On the Nonstationary Stokes System in a Cone ( L p Theory)

. The authors consider the Dirichlet problem for the nonstationary Stokes system in a threedimensional cone. In a previous work they had proved the existence and uniqueness of solution in weighted L 2 Sobolev spaces, where the weights are powers of the distance from the vertex of the cone. Now they extend these results to weighted L q,p Sobolev spaces.


Introduction
The present paper deals with the Dirichlet problem for the nonstationary Stokes system in a threedimensional cone K, i. e., we consider the problem u(x, t) = 0 for x ∈ ∂K, t > 0, u(x, 0) = 0 for x ∈ K.
Elliptic boundary value problems (including the stationary Stokes system) in domains with conical points or edges are well-studied in many papers (see e. g. the bibliography in the monographs [9,10,24,26]). For the stationary Stokes system, we refer also to [3,11,[21][22][23]29]. However, parabolic problems in domains with singularities on the boundary had not been researched as intensively until now. There are several papers dealing with the behavior of solutions of the heat equation near conical points and edges (see [2,8,14,19,25,30,31]). In [12,13,15], the solutions of second order parabolic equations with time-dependent coefficients in cones and wedges were studied. A theory for general parabolic equations (not systems) with time-independent coefficients in domains with conical points was developed in the papers [5][6][7]. This theory includes, in particular, existence, uniqueness and regularity theorems for solutions in weighted L 2 Sobolev spaces and a description of the behavior of the solutions near the conical point. However, the class of problems considered in [5][6][7] does not include the Stokes system, and an extension of the results to the Stokes system is not trivial. In our recent paper [16], we were able to prove an existence and uniqueness theorem for solutions of the nonstationary Stokes system in weighted L 2 Sobolev spaces. An analogous result for a two-dimensional angle was proved in [28]. The existence and uniqueness results in [16] and [28] show that there are essential differences between the Stokes system on one hand and the heat equation (and the class of parabolic problems considered in [5][6][7]) on the other hand. In the case of the heat equation, the bounds for the weight parameter β in the existence and uniqueness theorems depend on the eigenvalues of only one operator pencil. If one deals with the Stokes system, one has to consider the eigenvalues of two different operator pencils. In [17,18], we studied the behavior of the solutions of the Stokes system at infinity and near the vertex of the cone, respectively. It turns out that the asymptotics of solutions of the Stokes at infinity differs considerably from the asymptotics of solutions of the heat equation and other parabolic problems.
The goal of the present paper is to prove the existence and uniqueness of solutions in weighted L q,p Sobolev spaces and so generalize the results in [16]. As in [16], we consider first the problem s U − ΔU + ∇P = F, −∇ · U = G in K, U = 0 on ∂K (3) with the complex parameter s which arises if one applies the Laplace transform to the problem (1), (2). A large part of the paper (Sect. 2) is devoted to this parameter-depending problem. In [16] we proved an existence and uniqueness theorem for solutions of the problem (3) in weighted L 2 Sobolev spaces, regularity assertions and estimates for the solution. In Sects. 2.2-2.5, we extend these results to weighted L p Sobolev spaces. Let V l p,β (K) be the weighted Sobolev space with the norm where r = r(x) is the distance of x from the vertex of the cone. All other function spaces which are used in the present paper are introduced in Sect. 1. In particular, we obtain the following existence and uniqueness result (see Theorem 2.3 for s = 0 and [26, Chapter 3, Theorem 5.1] for s = 0). Suppose that Re s ≥ 0, F ∈ V 0 p,β (K), G ∈ V 1 p,β (K) and sG ∈ (V 1 p ,−β (K)) * , where 1 < p < ∞, p = p/(p − 1) and β satisfies the condition − min(μ 2 , λ 1 In the case s = 0, β + 3 p > 2, we assume that the integral of G over K is zero. Then the problem (3) has a uniquely determined solution (U, P ) in the space V 2 p,β (K) × V 1 p,β (K) which satisfies the estimate with a constant c independent of U , P and s. The uniqueness of the solution holds even for −(λ 1 + 1) < 2 − β − 3 p < μ 2 + 1 (see Theorem 2.2). In Sect. 2.6, we obtain a similar estimate for the derivative (U (s), P (s)) of the solution with respect to the parameter s. The numbers λ 1 and μ 2 in (5) are the smallest positive eigenvalues of certain operator pencils which are introduced in Sect. 1.3. In the case μ 2 < λ 1 + 1, the condition (5) on β is more restrictive than the analogous condition for the heat equation and other parabolic problems. But, as was shown in [16], the above condition on β is sharp. Theorem 2.3 with the condition (5) on β turns out to be insufficient to prove the existence and uniqueness of solutions of the time-dependent problem (1), (2) in weighted L p Sobolev spaces if p < 2. For this reason, we consider the parameter-depending problem (3) also for data F ∈ V 0 p,β (K) and G = 0, where β satisfies the weaker condition As is shown in Sect. 2.7, there exist solutions (U, P ) in a larger space V 2 p,β,γ (K) × V 1 p,β,γ (K) for such data. In Sect. 2.8, where we restrict ourselves to the case p = 2, we establish the asymptotics of the function U ∈ V 2 2,β,γ (K) at infinity. Here, the remainder is an element of the weighted space V 3/2 2,β− 1 2 (K) which is a subspace of V 0 p,β −2 (K) for arbitrary p ≥ 2 and β + 3 p = β + 3 2 . The second part of the paper (Sect. 3) deals with the time-dependent problem (1), (2). The goal of this part is to prove the existence and uniqueness of solutions in the space W 2,1 q,p;β (Q) × L q (R + ; V 1 p,β (K)) if β satisfies the condition (5). Here, Q = K × R + and W 2,1 q,p;β (Q) is a subspace of L q (R + ; V 2 p,β (K)) (see Sect. 1.2). The main result of the present paper is the following.
Suppose that f ∈ L q (R + ; V 0 p,β (K)), g ∈ L q (R + ; V 1 p,β (K)), g(x, 0) = 0 for x ∈ K, and ∂ t g ∈ L q (R + ; (V 1 p ,−β (K)) * ), where p, q ∈ (1, ∞) and β satisfies the inequalities (5). In the case β + 3 p > 2, we assume that g satisfies the condition K g(x, t) dx = 0 for almost all t. (8) Then there exists a uniquely determined solution (u, p) of the problem (1), (2) in the space W 2,1 q,p;β (Q) × L q (R + ; V 1 p,β (K)). In order to prove this result, we proceed as follows. First, we prove in Sect. 3.1 that for arbitrary g ∈ L q (R + ; V 1 p,β (K)), ∂ t g ∈ L q (R + ; (V 1 p ,−β (K)) * ), g(x, 0) = 0 for x ∈ K, β + 3 p ∈ {−1, 1, 2, 4}, there exists a vector function u ∈ W 2,1 q,p;β (Q) such that ∇ · u = g in Q, u(x, t) = 0 for x ∈ ∂K and u(x, 0) = 0 for x ∈ K. In the case 2 < β + 3 p < 4, we must assume for this that g satisfies the condition (8). This result allows us to restrict ourselves to the system (1), (2) with g = 0. In Sect. 3.2, we prove some a priori estimates for the solutions in weighted spaces, while Sects. 3.3-3.6 deal with the solvability of the problem (1), (2) in weighted L q,p Sobolev spaces for different p and q. In Sect. 3.3 we start with the case p = q ≥ 2. We prove that for arbitrary f ∈ L p (R + ; V 0 p,β (K)) and for g = 0, there exists a unique solution of the problem (1), (2) in the space W 2,1 p,p;β (Q) × L p (R + ; V 1 p,β (K)) if β satisfies the condition (5) (see Theorem 3.1). The method is similar to the proof of weighted L p estimates for elliptic problems in domains with edges in [20]. We use the estimates for solutions of the parameterdepending problem together with an extension of Mikhlin's Fourier multiplier theorem to operator-valued functions. In order to obtain the same existence and uniqueness result as in Theorem 3.1 for the case p = q < 2, it is not sufficient to apply a duality argument if μ 2 < λ 1 + 1. This is also a reason why we are interested in solutions of the problem (1), (2) for right-hand sides f ∈ L p (R + ; V 0 p,β (K)), g = 0, where p ≥ 2 and β satisfies the weaker condition (7). Under this condition on p and β, we show in Sect. 3.4 that there exists a solution of this problem satisfying the estimate with a constant c independent of f (see Lemma 3.11). For this, we use the results of Sects. 2.7 and 2.8 and the above mentioned Fourier multiplier theorem. Lemma 3.11 enables us to extend the result of Theorem 3.1 to the case p < 2 what is done in Sect. 3.5. In the closing Sect. 3.6 we prove the above given existence and uniqueness result in weighted L q,p Sobolev spaces for the case p = q.

