n )), which combined with Theorem 3.7 implies that e γt U t LpR,L qr n )) C n,p,q,µ e γt F LpR,L qr n )) for any γ. Analogously, considering that k 1 ɛ,λx) = k 2 ɛ,λx) = k 3 ɛ,λx) = we can show the following estimates: [ γe ɛ 2 P ) λ + µ 2 x); [ λ 1/2 e ɛ 2 j P ) λ + µ 2 [ j k e ɛ 2 P ) λ + µ 2 x), x); γ e γt U LpR,L qr n )) C n,p,q,µ e γt F LpR,L qr n )); λ 1/2 D j u LqR n ) C n,q,σ,µ f LqR n ), e γt Λ 1/2 γ D j U LpR,L qr n )) C n,p,q,µ e γt F LpR,L qr n ));

23 Maximal L p-l q regularity of the Stokes problem 583 D j D k u LqR n ) C n,q,σ,µ f LqR n ), e γt D j D k U LpR,L qr n )) C n,p,q,µ e γt F LpR,L qr n )) for any λ Σ σ, and γ, respectively. Therefore, we have proved the existence of solutions u and U to problems 3.9) and 3.1), which satisfy the estimates: λ u, λ 1/2 u, 2 u) LqR n ) C n,q,σ,µ f LqR n ), e γt U t, γu, Λ 1/2 γ U, 2 U) LpR,L qr n )) C n,p,q,µ e γt F LpR,L qr n )), 3.19) respectively. About the estimates of the pressure terms θ and Θ, we use the estimates 3.19) and the equations 3.9) and 3.1). The uniqueness of solutions follows from the existence of solutions to the dual problem. What F = when t < implies that U also vanishes when t <. In fact, we know the estimate γ e γt U LpR,L qr n )) C n,p,q,µ e γt F LqR,L qr n )) 3.2) for γ γ with some γ >. Since F = when t <, we have γ U Lp,),L qr n )) γ e γt U Lp,),L qr n )) γ e γt U LpR,L qr n )) C e γt F LpR,L qr n )) = C e γt F Lp, ),L qr n )) C e γt F Lp, ),L qr n )). Letting γ, we have U Lp,),L qr n )) =, which implies that U = when t <. This completes the proof of Theorem Solution formula of the model problem without surface tension in the half-space. In this section we consider the following generalized resolvent problem and non-stationary Stokes equations in R n +: λu µ u + θ = f, div u = g in R n +, Su, θ)n = h on R n, 4.1) U t µ U + Θ = F, div U = G in Q +, SU, Θ)n = H on Q, 4.2) subject to the initial condition: U t= =, respectively. In order to reduce 4.1)

24 584 Y. Shibata and S. Shimizu and 4.2) to the case where g = and G =, respectively, we use the following lemma. Lemma 4.1. Let 1 < p, q < and γ. 1) For any g Ŵ q 1 R+) n Wq 1 R+), n there exists a v Wq 2 R+) n n such that div v = g in Ω and there hold the estimates: v LqR n + ) C n,q g Ŵ 1 q R n + ), v LqR n + ) C n,q g LqR n + ), 2 v LqR n + ) C n,q g LqR n + ). 2) For any G L p,,γ R, Wq 1 R+)) n Wp,R, 1 Ŵ q 1 R+)), n there exists a V such that V L p,,γ R, W 2 q R n +) n) W 1 p,,γ R, Lq R n +) n) and div V = G in Q +. Moreover, for any γ γ there hold the estimates: e γt V t, γv ) LpR,L qr+ n )) C n,q e γt G t, γg) LpR,Ŵ 1 e γt Λ 1/2 γ V C LpR,L qr+ n )) n,q e γt Λ 1/2 γ G LpR,L qr+ n )), e γt 2 V LpR,L qr n + )) C n,q e γt G LpR,L qr n + )). q R n + )), Proof. 1) Throughout the paper, given function fx) defined on R n and F x, t) defined on R n + R, f e, F e and f o, F o denote their even and odd extensions, respectively, that is { fx) for xn > f e x) = fx, x n ) for x n <, { F x, t) for xn > F e x, t) = F x, x n, t) for x n <, { fx) for xn > f o x) = fx, x n ) for x n <, { F x, t) for xn > F o x, t) = F x, x n, t) for x n <, where x = x 1,..., x n 1 ). Setting v j x) = [ ij F [g o ) 2 x), vx) = v 1 x),..., v n x)),

