On the local wellposedness of free boundary problem for the Navier-Stokes equations in an exterior domain

This paper deals with the local well-posedness of free boundary problems for the Navier-Stokes equations in the case where the fluid initially occupies an exterior domain \begin{document} $Ω$ \end{document} in \begin{document} $N$ \end{document} -dimensional Euclidian space \begin{document} $\mathbb{R}^N$ \end{document} .


1.
Introduction. This paper deals with the local well-posedness of free boundary problems for the Navier-Stokes equations in an exterior domain. The problem is formulated as follows. Let Ω t be a domain in R N occupied at time t by an incompressible viscous fluid of density ρ and viscosity coefficient µ, where ρ and µ are positive constants. Assume that O t = R N \ Ω t is a bounded domain. Let Γ t and n t be the boundary of Ω t and the unit outer normal to Γ t , respectively. We consider the free boundary problem of the Navier-Stokes equations formulated by      ρ(∂ t u + u · ∇u) − Div (µD(u) − pI) = 0, div u = 0 in ∪ 0<t<T Ω t × {t}, (µD(u) − pI)n t = 0, V Γt = n t · u on ∪ 0<t<T Γ t × {t}, u| t=0 = u 0 , Ω t | t=0 = Ω 0 .
(1) Here, u = (u 1 , . . . , u N ) is an N -vector of functions describing the velocity field, where M denotes the transposed M , p a scalar function describing the pressure field, and u 0 = (u 01 , . . . , u 0N ) the prescribed initial data. Further, D(u) = ∇u + ∇u is the doubled deformation tensor, I is the N × N identity matrix, and V Γt is the evolution speed of Γ t along n t . Moreover, for any matrix field K with (i, j) th component K ij , the quantity Div K is an N vector whose i th component is N j=1 ∂ j K ij (∂ j = ∂/∂x j ), and for any N vector w = (w 1 , . . . , w N ), let div w = N j=1 ∂ j w j and the quantity w · ∇w be an N -vector with i th component N j=1 w j ∂ j w i . The unknowns are Ω t , u and p. We write Ω 0 , Γ 0 and n 0 as Ω, Γ and n, respectively, which are given.
As Ω t is unknown, Ω t is usually transformed to some fixed domain. One method is to use the Hanzawa transform (cf. [16, p.34]). However, this method is not applicable in our case. In fact, from the maximal regularity point of view, the W 3−1/q q regularity of a height function defining a free boundary is required, and such regularity is not guaranteed in our case. This regularity is, for example, guaranteed by the surface tension, but we do not take surface tension into account. Another known method is to use the Lagrange transformation (cf. Solonnikov, [30] and Shibata [21]). However, our final goal is to prove the global well-posedness, so this transformation does not fit our purpose. Let v be a velocity field in the Lagrangian coordinate. Nonlinear terms of the form: first derivatives of t 0 v(ξ, s) ds multiplied by second derivatives of v appear in the equations transformed from Eq. (1). As the reference domain Ω is unbounded, solutions of the linearized equations have only a polynomial decay rate (cf. Shibata -Shimizu [27]), which is not sufficient to control the term: first derivatives of t 0 v(ξ, s) ds times the second derivatives of v. To overcome this difficulty, our idea is to use the Lagrange transformation only near the boundary. Let R be a fixed positive number such that O = O 0 ⊂ B R/2 , where B L = {x ∈ R N | |x| < L}, and let κ be an element in C ∞ 0 (R N ) such that κ(x) = 1 for |x| ≤ R and κ(x) = 0 for |x| ≥ 2R. Let v(ξ, t) be a velocity field in the Lagrange coordinate ξ = (ξ 1 , . . . , ξ N ) with v ∈ H 1 p ((0, T ), L q (Ω)) ∩ L p ((0, T ), H 2 q (Ω)).
Remark 1. For small σ > 0, the transformation: x = ξ + t 0 κ(ξ)v(ξ, s) ds = X v (ξ, t) is injective. Thus, problem (1) admits unique solutions: To explain the reason why we use a partial Lagrange transform x = X v (ξ, t) with a cut-off function κ, we explain our strategy to prove the global well-posedness, which is the final goal of our study of Eq. (1). The global well-posedness is proved by the prolongation of local (in time) solutions of problem (13) to any time interval with the help of uniform estimations of several different L p -L q space-time norms of local (in time) solutions. To derive uniform estimates, we divide t 0 ∇(κv) ds into two parts as follows: because the second term of the right hand side gives the decay properties of nonlinear terms. By Taylor's formula, we divide V 0ij into two parts as follows: LetS(v, q) and div v be operators obtained from µD(u) − pI and div u by the change of variables: x = ξ + T 0 ∇(κv) ds, and then Eq.