Weighted Sobolev Spaces in a Cone
For nonnegative integer l and real p ∈ (1, ∞), β ∈ R, we define the weighted Sobolev space V l p,β (K) as the set of all functions (or vector functions) with finite norm (4). The space V 0 p,β (K) is also denoted by L p,β (K). The trace space of V l p,β (K), l ≥ 1, on the boundary ∂K is denoted by V l−1/p p,β (∂K). For a noninteger number s = l + σ, where l is an integer, l ≥ 0, and 0 < σ < 1, the space V s p,β (K) is defined as Note that the set C ∞ 0 (K\{0}) of all infinitely differentiable functions with compact support in K\{0} is dense in V l p,β (K) and V s p,β (K). Consequently, these spaces can be also defined as the completion of the set C ∞ 0 (K\{0}) with respect to the above norms. By or, what is the same, the space of all functions U ∈ V 1 p,β (K) which are zero on ∂K\{0}. By Hardy's inequality, there exists a constant c such that For U ∈ C ∞ 0 (K) this estimate is valid without the above restriction on β. This means that the Under the condition β = 1− 3 p , this is also true for the space V 1 p,β (K). Let l be a nonnegative integer, and let p and β be real numbers, 1 < p < ∞. Then E l p,β (K) is the weighted Sobolev space with the norm As is known, E l p,β (K) = V l p,β (K) ∩ V 0 p,β (K), and the E l p,β (K)-norm is equivalent to the norm The trace space for E l p,β (K), l ≥ 1, on the boundary is denoted by E l−1/p p,β (∂K). Furthermore, • E 1 p,β (K) is defined as the space of all functions u ∈ E 1 p,β (K) which are zero on the boundary. The function G in (3) is mostly considered as an element of the space where p ∈ (1, ∞) and p = p/(p−1). Note that there is also the equality [16] in the case p = 2).
Finally, we introduce the spaces with different weight parameters β and γ. Let ζ be a smooth (two times continuously differentiable) function on K with compact support, ζ(x) = 1 for |x| < 1, and let η = 1 − ζ. Then This means that the bigger parameter β in the space E 2 p,β,γ (K) controls the behavior of the functions near the vertex while the smaller parameter γ controls the behavior at infinity. The same is true for the space V 1 p,β,γ (K).