25 Maximal L p-l q regularity of the Stokes problem 585 we have div v = g o in R n. First, we prove that v LqR n + ) C n,q g Ŵ 1 q R n + ). From the proof of Lemma 3.2 we see that while we have v j LqR n ) C n,q g o Ŵ 1 q R n ), 4.3) g o Ŵ 1 q R n ) 2 g Ŵq 1 R+ n ). 4.4) In fact, we choose ϕ C R n ) arbitrarily and we observe that g o, ϕ) R n = R n + gx)ϕx) ϕx, x n )) dx. Since ϕx) ϕx, x n ) Ŵ 1 q, Rn +), we have g o, ϕ) R n 2 g Ŵ 1 q R+ n ) ϕ L q R+ n ) 2 g Ŵ 1 q R+ n ) ϕ L q R n ), which implies 4.4). Combining 4.3) and 4.4) yields that v j LqR+ n ) v j LqR n ) C n,q g o W 1 q R n ) 2C n,q g Ŵ 1 q R+ n ). By the Fourier multiplier theorem of S. G. Mihlin, we have v j LqR n ) C n,q g o LqR n ) 2C n,q g LqR n + ). Since D k g o = D k g) o for k = 1,..., n 1, we have D k v j LqR n ) C n,q D k g) o LqR n ) 2C n,q D k g LqR n + ) j = 1,..., n). Moreover, if we write D 2 nv k x) = [ 2 n F [D k g) o ) 2 x), we have D 2 n v k LqR n ) C n,q D k g) o LqR n ) 2C n,q D k g LqR n + ). 4.5)

26 586 Y. Shibata and S. Shimizu Since div v = g in R n, we have D 2 nv n = D n g n 1 k=1 D2 nv k in R n +, which combined with 4.5) yields that D 2 n v n LqR n + ) C n,q g LqR n + ). Summing up, we have proved the assertion 1). 2) Defining V j and V by the formulas: V j x, t) = [ ij F [G o, t) 2 x), V x, t) = V 1 x, t),..., V n x, t)), regarding t as a parameter and using the same argument as in the proof of the assertion 1), we have the assertion 2). This completes the proof of Lemma 4.1. Setting u = v + w, f = f λv µ Div Dv)), h = h µdv)n in 4.1) and U = V + W, F = F V t µ Div DV )), H = F µdv )n in 4.2), respectively, we have λw µ w + θ = f, div w = in R n +, Sw, θ)n = h on R n, 4.6) W t µ W + Θ = F, div W = in Q +, SW, Θ)n = H on Q, 4.7) subject to W t= =. By Lemma 4.1 we have f LqR n + ) f LqR n + ) + C n,q { λ g Ŵ 1 q R n + ) + µ g L qr n + ) }, λ 1/2 h LqR n + ) + h LqR n + ) λ 1/2 h LqR n + ) + h LqR n + ) + C n,q { λ 1/2 g LqR n + ) + µ g LqR n + ) }, e γt F LpR,L qr n + )) e γt F LpR,L qr n + )) { + C n,p,q e γt G t LpR,Ŵ q 1 R+ n )) + } µ e γt G LpR,L qr+ n )), e γt Λ 1/2 γ H, H) LpR,L qr+ n )) e γt Λ 1/2 γ H, H) LpR,L qr n + )) + C n,p,qµ e γt Λ 1/2 γ G, G) LpR,L qr n + )). Below, we consider 4.1) with g = and 4.2) with G =, respectively. First