Here,F(v),G(v),G(v) andH(v) denote nonlinear terms defined similarly to F(v), G(v), G(v), and H(v), respectively. The important point is that these nonlinear terms consist of To obtain uniform estimates of local (in time) solutions, it is essential to prove the so called L p -L q decay estimates: for any t > 0, hold for any solutions of the Stokes equations: provided that 1 < q ≤ p ≤ ∞ and q = ∞. One reason for using the cut-off function κ is that ∂ t w − J −1 DivS(w, r) = ∂ t w − Div (µD(w)−rI) and div w = div w outside of B R , so the system of equations in (18) can be regarded as a compact perturbation from the usual Stokes equations with a free boundary condition. The L p -L q decay properties corresponding to solutions of (17) have been proved by Shibata and Shimizu [27]. In Shibata [25], it is shown that choosing σ > 0 to be sufficiently small gives (17).
Another reason for using κ is that the nonlinear terms can be estimated as follows: for any r with 1 ≤ r ≤ q, where C R is a constant depending on R, r and q.
Here, the fact that the support of κ is contained in B 2R is essential. From these considerations, we finally drive a sufficiently uniform estimate of the local (in time) solutions to prove the global well-posedness of problem (13) [24]. The local and global well-posedness of free boundary problems of the Navier-Stokes equations have been studied in the bounded domain, half-space, exterior domains, and layer cases by many authors: (e.g., [1,3,4,5,6,10,11,12,13,14,15,16,18,21,23,30,31,34,35,36], and references therein). However, none of the papers mentioned above deal with global well-posedness in the exterior domain case.
To prove the global well-posedness of the quasilinear parabolic equations in unbounded domains such as the whole space or half space, where only polynomial decay properties can be obtained, it is essential to treat the problem in an L p -L q maximal regularity framework (cf. [18], [19], [24]). This is because L p integrability on the whole time interval (0, ∞) is guaranteed by choosing a suitably large index p with the help of the pointwise time decay property of solutions in the L q space. From this point of view, the maximal L p -L q regularity theorem for the Stokes equations with free boundary conditions is another subjects of this paper. This theorem is proved by combining the existence of R bounded solution operators for the corresponding generalized resolvent problem [20,22] with the Weis operator valued Fourier multiplier theorem [37].
The present paper is organized as follows. In Sect. 2, we give several preparatory results concerning R bounded solution operators, Bessel potential spaces, some rapidly decaying functions that attain given functions in B (Ω), and the maximal L p -L q regularity theorem for the free boundary problem of the Stokes equations in a uniform C 2 domain. In Sect. 3, the maximal L p -L q regularity theorem for the Stokes equations with free boundary conditions is proved in the exterior domain case. The key is to prove the unique existence theorem for solutions to the weak Dirichlet problem. In Sect. 4, the local well-posedness is proved. In Appendix A, some comments about the uniqueness of solutions of the weak Dirichlet problems are given, because there is a counter example for the uniqueness theorem in the strong Dirichlet problem case. In Appendix B, the unique existence theorem for the strong Dirichlet-Neumann problem (Lemma 3.4) is studied in the bounded domain case. Lemma 3.4 seems to be well-known, but the author could not find any proof, thus Lemma 3.4 is proved, because it gives an important step in our proof.
Notation. Finally, we explain symbols used in this paper. We denote the sets of all complex numbers, real numbers and natural numbers by C, R, and N, respectively.