Weighted Sobolev Spaces in Q
Let X be a space with the norm · X , and let q ∈ (1, ∞). Then L q (R + ; X ) is the Lebesgue space of functions (or vector functions) on the interval (0, ∞) with values in X and the norm Analogously, L q (0, T ; X ) is the Lebesgue space of functions on the interval (0, T ) with values in X . For p, q ∈ (1, ∞), we define L q,p;β (Q) = L q (R + ; L p,β (K)). Furthermore, W 2,1 q,p;β (Q) denotes the space of all vector functions u in Q = K × R + such that u ∈ L q (R + ; V 2 p,β (K)) and ∂ t u ∈ L q (R + ; V 0 p,β (K)). This space is equipped with the norm In the case q = p, we will always write L p;β (Q) and W 2,1 p;β (Q) instead of L p,p;β (Q) and W 2,1 p,p;β (Q), respectively.

Operator Pencils, the Eigenvalues λ 1 and μ 2
We introduce the following operator pencils L(λ) and N (μ) generated by the Dirichlet problem for the stationary Stokes system and the Neumann problem for the Laplacian in the cone K, respectively. For every complex λ, we define the operator L(λ) as the mapping where r = |x| and ω = x/|x|. Another description of the operator pencil L is given in Sect. 3.1. As is known, the strip −1 ≤ Re λ ≤ 0 is free of eigenvalues of the pencil L(λ) (see [11] or [10, Theorem 5.5.6]) and the strip −2 ≤ Re λ ≤ 1 contains only real eigenvalues (see [10,Theorem 5.3.1]). We denote the smallest positive eigenvalue of this pencil by λ 1 . Then λ −1 = −1 − λ 1 is the greatest negative eigenvalue. The numbers λ = 1 and λ = −2 are always eigenvalues. Thus, 0 < λ 1 ≤ 1. The operator N (μ) is defined for any complex μ as where δ ω denotes the Beltrami operator on the sphere S 2 and n is the exterior normal vector to ∂Ω. The eigenvalues of the pencil N (μ) are real. The spectrum contains, in particular, the simple eigenvalues μ 1 = 0 and μ −1 = −1 with the eigenfunction φ 1 = const. By μ 2 , we denote the smallest positive eigenvalue of the pencil N (μ).