27 Maximal L p-l q regularity of the Stokes problem 587 of all, we reduce these problems to the case where f = and F =. For this purpose, setting ιf = f o 1,..., f o n 1, f e n) and ιf = F o 1,..., F o n 1, F e n), let us define vx), τx)) and V x, t), Υx, t)) by the formulas: vx) = [ P )F [ιf) λ + µ 2 x), V x, t) = L 1 λ Υx, t) = τx) = 2 [ i F [ιf) 2 [ P )L F [ιf, λ) λ + µ 2 x, t), [ i F [ιf, t) x). Employing the same argument as in Section 3, we have x), v, τ) W 2 q R n ) n Ŵ 1 q R n ), λv µ v + τ = ιf, div v = in R n, λ v, λ 1/2 v, 2 v, τ) LqR n ) C n,q,σ,µ ιf LqR n ) 2C n,q,σ,µ f LqR n + ) 4.8) for any λ Σ σ, and < σ < π/2. And also, when F L p, R, L q R n +) n ) we have V L p, R, W 2 q R n ) n) W 1 p, R, Lq R n ) ), Υ L p, R, Ŵ 1 q R n ) ), V t µ V + Υ = ιf, div V = in R n, ), V t= =, e γt V t, γv, Λ 1/2 γ V, 2 V, Υ) LpR,L qr n )) C n,p,q,µ e γt F LpR,L qr n + )) 4.9) for any γ, where we have used e γt ιf LpR,L qr n )) 2 e γt F LpR,L qr n + )). Moreover, from the definition of ιf and ιf it follows that D n v n R n =, τ R n =, D n V n R n =, Υ R n = cf. Shibata and Shimizu [31, [34). Now, setting u = v + w, θ = τ + κ, h = h µdv)n in 4.1) with g = and U = V + W, Θ = Υ + Ξ, H = H µdv )n in 4.2) with G =, respectively, we have

28 588 Y. Shibata and S. Shimizu λw Div Sw, κ) =, div w = in R n +, Sw, κ)n = h on R n, 4.1) W t Div DW, Ξ) =, div W = in Q +, SW, Ξ)n = H on Q, 4.11) subject to W t= =. By 4.8) and 4.9) we have λ 1/2 h, h) LqR n + ) λ 1/2 h, h) LqR n + ) + C n,q,σ,µ f LqR n + ), e γt Λ 1/2 γ H, H ) LpR,L qr+ n )) e γt Λ 1/2 γ H, H ) LpR,L qr n + )) + C p,q,n,µ e γt F LpR,L qr n + )) for any λ Σ σ, and γ, and H = for t <. Therefore, in what follows we consider 4.1) and 4.2) under the conditions that f =, g = and F =, G =, respectively. Since Div Su, θ) = µ u θ when div u =, in what follows we consider the problems: λu µ u + θ =, div u = in R n +, µd n u j + D j u n ) = h j j = 1,..., n 1) on R n, 2µD n u n θ = h n on R n, 4.12) U t µ U + Θ =, div U = in Q +, µd n U j + D j U n ) = H j j = 1,..., n 1) on Q, 2µD n U n Θ = H n on Q, 4.13) subject to U t= = under the conditions that h Wq 1 R+) n and H L p, R, Wq 1 R+)) n H 1/2 p, R, L qr+)). n To get the solution formula to 4.12), we apply the partial Fourier transform with respect to x = x 1,..., x n 1 ) that is defined by the formula: ˆv, x n ) = R n 1 e ix to 4.12) and therefore, setting vx, x n ) dx, = 1,..., n 1 ) 4.14) we have A =, B = λµ 1 + 2, 4.15)