In particular, for scalars, θ, and N -vectors, u = (u 1 , . . . , u N ), functions and n ∈ N 0 , we set ∇ n θ = (∂ α x θ | |α| = n) and ∇ n u = (∇ n u j | j = 1, . . . , N ). In particular, ∇ 0 θ = θ, ∇ 1 θ = ∇θ, ∇ 0 u = u and ∇ 1 u = ∇u. We use bold lowercase letters to denote N -vectors and bold capital letters to denote N × N matrices. For an N vector a, a i denotes the i th component of a and for an N × N matrix A, A ij denotes the (i, j) th component of A, and moreover, the N × N matrix whose (i, j) th component is K ij is written as (K ij ). Let δ ij be the Kronecker delta symbol, that is δ ii = 1 and δ ij = 0 for i = j. In particular, I = (δ ij ) is the N × N identity matrix. For any N -vectors a and b, let a · b =< a, b >= N j=1 a j b j . For any N -vector a, let a τ = a− < a, n > n. Given 1 < q < ∞, let q = q/(q − 1). For L > 0, let B L = {x ∈ R N | |x| < L} and S L = {x ∈ R N | |x| = L}. Throughout this paper, R > 0 is a fixed positive number , and B s q,p (G) be the standard Lebesgue, Sobolev, and Besov spaces on G, and let · Lq(G) , · H m q (G) , and · B s q,p (G) denote their respective norms. We write L q (G) as H 0 q (G), and B s q,q (G) as simply W s q (G). For a Banach space X with norm · X , let where dω denotes the surface element on ∂G. For 1 ≤ p ≤ ∞, L p ((a, b), X) and H m p ((a, b), X) denote the standard Lebesgue and Sobolev spaces of X-valued functions defined on an interval (a, b), and · Lp((a,b),X) , · H m p ((a,b),X) denote their respective norms. Let C ∞ 0 (G) be the set of all C ∞ functions whose supports are compact and contained in G. For two Banach spaces X and Y , X + Y = {x + y | x ∈ X, y ∈ Y }, L(X, Y ) denotes the set of all bounded linear operators from X into Y and L(X, X) is written simply as L(X). For a domain U in C, Hol (U, L(X, Y )) denotes the set of all L(X, Y )-valued holomorphic functions defined on U . Let R L(X,Y ) ({T (λ) | λ ∈ U }) be the R norm of the operator family T (λ) ∈ Hol (U, L(X, Y )). Let Moreover, the letter C denotes a generic constant and C a,b,c,··· denotes that the constant C a,b,c,··· depends on a, b, c, · · · . The value of C and C a,b,c,··· may change from line to line.

2.
Preliminaries. In this section, Ω is assumed to be a uniform C 2 domain and n denotes the unit outer normal to the boundary Γ of Ω.
2.1. R bounded solution operators. In this subsection, we quote a result given by Shibata [20,22] concerning the R bounded solution operators for the generalized resolvent problem of the Stokes equations with free boundary conditions formulated as follows: To state results concerning the R-bounded solution operators, we first consider the weak Dirichlet problem: whereĤ 1 r,0 (Ω) = {u ∈ L r,loc (Ω) | ∇u ∈ L r (Ω) N , u| Γ = 0} for r ∈ (1, ∞). Definition 2.1. Let 1 < q < ∞. We say that the weak Dirichlet problem is uniquely solvable with an exponent q ∈ (1, ∞), if the following assertion holds: For any f ∈ L q (Ω) N there exists a unique θ ∈Ĥ 1 q,0 (Ω) that satisfies the variational equation (20) and the estimate: ∇θ Lq(Ω) ≤ C q f Lq(Ω) for some constant C q independent of f , θ and ϕ.
Definition 2.2. Let X and Y be two Banach spaces. A family of operators T ⊂ L(X, Y ) is said to be R-bounded on L(X, Y ), if there exist constants C > 0 and q ∈ [1, ∞) such that for each n ∈ N, {T j } n j=1 ⊂ T , {f j } n j=1 ⊂ X and for all sequences {r j (u)} n j=1 of independent, symmetric, {−1, 1}-valued random variables on [0, 1], the following inequality holds: The smallest such C is called the R-bound of T on L(X, Y ), which is denoted by The following theorem was given by Weis [37].
Theorem 2.3. Let X and Y be two UMD Banach spaces and 1 < p < ∞. Let M be a function in C 1 (R\{0}, L(X, Y )) such that for some constant κ. Then, the operator 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(Lp(R,X),Lp(R,Y )) ≤ Cκ for some positive constant C depending on p, X and Y . Here, F and F −1 denote the Fourier transform and inverse Fourier transform, and S(R, X) is the Schwartz space of X-valued functions defined on R.