The Parameter-Depending Problem
We consider the parameter-depending problem (3). In [16] we obtained an existence and uniqueness result for solutions in the space E 2 2,β (K) × V 1 2,β (K). One goal of this section is to extend this result to weighted L p Sobolev spaces. V. Kozlov and J. Rossmann JMFM

Some Imbeddings and Estimates for Weighted Spaces in K
Let k, l be integers, 0 ≤ k ≤ l. Then it follows directly from the definitions of the norms in V l p,β (K) and The operators of these imbeddings are continuous. Using the inequality one can easily show that Hence, the imbeddings hold for arbitrary σ, 0 ≤ σ < 1, and arbitrary β, γ, β − 1 + σ ≤ γ ≤ β. Let ζ ν = ζ ν (x) be two times continuously differentiable functions on K depending only on r = |x| such that for all U ∈ V l p,β (K). An analogous result is true for the norms in V s p,β (K), E l p,β (K), in the trace spaces V l−1/p p,β (∂K) and E l−1/p p,β (∂K) and in the dual space (V l p,β (K)) * . Using (10), we can prove the following inequality.
Lemma 2.1. Suppose that 0 ≤ s < l, 1 < p < ∞ and β ∈ R. If ε is an arbitrary real number, then there exists a constant c(ε) such that for all U ∈ V l p,β (K).
Since the weighted and nonweighted Sobolev norms are equivalent on the set of functions with support in the set 1/2 ≤ |x| ≤ 2, it follows from Ehrling's lemma (see, e.g. [33, Chapter 1, with a constant C(ε) independent of ν. Using the coordinate transformation 2 ν x = y, we get with the same constants ε and C(ε). Summing up over all integer ν and using (10), we obtain (11).
The following lemma on the traces of V s p,β (K)-and E l p,β (K)-functions is essentially proved in [20]. Lemma 2.2. There exists a constant c such that for all U ∈ E l p,β (K), l ≥ 1. Proof. For the estimate (13), we refer to [20,Lemma 1.6]. In the case of integer s ≥ 1, the estimate (12) is given in [20,Lemma 1.4]. If s = l + σ ≥ 1, then the inequality (12) follows from the imbedding V l+σ p,β (K) ⊂ V 1 p,β−l−σ+1 (K) and from [20,Lemma 1.4]. It remains to prove (12) for 1 p < s < 1. Let U ν and ζ ν be defined as in the proof of Lemma 2.1. Since the W s p and V s p,β norms are equivalent on the subset of functions with the support in the set 1/2 ≤ |x| ≤ 2, we have . Using the coordinate transformation 2 ν x = y, we get this inequality (with the same constant c) for the function ζ ν U . Therefore, summing up this inequality over all integer ν, we obtain (12).
In the next lemma, we give some weighted L q estimates for functions U ∈ E 1 p,β (K), p ≥ q.
In the case q = p, this is even true β ≤ γ ≤ β + 1 − 1 p . Proof. For the case q = p, we refer to (9) and (13). Suppose that q < p.
(1) Let a(r) = r for r < 1 and a(r) = r −1 for r ≥ 1. Then Hölder's inequality implies is finite for positive ε. We can choose ε such that Then we get By Hölder's inequality, is finite for positive ε. Let ε be such that β + 3 p ≤ γ + 3 q − ε pq and γ + 3 q + ε pq ≤ β + 1 + 2 p . Then, by means of Lemma 2.2, we obtain . This proves the second assertion V. Kozlov and J. Rossmann JMFM The case p ≤ q is considered in the next lemma.
Proof. (1) Let U ∈ V s p,β (K), and let U ν andζ ν be the same functions as in the proof Lemma 2.1. Then it follows from the continuity of the imbedding W s p ⊂ W l q for bounded domains that with a constant c independent of U and ν. Using the coordinate transformation 2 ν x = y, we get the same estimate (with the same constant c) for the function ζ ν U . Since p ≤ q this implies (2) Obviously, the estimate is valid for all functions with support in the set {x ∈ K : 1 2 < |x| < 2}. In particular, the functionsζ ν U ν satisfy this estimate. Thus, by means of the coordinate transformation 2 ν x = y, we get the estimate with a constant c independent of ν. Hence, . This proves the lemma.
Finally, the following lemma can be easily proved by means of Hölder's inequality.
then F ∈ L s,δ (K) and by means of Hölder's inequality. This proves the lemma.
It follows from the last lemma that V l p,β (K) ∩ V l q,γ (K) ⊂ V l s,δ (K) for arbitrary l ≥ 0 under the conditions of Lemma 2.5 on p, q, s and β, γ, δ.

Estimates for Solutions of the Parameter-Depending Problem
In the sequel, p is always a real number in the interval 1 < p < ∞ and p = p/(p − 1). The following lemma can be easily deduced from [32, , where c is a constant independent of U , P and s.
Proof. Obviously, where By (15), we have where Φ(x) = 2 2ν F (2 ν x) and Ψ(x) = 2 ν G (2 ν x). Since V and Q vanish outside the set K 0 = {x ∈ K : 1/2 < |x| < 2}, it follows from Lemma 2.6 that , where the constant c is independent of ν and s. Here, D 2 V denotes the vector of all second order derivatives of V . One can easily check that . Furthermore, we obtain the estimates Thus, we get (14).
The last norm can be estimated by means of the following lemma.
, where σ is an arbitrary real number greater than p −1 and c is independent of U , P and s.
Now it is easy to deduce the following estimate from Lemma 2.7.
where c is independent of U , p and s.
Proof. Summing up the inequality (14) over all integer ν and using Lemma 2.8, we obtain (16) with the additional term on the right-hand side, where σ is an arbitrary real number, p −1 < σ < 1. By (11), we have Here, ε can be chosen arbitrarily small. This proves (16).