29 Maximal L p-l q regularity of the Stokes problem 589 λû j, x n ) + µa 2 û j, x n ) µdnû 2 j, x n ) + i j ˆθ, x n ) = x n > ), λû n, x n ) + µa 2 û n, x n ) µdnû 2 n, x n ) + D n ˆθ, x n ) = x n > ), n 1 i j û j, x n ) + D n û n, x n ) = x n > ), j=1 µ D n û j, ) + i j û n, ) ) = ĥj, ), 2µD n ĥ n, ) ˆθ, ) = ĥn, ), 4.16) where j runs through 1 to n 1. Setting û j, x n ) = α j e Axn + β j e Bxn, ˆθ, x n ) = γe Axn and inserting these formulas into 4.16), we have µα j B 2 A 2 ) + i j γ =, µα n B 2 A 2 ) Aγ =, n 1 i k α k Aα n =, k=1 n 1 i k β k Bβ n =, k=1 µaα j + Bβ j + i j α n + β n )) = ĥj, ), 2µAα n + Bβ n ) + γ = ĥn, ), where j runs through 1 to n 1. Solving 4.17) and setting we have α j = β j = { i j 2iB µb A)DA, B) 4.17) DA, B) = B 3 + AB 2 + 3A 2 B A 3, 4.18) n 1 k=1 { i j A 2 + B 2 4AB) µbb A)DA, B) + 1 µb ĥj, ), 1 α n = µb A)DA, B) { n 1 2AB } k ĥ k, ) A 2 + B 2 )ĥn, ), k=1 n 1 k=1 } i k ĥ k, ) + 2AB 2 ĥ n, ) } i k ĥ k, ) A 2 + B 2 )Aĥn, ),

30 59 Y. Shibata and S. Shimizu β n = { 1 A 2 + B 2 ) µb A)DA, B) γ = A + B { 2B DA, B) n 1 k=1 } i k ĥ k, ) 2A 3 ĥ n, ), n 1 } i k h k, ) A 2 + B 2 )ĥn, ), k=1 where j runs through 1 to n 1. Therefore, setting we have e M A, B, x n ) = e Bxn Axn, 4.19) B A [ u j x) = 2ij A µda, B) M A, B, x n) i ĥ, ) Bĥn, ) ) x ) [ e Bx n B A) j µbda, B) + u n x) = ĥ, ) x ) [ 2ij e Axn i µda, B) ĥ, ) + B A)ĥn, ) ) x ) x ) j = 1,..., n 1), 4.2) [ e + Bx n µb ĥj, ) [ A µda, B) M A, B, x n) 2Bi ĥ, ) A 2 + B 2 )ĥn, ) ) x ) [ e + Bx n B A)i µda, B) ĥ, ) + AA + B)ĥn, ) ) x ), 4.21) [ A + B)e θx) = Ax n 2Bi DA, B) ĥ, ) A 2 + B 2 )ĥn, ) ) x ), 4.22) where we have set ĥ = n 1 k=1 kĥk for the notational simplicity and denotes the Fourier inverse transform with respect to = 1,..., n 1 ), that is [g )x 1 ) = 2π) n 1 e ix g ) d. R n 1 Using the partial Fourier transform with respect to x and the Laplace trans-

31 Maximal L p-l q regularity of the Stokes problem 591 form with respect to t, we have the following solution formula to 4.13) as follows: [ U j x, t) = L 1 λ 2ij AM A, B, x n ) i L F [H,, λ) µda, B) BL F [H n,, λ) ) x, t) [ e L 1 λ Bx n B A) j L F [H,, λ) x, t) µbda, B) [ + L 1 λ 2ij e Axn i L F [H,, λ) µda, B) + B A)L F [H n,, λ) ) x, t) [ e + L 1 λ Bx n µb L F [H j,, λ) x, t) j = 1,..., n 1), [ AM A, B, U n x, t) = L 1 λ xn ) 2Bi L F [H,, λ) µda, B) 4.23) A 2 + B 2 )L F [H n,, λ) ) x, t) [ e + L 1 λ Bx n B A)i L F [H,, λ) µda, B) + AA + B)L F [H n,, λ) ) x, t), 4.24) [ A + B)e Θx, t) = L 1 λ Ax n DA, B) 2Bi L F [H,, λ) A 2 + B 2 )L F [H n,, λ) ) x, t), 4.25) where we have set L F [H, x n, λ) = R n e λt+i x Hx, x n, t) dtdx = F t F x [e γt H, x n, ), τ) λ = γ + iτ), L 1 λ [G, x n, λ)x, t) = 1 x 2π) n G, x n, λ) dτd = e γt τ R n e λt+i [G, x n, γ + iτ)x, t).