Remark 3. For the definition of UMD space, we refer to a book written by Amann [2, p.141]. For 1 < q < ∞, L q (Ω) and H m q (Ω) (m ∈ N) are both UMD spaces. Finally, we consider the divergence equation: div u = g. Let DI q (Ω) be a space defined by where H 1 r,0 (Ω) = {ϕ ∈ H 1 r (Ω) | ϕ| Γ = 0}. Let G(g) = {H ∈ L q (Ω) N | div G = div H} and [G(g)] denotes the representative elements of the set G(g). Where there is no possibility of confusion, we write [G(g)] as G(g) for simplicity. DI q (Ω) is the data space for the divergence equation: div u = g in Ω. As C ∞ 0 (Ω) ⊂ H 1 q ,0 (Ω), div G(g) = g in a distribution sense. Moreover, if g = div G for some G ∈ L q (Ω) N , then G = G(g).
The following theorem, which has been proved by Shibata [20,22], states the existence of an R bounded solution operator for problem (19).
Theorem 2.4. Let 1 < q < ∞ and 0 < < π/2. Assume that Ω is a uniform C 2 domain and that the weak Dirichlet problem is uniquely solvable with exponents q and q . Let Then, there exist a constant λ 0 ≥ 1 and operator families: A(λ) and P(λ) with (18), and with a constant γ * > 0 for = 0, 1. Here and in the following, λ represents a complex number with λ = γ + iτ ∈ C.
Remark 4. In Theorem 2.4, F 1 , F 2 , F 3 , F 4 , F 5 , and F 6 are variables corresponding to f , λG(g), λ 1/2 g, g, λ 1/2 h, and h, respectively. The norms · Xq(Ω) and · Xq(Ω) of the spaces X q (Ω) and X q (Ω) are defined by . Let X be a UMD Banach space. For any 1 < p < ∞ and α ∈ R, let where Using the Weis operator valued Fourier multiplier theorem and employing the same argument as in Calderón [7], we can extend the properties of Bessel potential spaces on R to the X-valued case. For example, we have where (·, ·) [σ] denotes a complex interpolation functor.
). In the following, we prove Proposition 1. A lemma given by Enomoto-Shibata [8,Theorem 3.3] is used to prove the R-boundedness of operator families.
Lemma 2.5. Let 1 < q < ∞ and let Σ be a set in C. Let m(λ, ξ) be a function defined on Σ × (R N \ {0}) such that for any multi-index α ∈ N N 0 , there exists a constant C α depending on α and Σ such that for some constant C depending solely on q and N .
To prove Proposition 1, in view of Theorem 2.3, it is sufficient to prove that there for some constant γ. In fact, setting (28) and using Theorem 2.3 gives , which shows Proposition 1.
In the following, we construct Φ k 1 and Φ k 2 . First, we consider the case where where ∆f = N j=1 ∂ 2 j f , and F and F −1 ξ denote the Fourier transform and inverse Fourier transform on R N defined by for some constant γ 0 . Next, we consider the case where Ω = R N and then by (31) where γ 1 = Cγ 0 for some constant C > 0. In fact, the second formula in (31) gives which yields the second formula with = 0 in (32). Other formulas in (32) can be obtained analogously. Finally, we consider the case where Ω is a uniform C 2 domain. We need the following lemma, which can be proved by employing the same argument as that in the appendix of [8].

Proposition 2.
Let Ω be a uniform C 2 domain in R N . Then, there exist constants M > 0, d > 0, c 0 > 0, an open set U ⊂ Ω, at most countably many N -vectors of functions Ψ j ∈ C 2 (R N ) N and points x j ∈ Γ such that the following assertions hold: (i) The maps: There exists an integer L ≥ 2 such that any L + 1 distinct sets of {B j | j = 1, 2, 3, . . .} have an empty intersection, thus for any r ∈ [1, ∞) there exists a constant C r,L such that Let Ψ −1 j = (a j1 , . . . , a jN ), and then (31) and (32), By the Minkowski inequality and (33), which shows that Analogously, we can show that the rest of (29) and (30) hold, which completes the proof of Proposition 1.

An extension map defined on
For a given g ∈ B (Ω) . Thus, it suffices to prove (34) in the case where Ω = R N . Let |t| , by the Fourier multiplier theorem we have We observe that Analogously, by (35) we have . Thus, we have (34).