Normal Solvability of the Parameter-Depending Problem
Our goal is to show that (under certain conditions on p and β) the operator of the problem (3) has closed range if s = 0. For this, it is sufficient to show that where S is a bounded subdomain of K with positive distance from the vertex of K. where We define Then Obviously, Φ 1 and Φ 2 are functionals on Furthermore, the following assertion is true for the functional Φ 2 .
defines a linear and continuous functional on V 2 p ,1−γ (K), and the estimate is valid with a constant c independent of U and s. V. Kozlov and J. Rossmann JMFM Proof. Suppose that |s| = 1. One can easily show (cf. [16,Lemma 2.5]) that and, consequently, By Lemma 2.2, and . This proves the estimate for the (V 2 p ,1−γ (K)) * -norm of Φ 2 in the case |s| = 1. In the case |s| = 1, we consider the functions we obtain the desired estimate for the (V 2 p ,1−γ (K)) * -norm of Φ 2 in the case |s| = 1. The following lemma was proved in [16] for the case p = 2. Similarly, we can prove this lemma for arbitrary p.
is not an eigenvalue of the pencil N (μ). Then there exists a unique solution P ∈ V 1 p,β (K) of the problem (19).
where μ j are the eigenvalues of the pencil Proof. (1) The first assertion can be found e.g. in [24,Theorem 7.7.3].
(2) If 1 − γ − 3 q is not an eigenvalue of the pencil N (μ), then the operator is an isomorphism. Consequently, the adjoint operator is also an isomorphism. (3) The proof of the third assertion proceeds analogously to [16, Lemma 2.6].
As a simple consequence of the last lemma, we obtain the following regularity assertion for solutions of the problem (19).
Proof. The first assertion is obvious.
By the well-known formula for the coefficients c j in (22) which can be found, e. g., in [26, Chapter 3, Theorem 5.8], the coefficient c −1 is a multiple of Φ, 1 . Thus, (20)), the condition Φ, 1 = 0 in Corollary 2.1 means that Proof. We prove the estimate (18) for solutions of the problem (3) . If the support of (U, P ) is contained in the ball |x| < ε, then and it follows from well-known results for the stationary Stokes system that . Assume now that U (x) and P (x) are zero for |x| < N and N is sufficiently large. Then is not an eigenvalue of the pencil N (μ), then the problem (19) has a uniquely determined solution P = P 1 + P 2 ∈ V 1 p,β (K), where P j , j = 1, 2, are the (uniquely determined) solutions of the problems and Φ j are the functionals (20). Here, (see (21)). Suppose that β < γ ≤ β + 1 p and that there are no eigenvalues of the pencil N (μ) in the Then it follows from Lemmas 2.10 and 2.11 that Thus, Lemma 2.9 yields (3) with arbitrary support in K, and let S denote the . As was shown above, the vector functions (U 2 , P 2 ) and (U 3 , P 3 ) satisfy the estimate for k = 2 and k = 3 if ε is sufficiently small and N is sufficiently large. Combining the last inequalities, we obtain (18). This proves the theorem.

A Regularity Assertion for the Solution of (3)
The following lemma was proved for p = q = 2 in [16, Lemma 2.9]. We prove this for arbitrary p, q ∈ (1, ∞).
. We assume that one of the following three conditions is satisfied: is not an eigenvalue of the pencil N (μ), and the line Proof.
(1) Suppose that p = q and the condition (i) is satisfied. First, let γ < β + 1 − 1 p . Then P is a solution of the problem (19) (2) Suppose that p = q and the condition (ii) is satisfied. Then we consider first the case that γ ≥ β − 2.
Let ζ be a smooth (two times continuously differentiable) function on R + with support in [0, 1] which is equal to one on the interval (0, 1 2 ), and let η = 1−ζ. The functions ζ, η can be considered as smooth where . Analogously, we can show by induction that U ∈ E 2 p,γ (K) and Thus, the lemma is proved for the case p = q, (3) Suppose that p > q. Then it can be easily shown by means of Hölder's inequality that ζ(U, P ) ∈ are the same functions as in part 2) of the proof), it follows from parts 1) and 2) of the proof that ζU ∈ E 2 q,γ (K) and ζP ∈ V 1 q,γ (K). Analogously, we obtain ηU ∈ E 2 q,γ (K) and ηP ∈ V 1 q,γ (K). (4) We consider the case p < q.
Remark 2.1. The assertion of the last lemma is also true if β + 3 p < 2 < γ + 3 q and the interval 1 − γ − 3 q ≤ λ ≤ 1 − β − 3 p contains only the eigenvalue λ = −1 of the pencil N (μ). However, then the function G must satisfy the condition (23). In order to show this, one has to apply Corollary 2.1 instead of Lemma 2.11.