32 592 Y. Shibata and S. Shimizu 5. Technical lemmas. In this section, we show several estimates of Fourier multipliers, which will be used to estimate solution formulas obtained in Section 4. First of all, we introduce two classes of multipliers. Let < ɛ < π/2 and γ. Let mλ, ) be a function defined on Σ ɛ,γ which is infinitely many times differentiable with respect to τ and when λ = γ + iτ Σ ɛ,γ and R n 1 \ {}. If there exists a real number s such that for any multi-index α = α 1,..., α n 1 ) N n 1 and λ, ) Σ ɛ,γ R n 1 \ {}) there hold the estimates: Dα D α mλ, ) C α,ɛ,γ,µ λ 1/2 + A) s α, τ m ) τ λ, ) C α,ɛ,γ,µ λ 1/2 + A) s α 5.1) for some constant C α,ɛ,γ,µ depending on α, ɛ, γ and µ only, then mλ, ) is called a multiplier of order s with type 1. If there exists a real number s such that for any multi-index α = α 1,..., α n 1 ) N n 1 and λ, ) Σ ɛ,γ R n 1 \{}) there hold the estimates: Dα D α mλ, ) C α,ɛ,γ,µ λ 1/2 + A) s A α, τ m ) τ λ, ) C,ɛ,γ,µ λ 1/2 α + A) s A α 5.2) for some constant C α,ɛ,γ,µ depending on α, ɛ, γ and µ only, then mλ, ) is called a multiplier of order s with type 2. In what follows, we denote the set of all multipliers defined on Σ ɛ,γ R n 1 \ {}) of order s with type l l = 1, 2) by M s,l,ɛ,γ. For example, the Riesz kernel j / belongs to M,2,ɛ, j = 1,..., n 1). A function λ s = γ 2 + τ 2 ) s/2 belongs to M 2s,1,ɛ, when s. A function λ λ 1/2 belongs to M 1,1,ɛ,γ. The following lemma follows from the definition of M s,l,ɛ,γ and the Leibniz rule. Lemma 5.1. Let s 1, s 2 R. 1) Given m i M si,1,ɛ,γ i = 1, 2), we have m 1 m 2 M s1+s 2,1,ɛ,γ. 2) Given l i M si,i,ɛ,γ i = 1, 2), we have l 1 l 2 M s1+s 2,2,ɛ,γ. 3) Given n i M si,2,ɛ,γ i = 1, 2), we have n 1 n 2 M s1+s 2,2,ɛ,γ. From now on, we show several lemmas which will be used to estimate solution formulas given in Section 4.

33 Maximal L p-l q regularity of the Stokes problem 593 Lemma 5.2. Let s R and < ɛ < π/2. Let A, B and DA, B) be symbols defined in 4.15) and 4.18), respectively. Then, there exists a positive constant c depending on ɛ and µ such that c λ 1/2 + A) Re B B µ 1 λ ) 1/2 + A, 5.3) c λ 1/2 + A) 3 DA, B) 6µ 1 λ ) 1/2 + A) ) Moreover, we have B s M s,1,ɛ,, A + B) s M s,2,ɛ, and DA, B) s M 3s,2,ɛ, for any s R. If s, then A s M s,2,ɛ,. Proof. The inequalities 5.3) and 5.4) were proved in Shibata and Shimizu [31, Lemma 4.4. Now, we prove that B s M s,1,ɛ,. Employing the same argument as in the proof of Lemma 3.4, we have D α Bs Cα,ɛ,µ,s λ 1/2 + A) s α. Using the formula: τ τ B s = i2µ) 1 sτλµ 1 + A 2 ) s/2 1, we have Dα ) τ Bs C α,ɛ,µ,s τ λ 1/2 + A) s 2 α τ λ C α,ɛ,µ,s λ 1/2 + A) 2 λ 1/2 + A) s α C α,ɛ,µ,s λ 1/2 + A) s α. Combining these estimates implies that λ 1/2 + A) s M s,1,ɛ,. By the Bell formula, we have also D α As Cα,sA s α. 5.5) Since A s λ 1/2 + A) s when s, it follows from 5.5) that A s M s,2,ɛ, when s. Setting ft) = t s for t >, by the Bell formula we have α D α A + B)s Cα f l) A + B) l=1 α 1 + +α l =α α i 1 D α 1 A + B) A + B) D α l