Then, A generates a C 0 analytic semi-group {T (t)} t≥0 on J q (Ω). As < (µD(u) − K(u)I)n, n >= div u = 0 on Γ, D q (Ω) is given by for any t > 0 with some constants C and γ. Thus, by a real interpolation method similar to that in Subsec. 2.3 we have the following theorem (cf. [28]). Theorem 2.6. Let 1 < p, q < ∞ and assume that the weak Dirichlet problem is uniquely solvable for an index q. Let D p,q (Ω) = (J q (Ω), D q (Ω)) 1−1/p,p . Then, for any u 0 ∈ D p,q (Ω), problem (39) admits unique solutions u and p with q ,0 (Ω)) for any T > 0 satisfying the estimate: for any γ ≥ γ 0 for some γ 0 , where the constant C is independent of γ as far as γ ≥ γ 0 .
Next, we consider the initial-boundary value problem: We then have the following theorem.
Theorem 2.7. Let 1 < p, q < ∞ with 2/p + 1/q = 1 and 0 < T < ∞. Assume that the weak Dirichlet problem is uniquely solvable for an index q. Let which satisfy the compatibility condition: and, in addition, for 2/p + 1/q < 1. Then, problem (45) admits unique solutions u and p with for some positive constants C and γ.
Proof. Let ψ(t) be a function in C ∞ (R) such that ψ(t) = 1 for t > −1 and ψ(t) = 0 for t < −2. Let F be the zero extension of f , that is F(·, t) = f (·, t) for t ∈ (0, T ) and F(·, t) = 0 for t ∈ (0, T ). Let G = ψ(t)g and H = ψ(t)h, and then for any because G(G) = ψ(t)G(g). We consider the equations: where L and L −1 denote the Laplace transform and the inverse Laplace transform defined by As A(λ) and P(λ) are the R bounded operator families stated in Theorem 2.4, we have from Theorem 2.3 that there exists some γ 0 such that for any γ ≥ γ 0 the following holds: for any S > 0 with some constant C γ that depends on γ and is independent of S > 0. As F = f on (0, T ), v and q satisfy Eq. (45), except for the initial condition. Thus, we have to compensate the intial condition. Let w and r be solutions of the Cauchy problem: in Ω.
By (2) and (53) with S = 1, we know that v| t=0 ∈ B . By the compatibility condition in (47), Moreover, in the case where 2/p + 1/q < 1, by the compatibility condition in (48) Thus, u 0 − v| t=0 ∈ D q,p (Ω), and so by Theorem 2.4, there exist w and r with (Ω) . Summing up, u = v+w and p = q+r satisfy Eq. (45) and the estimate in (50). The uniqueness follows from the existence of solutions to the dual problem (cf. [28]), and so the proof of Theorem 2.7 is completed.
3. Maximal L p -L q regularity in exterior domains. In this section, let Ω be an exterior domain whose boundary Γ is a compact hypersurface of C 2 class. We consider the non-stationary Stokes equations with free boundary conditions: Then, we will show the following theorem.
As the unique solvability of the weak Dirichlet problem is the core of the maximal L p -L q theorem for the Stokes equations with free boundary conditions, in the following we give a new proof of Theorem 3.2, whose idea is completely different than that in all the above studies.
A proof of Theorem 3.2. First, we consider the variational problem: To construct a solution of the variational problem (59), we may assume that f ∈ and then u satisfies (59) by the divergence theorem of Gauß. Moreover, adding some constant, if necessary, we see that u satisfies the estimate: by the Fourier multiplier theorem of the Marcinkiewicz-Mikhlin-Hörmander type and Poincaré's inequality. Let S ∞ be an operator acting on f defined by S ∞ f = u, and then we have the following lemma. Next, we consider the variational equation: whereĤ 1 q ,0 (Ω 5R ) = {v ∈ H 1 q (Ω 5R ) | v| Γ = 0}, and then we have Lemma 3.4. Let 1 < q < ∞. Then, there exists an operator S 0 : L q (Ω 5R ) N → H 1 q,0 (Ω 5R ) such that for any f ∈ L q (Ω 5R ) N , u = S 0 f is a unique solution of the variational equation (62) possessing the estimate Remark 8. For the sake of completeness of the paper as much as possible, we give a sketch of proof of Lemma 3.4 in Appendix B below.