Bijectivity of the Operator (17)
First, we prove the following generalization of [ Proof. In the case s = 0, the operator (U, as is known from the elliptic theory. Let (U, P ) be a solution of the problem U ∈ E 2 p,β (K), P ∈ V 1 p,β (K), Re s ≥ 0, s = 0. We have to show that U = 0 and P = 0 under the conditions of the lemma. We consider the following cases.
(1) p = 2, 0 ≤ β ≤ 1. Then the solution coincides with the variational solution in the space . Since this solution is unique by [16, Theorem 1.1], it follows that U = 0 and P = 0.

An Estimate for the Derivatives of the Solution
Suppose that Re s ≥ 0, s = 0, and that β satisfies the condition (5). Then for arbitrary F ∈ V 0 p,β (K), there exists a uniquely determined solution (U (s), P (s)) ∈ E 2 p,β (K) × V Applying Theorem 2.3, we obtain the following estimate for the solution (U, P ) and its derivative (U , P ).
Proof. For k = 0, we refer to Theorem 2.3. Obviously, the functions U h (s) = h −1 (U (s + h) − U (s)) and P h (s) = h −1 (P (s + h) − P (s)) are elements of the spaces E 2 p,β (K) and we obtain the estimate for small |h|. This proves (27) for k = 1.
In the following lemma, we denote by ζ ν the same functions on K as in Sect. 2.1.
with a certain positive ε is satisfied for k = 0 and k = 1. The constant c in this estimate is independent of μ, ν and F .

Lemma 2.16.
Suppose that Re s ≥ 0, s = 0 and Then for arbitrary F ∈ V 0 p,β (K), there exists a unique solution (U, P ) ∈ E 2 p,β,γ (K) × V 1 p,β,γ (K) of the problem (26). This solution is independent of the choice of γ in the given interval. Furthermore, the estimate holds with a constant c independent of F and s. In particular, U ∈ V 0 p,β−2 (K) and where c is independent of F and s.
Proof. We prove the uniqueness. Suppose that (U, P ) is a solution of the problem (25) in the space . Under the conditions on β and γ, the strip 2−β− 3 is free of eigenvalues of the pencil L(λ). Thus, it follows from Lemma 2.12 that ζU ∈ E 2 p,γ (K) and ζP ∈ V 1 p,γ (K). Consequently, U ∈ E 2 p,γ (K) and P ∈ V 1 p,γ (K). Applying Theorem 2.3, we conclude that U = 0 and P = 0.
Next, we prove an estimate for the derivatives U (s) and P (s).
The function χ s introduced above is not differentiable with respect to the complex variable s. For this reason, we replace it by a holomorphic function Ψ s which is defined as follows. Let ψ be a C ∞ -function on (−∞, +∞) with support in the interval [0, 1] satisfying the conditions where N is a sufficiently large integer. By Ψ, we denote the Laplace transform of ψ. The function Ψ is analytic in C and satisfies the conditions Ψ(0) = 1, Ψ (j) (0) = 0 for j = 1, . . . , N. Hence, for Re s ≥ 0. Since s n Ψ (j) (s) is the Laplace transform of the function (−1) j d n dt n t j ψ(t) , it follows that Ψ (j) (s) ≤ c j,n |s| −n for every j ≥ 0 and n ≥ 0, where c j,n is independent of s, Re s ≥ 0. We set Ψ s (x) = Ψ(sr 2 ). Obviously, (Ψ s − χ s ) U ∈ E 2 p,β (K) ∩ E 2 p,γ (K) for arbitrary U ∈ E 2 p,β,γ (K). Hence, the cut-off function χ s can be replaced by Ψ s in Lemma 2.16.
has the form 2,β−1/2 (K). For |s| = 1, we obtain the estimate with a constant c independent of s. We prove the formula (38) for the coefficients c j,k (s). Since μ j < λ 1 ≤ 1 for j ∈ I β , the coefficients c j,k (s) are given by the formula k) ) is the solution of the problem (36) introduced above, i. e.,
. Then by means of the estimate in Theorem 2.3, we get with a constant c independent of s. The same estimate can be easily proved for the derivative with respect to s of the functions s) and, in the case μ j > β − 3/2, for the derivatives of the s). Furthermore, it follows from Lemmas 2.16 and 2.18 that for |s| = 1. Since V (s) is given by (42) and the spaces E 2 2,β+ε−1/2 (K) and V 2 2,β (K) are subspaces of V 3/2 2,β−1/2 (K), it follows that V (s) ∈ V 3/2 2,β−1/2 (K). In the case |s| = 1, we obtain the estimate with a constant c independent of F and s. It remains to prove the estimates (39) and (40) for arbitrary s = 0, Re s ≥ 0. Let (U, P ) be the solution of the problem (25). For an arbitrary positive real number a, we define U a (x, s) = U (a −1/2 x, as), P a (x, s) = a −1/2 P (a −1/2 x, as), F a (x, s) = a −1 F (a −1/2 x, as). Then and U a = 0 on ∂K. As was shown above, the function U a admits the decomposition For |s| = a, the functions V a (·, a −1 s) and d j,k (s, a) satisfy the estimates where V a (x, s) = ∂ s V a (x, s). It can be easily seen that Consequently, with the substitution a −1/2 x = y, we obtain a 1/2 y, a −1 s).
The last equality together with (37) imply V (y, s) = V a (a 1/2 y, a −1 s). Using (44) and the equalities and we obtain (39) and (40). The proof is complete.
Let ζ ν be the same smooth functions on K as in Sect. 2.1. Analogously to Lemma 2.15, we obtain the following assertion.