34 594 Y. Shibata and S. Shimizu α C α,ɛ,µ,s λ 1/2 + A) s l λ 1/2 + A) l A α l=1 where we have used 5.5) and C α,ɛ,µ,s λ 1/2 + A) s A α, D α B Cα,ɛ,µ λ 1/2 + A) 1 α C α,ɛ,µ,s λ 1/2 + A)A α, c λ 1/2 + A) Re B Re A + B) A + B 2 λ µ 1 ) 1/2 + A ). Since τ τ A + B) s = 2 1 sa + B) s 1 τµ 1 λµ 1 + A 2) 1/2, by the Leibniz rule we have Dα τ ) A + B)s τ s 2 τ α! β!γ D β A + B) s 1 D γ! B 1 β +γ =α C α,ɛ,µ,s λ λ 1/2 + A) s 1 A β λ 1/2 + A) 1 A γ β +γ =α C α,ɛ,µ,s λ λ 1/2 + A) 2 λ 1/2 + A) s A α C α,ɛ,µ,s λ 1/2 + A) s A α. Combining these estimates implies that A + B) s M s,2,ɛ,. Since A, B M 1,2,ɛ, and DA, B) is a cubic polynomial with respect to A and B, by Lemma 5.1 we have DA, B) M 3,2,ɛ,. Setting ft) = t s for t >, by the Bell formula and 5.4) we have D α DA, B)s α C α f l) DA, B)) l=1 α α 1 + +α l =α α i 1 C α,ɛ,µ,s λ 1/2 + A) 3s l) l=1 C α,ɛ,µ,s λ 1/2 + A) 3s A α. D α 1 DA, B) λ α 1 + +α l =α α i 1 DA, B) D α l 1/2 + A) 3 A α 1 λ 1/2 + A) 3 A α l

35 Maximal L p-l q regularity of the Stokes problem 595 We have τ τ DA, B) s = sτda, B) s 1 EA, B) with EA, B) = i3/2)µ 1 B + iµ 1 A + i3/2)µ 1 B 1 A 2. Since EA, B) M 1,2,ɛ, as follows from Lemma 5.1, by the Leibniz formula we have ) DA, B)s τ τ C α s τ Dα β +γ =α D β DA, B) s 1 D γ EA, B) C α,ɛ,µ,s λ λ 1/2 + A) 3s 1) A β λ 1/2 + A)A γ β +γ =α C α,ɛ,µ,s λ 1/2 + A) 3s A α. Combining these estimates implies that DA, B) s M 3s,1,ɛ,. This completes the proof of the lemma. Lemma 5.3. Let l =, 1 and < ɛ < π/2. We use the symbols defined in 4.15) and 4.19). Then, for any multi-index α N n 1 and λ,, x n ) Σ ɛ,γ R n 1 \ {}), ), we have { D α τ τ ) l e Bxn} Cα,ɛ,µ λ 1/2 + A) α e d λ 1/2 +A)x n, D α e Axn Cα A α e 1/2)Axn, { D α τ τ ) l M A, B, x n ) } Cα,ɛ,µx n or λ 1/2 )e daxn A α, where d is a positive constant which depends on ɛ and µ but is independent of α. Proof. We write 1 M A, B, x n ) = x n e 1 θ)a+θb)xn dθ. Setting ft) = e txn, by the Bell formula we have