On the other hand, for any v ∈Ĥ 1 For where |D R1,R2 | means the Lebesgue measure of D R1,R2 , by Poincaré's inequality, we have which yields that . In this case, and so In view of (68), (69), (70), and (72), Rf ∈ H q (R N ) and for any f ∈ L q (Ω). In the following, for any f ∈ H q (R N ) we will look for u ∈Ĥ 1 q,0 (Ω) that is a solution of the variational equation: possessing the estimate: for some constant C > 0. Let E N (x) be a fundamental solution of −∆, that is where ω N is the area of the unit sphere S 1 = {ω ∈ R N | |ω| = 1}. Note that and then, −∆T ∞ f = f in R N . Moreover, In fact, by Sobolev's theorem for the weak singular operator (cf. [9, Theorem 9.2 and Theorem 9.3 in Sect. II.9]), we have which yields that SinceT ∞ (div g) = div (T ∞ g) ∈ L q,loc (R N ) as follows from Sobolev's theorem for the weak singular operator and since ∇T ∞ (div g) L q (R N ) ≤ C g L q (R N ) as follows from the boundedness of the singular integral operator due to Calderón and Zygmund (cf. [9,Theorem 9.4 in Sect. II.9]), we have . By the boundedness of the singular integral operator, we also have Using the facts that Next, let u ∈ H 2 q (Ω 5R ) be a unique solution of the strong Dirichlet-Neumann problem: where ∂u ∂ω = (x/|x|) · ∇u. Let T 0 be an operator defined by T 0 f = u, and then we have For f ∈ H q (R N ), let c f be a constant such that Moreover, by (80) and (84),  (84), we have T f ∈Ĥ 2 q,0 (Ω), for any f ∈ H q (R N ). Moreover, noting that ∇ϕ ∞ = −∇ϕ 0 , ∆ϕ ∞ = −∆ϕ 0 , ∆T ∞ f = ∆T 0 f = −f , and using (83), for any v ∈Ĥ 1 and then we have Since 1 ∈Ĥ 1 q (R N ) and f ∈ H q (R N ), we have (M, 1) R N = 0, and so Mf ∈ H q (R N ). Moreover, M is a compact operator on H q (R N ). In fact, let {f j } ∞ j=1 be a bounded sequence in H q (R N ), that is f j ∈ H q (R N ) and f j Lq(R N ) ≤ K with some constant K. By (89) and the Rellich compactness theorem, passing to a subsequence and writing it {j} again, we see that there exists an The following lemma is a key to prove the solvability of the variational equation (76).
Lemma 3.5. Let 1 < q < ∞. Let T be the operator defined by (87). Assume that if f ∈ H q (R N ) satisfies f + Mf = 0, then T f = 0. Then, for any f ∈ H q (R N ) problem (76) admits a solution u ∈Ĥ 2 q,0 (Ω) satisfying the estimate: Proof. In view of (88), to prove Lemma 3.5 it suffices to prove the existence of the inverse operator (I + M) −1 , which is a bounded linear operator on H q (R N ). Since M is a compact operator on H q (R N ), in view of the Riesz-Schauder theorem, in particular the Fredholm alternative principle, it suffices to prove that the kernel of the operator I + M is trivial. Thus, let f be an element in H q (R N ) such that (I+M)f = 0. Our task is to prove that f = 0. By the assumption, we have T f = 0, which implies that where we have used the definition: ϕ ∞ = 1 − ϕ 0 . By (64) and (94), and then noting (83) and (84), we see that w ∈ H 2 q (B 5R ) and that w satisfies the Neumann problem: On the other hand, what and T ∞ f satisfies the Neumann problem (95), which, combined with the uniqueness, leads to T ∞ f = T 0 f + c in Ω with some constant c. However, (85) holds, and so c = 0. Namely, Proof. First, we solve (76). In view of Lemma 3.5, it is sufficient to prove that T f = 0 provided that f ∈ H q (R N ) and f +Mf = 0. Let ω be a function in C ∞ 0 (R N ) such that ω(x) = 1 for |x| ≤ 1 and ω(x) = 0 for |x| ≥ 2, and let ω L (x) = ω(x/L). Since 2 ≤ q < ∞, by (89) T f ∈ H 2 q ,loc , which, combined with T f | Γ = 0, leads to ω L T f ∈Ĥ 1 q ,0 (Ω). Thus, by (88) Let L > 7R , and then supp ∇ϕ ∞ ∩ supp ∇ω L = ∅. Recalling that as L → ∞, which leads