The Time-Dependent Problem
We pass to the problem (1), (2).

Existence of Solutions of the Equation ∇ · u = g
Suppose that g ∈ L q (R + ; V 1 p,β (K)), ∂ t g ∈ L q (R + ; (V 1 p ,−β (K)) * ) and g(x, 0) = 0 for x ∈ K. Our goal is to show that there exists a function u ∈ W 2,1 q,p;β (Q) satisfying the equation ∇ · u = g in Q, the initial condition u(x, 0) = 0 for x ∈ K and the boundary condition u(x, t) = 0 for x ∈ ∂K, t > 0.
We show this by means of solvability and regularity results for the stationary Stokes system in a thin cone. But for this, it is necessary to show that the eigenvalues of the pencil L, except λ = 1 and λ = −2 lie outside a wide strip |Re λ + 1 2 | < m of the complex plane if the domain Ω on the sphere S 2 is small. Let r, ϕ, θ denote the spherical coordinates of the point x, and let u r , u ϕ , u θ be the spherical components of the vector function u = (u 1 , u 2 , u 3 ). By [10, Lemma 3.2.1], the vector function u belongs to is the completion of the set C ∞ 0 (Ω) with respect to the norm Here, Q denotes the sesquilinear form where dω = sin θ dθ dϕ. We denote by the spherical divergence of u ω and spherical gradient of u r , respectively. As was shown in [11] (see also [10,Subsection 5.2.1]), the eigenvectors (u r , u ω , p) of the pencil L corresponding to the eigenvalue λ satisfy the integral identity (Ω) and q ∈ L 2 (Ω). We may assume without loss of generality that Ω = K ∩ S 2 does not contain the south pole (0, 0, −1). Then we define the domain Ω δ on the unit sphere S 2 as follows. Let G be the set of all (ϕ, θ) ∈ [0, 2π)×[0, π) such that (cos ϕ sin θ, sin ϕ sin θ, cos θ) ∈ Ω. For arbitrary positive δ < 1, we define Ω δ as the set of all ω ∈ S 2 which have the form ω = (cos ϕ sin θ, sin ϕ sin θ, cos θ), where (ϕ, δ −1 θ) ∈ G (see Fig. 1).
The above introduced operator pencil for the domain Ω δ is denoted by L δ . Using the representation (45), we can prove the following lemma on the eigenvalues of the pencil L δ . Proof. First note that for arbitrary g ∈ L 2 (Ω δ ), Ω δ g dω = 0, there exists a function u ω ∈ Here, the constant c can be chosen independent of δ. Indeed, we may assume that θ < π − ε with a certain positive ε for (ϕ, θ) ∈ G. Then the function h(ϕ, θ) = g(ϕ, δθ) (sin θ) −1 sin(δθ) belongs to L 2 (Ω) for arbitrary g ∈ L 2 (Ω δ ) and with a constant c independent of δ. Furthermore, the integral of h over Ω is zero.
Furthermore, one can easily verify that with a constant c independent of δ. In particular, it follows that Ω δ u r dω = 0.
Thus, for v r = u r and v ω = 0, the integral identity (45) implies wherep is the mean value of p. This implies where We estimate u ω and p −p. For v r = 0, the identity (45) takes the form Let w ω be a function and the estimate (see the statement at the beginning of the proof) For small δ, the L 2 (Ω δ )-norm of u ω is small compared to Q(u ω , u ω ). Consequently, with a constant C 3 independent of δ. This together with the above estimate for the norm of u ω yields For u r = 0, this contradicts (46) if − 1 2 < λ < m − 1 2 and Λ δ is large, i.e., δ is small. If u r = 0, then it follows that p −p = 0 and u ω = 0. In this case, (45) holds for arbitrary (v r This proves the lemma.
The last lemma together with regularity result for solutions of the stationary Stokes system allow us to prove the following assertion.
with a constant c depending on δ. As was shown above, there exists a vector function v on K δ satisfying the equation ∇ · v = h and the estimate . This means that the spherical components v r , v ϕ and v θ of v satisfy the equation Then the functions and u ϕ (r, ϕ, θ, t) = v ϕ (r, ϕ, δθ, t) satisfy the equation This means that the vector function u with the spherical components u r , u ϕ and u θ is a solution of the equation div u = g. Furthermore, there exists a constant c depending on δ such that u W 2,1 q,p;β (Q) ≤ c v W 2,1 q,p;β (Q) . Thus, u satisfies (48).