36 596 Y. Shibata and S. Shimizu D α e 1 θ)a+θb)xn = D α f1 θ)a + θb) α C α f l) a θ)a + θb) l=1 α α 1 + +α l =α α i 1 D α 1 1 θ)a + θb) 1 θ)a + θb) D α l C α x l ne 1 θ)a+cθ λ 1/2 +A))x n 1 θ)a 1 α 1 + θ λ 1/2 + A) ) 1 α 1 l=1 1 θ)a 1 α l + θ λ 1/2 + A) 1 α l ), where we have used e 1 θ)a+θb)xn = e 1 θ)a+θre B)xn and 5.3). When θ =, we have α D α e Axn Cα x l ne Axn A l α C α e 1/2)Axn A α. 5.6) When θ = 1, we have l=1 α D α e Bxn Cα x l ne c λ 1/2 +A)x n λ 1/2 + A) l α l=1 C α e c/2) λ 1/2 +A)x n λ 1/2 + A) α. 5.7) For general < θ < 1, since we may assume that < c 1 without loss of generality, we have D α e 1 θ)a+θb)xn α C α x l ne 1 θ)a+θc λ 1/2 +A))x n 1 θ)a + θ λ 1/2 + A) ) l A α l=1 α C α x l ne c1 θ)a+θ λ 1/2 +A))x n 1 θ)a + θ λ 1/2 + A) ) l A α l=1 ) C α e c/2) 1 θ)a+θ λ 1/2 +A) x n A α, 5.8) which implies that

37 D α M A, B, x n) C α Maximal L p-l q regularity of the Stokes problem 597 = C α 1 1 e c/2)1 θ)a+θ λ 1/2 +A))x n dθx n A α e c/2)axn e θc/2) λ 1/2 x n dθx n A α. On the one hand, integrating the last formula with respect to θ, we have D α M A, B, x n) C α c/2) 1 λ 1/2 e c/2)axn A α, 5.9) but on the other hand, using the estimate: e θc/2) λ 1/2 x n 1, we have D α M A, B, x n) C α x n e c/2)axn A α. 5.1) Since τ e Bxn = i2µ) 1 x n B 1 e Bxn, by the Leibniz formula, Lemma 5.2 and 5.7) D α τ τ e Bxn) Cα x n Since β +γ =α D β τ τ B) D γ e Bxn C α x n λ 1/2 + A) 1 β e c λ 1/2 +A)x n λ 1/2 + A) γ β +γ =α C α e c/2) λ 1/2 +A)x n λ 1/2 + A) α. τ τ M A, B, x n) = i µ 1 τx 2 n 2 1 by the Leibniz rule, Lemma 5.2 and 5.8) we have D α τ τ M A, B, x n )) C α λ β +γ =α 1 λ C α x n λ 1/2 + A) 2 θb 1 e 1 θ)a+θb)xn dθ, θ λ 1/2 + A) 1 β e c/2)1 θ)a+θ λ 1/2 +A))x n A γ dθx 2 n 1 θ λ 1/2 + A)x n e c/2)1 θ)a+θ λ 1/2 +A))x n A α dθ

38 598 Y. Shibata and S. Shimizu 1 C α x n e c/4)1 θ)a+θ λ 1/2 +A))x n dθa α 1 = C α x n e c/4)a+θ λ 1/2 )x n dθa α. Therefore, by the same argument as in obtaining 5.9) and 5.1) we have D α τ τ M A, B, x n )) C α x n or λ 1/2 )e c/4)axn A α. This completes the proof of the lemma. Lemma 5.4. Let < ɛ < π/2, 1 < q < and γ and we use the symbols defined in 4.14), 4.15) and 4.19). Let m i M,i,ɛ,γ i = 1, 2), and we define the operators K j λ) j = 1, 2, 3, 4, 5) for λ Σ ɛ,γ by the formulas: [K 1 λ)gx) = [K 2 λ)gx) = [K 3 λ)gx) = [K 4 λ)gx) = [K 5 λ)gx) = [ m1 λ, ) λ 1/2 e Bxn+yn) ĝ, y n ) x ) dy n, [ m2 λ, )Ae Bxn+yn) ĝ, y n ) x ) dy n, [ m2 λ, )Ae Axn+yn) ĝ, y n ) x ) dy n, [ m2 λ, )A 2 M A, B, x n + y n )ĝ, y n ) x ) dy n, [ m2 λ, ) λ 1/2 AM A, B, x n + y n )ĝ, y n ) x ) dy n. Then, for l = 1, 2 and j = 1, 2, 3, 4, 5, the sets {τ τ ) l K j λ) λ Σ ɛ,γ } are R-bounded families in L L q R n +)), whose R bounds do not exceed some constant C n,q,ɛ,γ,µ depending essentially only on n, q, ɛ, γ and µ. Proof. In what follows, we say that the family of operator {Aλ) λ Σ ɛ,γ } has the required properties if {τ τ ) l Aλ) λ Σ ɛ,γ } are R-bounded families in L L q R n )), whose R bounds do not exceed some constant C n,q,ɛ,γ,µ which depends essentially only on n, q, ɛ, γ and µ. First, we consider K 1 λ). Setting k 1,λ x) = [m 1 λ, ) λ 1/2 e Bxn x ), we have [K 1 λ)gx) = R n + k 1,λ x y, x n + y n )gy) dy.