to ∇T f L2(Ω) = 0. Thus, T f is a constant. However, T f | Γ = 0, and so T f = 0. Namely, in the case 2 ≤ q < ∞, the assumption in 4. Local well-posedness. In this section, we prove Theorem 1.1. Let T and L be positive numbers determined later and let Since T > 0 is chosen small enough eventually, we may assume that 0 < T ≤ 1. Given w ∈ I T , let v be a solution of linear equations: To solve (100), we use Theorem 2.7. To this end, we have to extend the right hand side of the equations (100) for all t ∈ R. Let h be a function defined on (0, T ) such that h| t=0 = 0, and then an operator e T acting on h is defined by Since h| t=0 , we have Let T (t) be the operator given in Subsect. 2.3, and let ψ(t) be a function in C ∞ (R) such that ψ(t) = 1 for t > −1 and ψ(t) = 0 for t < −2. Let w ∈ I T , and let Note that r T w(·, t) = w(·, t) for 0 < t < T . For simplicity, we write G(w), G(w) and H(w) given in Eq. (13) symbolically as where < ·, · > 1 and < ·, · > 2 denote the standard inner products in R N 2 and R N , respectively. We may think that v 1 (k) is an N 2 vector of some smooth functions for |k| < 2σ, v 2 (k) an N × N matrix of some smooth functions for |k| < 2σ, and v 3 (k) an N × N 2 matrix of smooth functions for |k| < 2σ, and we may assume that v 1 (0) = 0, v 2 (0) = 0 and v 3 (0) = 0. Let t ∈ (0, T ), g = e T (G(w)), g = e T (G(w)), h = e T (H(w)).
In particular, we have Since as follows from r T w = w for 0 < t < T and since div G(w) = G(w) for 0 < t < T , we have div g = g for any t ∈ R, that is G(g) = g.
Let v and q be solutions of the linear equations: and then v and q are also solutions of the equations (100), because f = F(w), g = G(w), g = G(w) and h = H(w) for t ∈ (0, T ). In the following, using the Banach fixed point theorem, we prove that there exists a unique v ∈ I T such that v = w, which is a required solution of Eq. (13). Applying Theorem 2.7 gives that provided that the right hand side in (104) is finite. In the following, C denotes generic constants independent of T and L. Note that Recalling the definition of F 1 and F 2 given in (11) and (12), by (105) and (5) we have w(·, s) H 2 q (Ω) ds ∇w(·, t) L∞(Ω) + w(·, t) L∞(Ω) ∇w(·, t) Lq(Ω) .

By the Sobolev inequality and
(Ω) ), which, combined with (119) and (108), leads to Analogously, Finally, we estimate ∂ t g. To this end, we write where v 2 (k) = ∇ k v 2 (k). By (108), Choosing T > 0 so small that (Ω) LT 1/p , and so choosing T > 0 in such a way that C q κ H 1 ∞ (Ω) LT 1/p ≤ σ, we have Thus, v ∈ I T . Let Q be a map defined by Qw = v, and then Q maps I T into itself. Analogously, we can show that for any w 1 , w 2 ∈ I T , holds. Choosing T smaller if necessary, we may assume that C(L + S)(T 1 p + T 1 p + T q−N pq ) ≤ 1/2, and so Q is a contraction map on I T . By the Banach fixed point theorem, there exists a unique v ∈ I T such that Qv = v, which is a required unique solution of problem (13). This completes the proof of Theorem 1.1.
Analogously, we can prove the following theorem, which is used to prove the global well-posedness.
Theorem 4.2. Let 2 < p < ∞, N < q < ∞ and T > 0. Let Ω be an exterior domain in R N (N ≥ 2), whose boundary Γ is a C 2 compact hypersurface. Assume that 2/p + N/q < 1. Then, there exists an 0 > 0 depending on T such that if initial data u 0 ∈ B Here, C is a constant independent of 0 and T .
Appendix A. Remark on the uniqueness of the weak Dirichlet problem.
The results in the half-space can be extended to the bent half space case. Thus, by constructing a parametrix with the help of partition of unity, we can prove Theorem B.2, which completes the proof of Theorem B.2.