A Local Estimate
The following lemma can be easily derived from solvability and uniqueness assertions for the Stokes system in bounded domains with smooth boundary which can be found in [32].
In the case a = 1, we introduce the new coordinates x = a −1 x, t = a −2 t. Obviously, the functions v(x , t ) = u(ax , a 2 t ) and q(x , t ) = a p(ax , a 2 t ) are zero outside the area 1 2 < |x| < 2 and satisfy the equations and the initial and boundary conditions (2). Thus v, q satisfy the estimate (51), where f(x , t ), g(x , t ) are replaced by the functions a 2 f(ax , a 2 t ) and a g(ax , a 2 t ), respectively. Using the equality and analogous relations for the norm of p, f and g, we obtain (51) with a constant c independent of a.

Solvability in Weighted L p Sobolev Spaces, the Case p ≥ 2
In this subsection, we assume always that β satisfies (5) and γ = β + 3 p − 3 2 , i. e., Using Lemma 2.14 and an extension of Mikhlin's Fourier multiplier theorem to operator-valued functions (see [1, Theorem 6.1.6]), we can prove the following lemma.
Proof. For the first assertion, we refer to [16,Theorem 3.1]. Let U (x, s), P (x, s) and F (x, s) be the Laplace transforms of u(x, t), p(x, t) and f(x, t), respectively. Then We denote the operator V 0 2,γ (K) F → U V 2 2,γ (K) of the problem (26) by Φ(s). If we extend f by zero to K × (−∞, 0), we have the representation where F t→τ denotes the Fourier transform with respect to t. By Lemma 2.14, there is the estimate Φ (k) (s) V 0 2,γ (K)→V 2 2,γ (K) ≤ c |s| −k for Re s ≥ 0, s = 0 if k = 0 or k = 1. Using the above mentioned extension of Mikhlin's multiplier theorem, we conclude that F −1 τ →t Φ(iτ ) F t→τ is a continuous operator from L p (R; V 0 2,γ (K)) into L p (R; V 2 2,γ (K)) and with a constant c independent of f. Analogously, the estimate for the L p (R + ; V 1 2,γ (K))-norm of p holds. Finally, the desired estimate for the L p,2;γ (Q)-norm of ∂ t u follows from the equation ∂ t u = f + Δu − ∇p.
Let ζ ν be the same smooth functions on K as in Sects. 2.1 and 3.2. We consider the solution (u ν , p ν ) of the problem u(x, t) = 0 for x ∈ ∂K, t > 0, u(x, 0) = 0 for x ∈ K.
Analogously to Lemma 3.5, we obtain the following assertion.
Using the inequalities (56), we can deduce the following result from Lemma 3.7.
with a constant c independent of f.
With the substitution |x|/ √ t − τ = s, we obtain This proves the lemma.
Let (u, p) be the solution of the problem (54), (55) given in Lemma 3.9. By Lemma 3.9, there exist also solutions (u ν , p ν ) of the problem (58), (59), where u ν has the representation with a constant c independent of f.
Proof. It suffices to prove the estimate (68) for smooth vector functions (u, p) with compact support in (K\{0}) × R + . Let also h be a continuous vector function with compact support in (K\{0}) × R + . From the condition on p and β, it follows that 2 − λ 1 < 2 − β + 3 p < λ 1 + 3. Thus by Lemma 3.11, there exists a solution (v, q) of the system with the initial and boundary conditions (55) satisfying the estimate v L p ;−β (Q) ≤ c h L p ;2−β (Q) .
Next, we prove the following estimate for the pressure p.
where c is independent of u and p.

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