39 Maximal L p-l q regularity of the Stokes problem 599 We prove that there exists a constant C n,ɛ,γ,µ depending essentially only on n, ɛ, γ and µ such that k 1,λ x) C n,ɛ,γ,µ x n λ Σ ɛ,γ, x R n \ {}), 5.11) τ τ k 1,λx) C n,ɛ,γ,µ x n λ Σ ɛ,γ, x R n \ {}). 5.12) By the assumption, the Leibniz rule and Lemma 5.3 we have Using the identity: D α m 1λ, ) λ 1/2 e Bxn ) C α,ɛ,γ,µ λ 1/2 λ 1/2 + A) α e d λ 1/2 +A)x n. 5.13) e ix = k 1,λ x) can be written in the form: k 1,λ x) = α =n ix x 2 n 1 j=1 x j i x 2 e ix, j ) α ) n 1 1 e ix D α 2π m1 λ, ) λ 1/2 e Bxn) d. R n 1 Applying 5.13) to the above formula and using the change of variables: = λ 1/2 η imply that k 1,λ x) C n,ɛ,γ,µ x n R n 1 λ 1/2 λ 1/2 + ) n d = C n,ɛ,γ,µ x n R n η ) n dη. Moreover, by 5.13) we have ) n 1 1 k 1,λ x) C ɛ,γ,µ λ 1/2 e d λ 1/2 + )x n d 2π R n 1 ) n 1 1 C ɛ,γ,µn! 2π dx n ) n λ 1/2 λ 1/2 + ) n d R n 1

40 6 Y. Shibata and S. Shimizu = ) n 1 1 C ɛ,γ,µn! 2π dx n ) n 1 + η ) n dη. R n 1 Combining above two estimations implies 5.11). Recall that τ τ k 1,λ x) = [τ τ m 1 λ, ) λ 1/2 e Bxn )x ). Noting that τ τ m1 λ, ) λ 1/2 e Bxn) = τ m 1λ, ) λ 1/2 e Bxn + τ 2 ) τ 2 λ 3 2 e Bx n + m 1 λ, ) λ 1/2 τ e Bxn, τ by the Leibniz rule, the assumption and Lemma 5.3 we have τ λ 1/2 e Bxn)} τ { C α,ɛ,γ,µ λ 1/2 + A) β λ 1/2 λ 1/2 + A) γ e d λ 1/2 +A)x n β +γ =α { Dα + τ 2 } λ 3/2 λ 1/2 + A) α e d λ 1/2 +A)x n C α,ɛ,γ,µ λ 1/2 λ 1/2 + A) α e d λ 1/2 +A)x n. 5.14) Employing the same argument as in proving 5.11) by 5.13), we have 5.12) by using 5.14). Now, using Proposition 2.7, we prove that K 1 λ) has the required properties. For this purpose, in view of 5.11) and 5.12) we set k x) = C n,ɛ,γ,µ x n and we define the operator K by the formula: [K gx) = R n + k x y, x n + y n )gy) dy. We prove that K is a bounded linear operator on L q R n +), whose bound does not exceed a constant C n,ɛ,γ,µ. By the Young inequality we have K [g, x n ) LqR n 1 ) C n,ɛ,γ,µ k, x n + y n ) L1R n 1 ) g, y n ) LqR n 1 ) dy n g, y n ) LqR n 1 ) x n + y n dy n. 5.15)