Time-periodic Stokes equations with inhomogeneous Dirichlet boundary conditions in a half-space

The time-periodic Stokes problem in a half-space with fully inhomogeneous right-hand side is investigated. Maximal regularity in a time-periodic Lp setting is established. A method based on Fourier multipliers is employed that leads to a decomposition of the solution into a steady-state and a purely oscillatory part in order to identify the optimal function spaces.


INTRODUCTION
We investigate the  time-periodic Stokes problem in a half-space R n + of dimension n ≥ 2. Here, u ∶ R×R n + → R n denotes the velocity field and p ∶ R×R n + → R the pressure term. As customary in the formulation of time-periodic problems, the time-axis is taken to be the whole of R. Variables in the time-space domain R × R n + are denoted by (t, x). The time-period  > 0 shall remain fixed. Data ∶ R × R n + → R n , g ∶ R × R n + → R and h ∶ R × R n + → R n that are also  time-periodic are considered. The time-periodic Stokes equations play a fundamental role in a wide range of problems in fluid mechanics. Although comprehensive L p estimates of maximal regularity type are available in the whole-space case, 1 similar estimates in the more complicated half-space case were only established recently by Maekawa and Sauer. 2 The analysis in Maekawa and Sauer, 2 however, does not include inhomogeneous boundary data h ≠ 0. In the following, we shall establish maximal regularity L p estimates that include the case h ≠ 0. Such estimates are crucial in a number of applications. For example, a classical approach to free boundary problems in fluid-structure interaction relies heavily on maximal regularity frameworks that include inhomogeneous boundary data. When time-periodic driving forces are studied in such settings, time-periodic Stokes equations appear in the linearization. We mention as a particular example the investigation of resonance in thin elastic structures that interact with liquids (see, for example, the previous studies [3][4][5][6][7]. Mathematically, this problem can be studied as a coupled fluid-structure system where the fluid occupies a half-space and the thin elastic structure its boundary. If a viscous fluid governed by the Navier-Stokes equations is considered, resonance under time-periodic forcing in the hyperbolic elastic structure may be avoided due to energy dissipation in the fluid. The mathematical investigation hereof requires a function analytic framework for the linearized equations of motion, which leads us to the time-periodic Stokes system (1) analyzed in the following. We believe the L p framework and the maximal regularity established in Theorem 1 may lead to improvements in this field of time-periodic fluid-structure interaction, which is otherwise typically studied in an L 2 framework.
The nature of the Stokes problem does not allow for the treatment of inhomogeneous boundary data by a simple lifting argument. Introduction of a vector field H ∶ R × R n + → R n that coincides with h on the boundary R × R n + may reduce the problem to one of homogeneous boundary data by consideringũ ≔ u − H instead of u, but it will then be difficult to treat the additional term divH on the right-hand side of the equation divũ = g − div (H). Indeed, if a vector field H is chosen from the canonical trace space of the solution, the term divH is not admissible as right-hand side in Maekawa and Sauer. 2 Consequently, an extension of the results in Maekawa and Sauer 2 to include the case h ≠ 0 is by no means trivial. In the following, we employ a different approach than the reflection type argument used in Maekawa and Sauer. 2 Instead, we use the Fourier transform ℱ T×R n−1 to reduce (1) to an ordinary differential equation in the variable x n . Here, T denotes the torus R∕ Z. The L p estimates are then established with arguments based on Fourier multipliers and interpolation techniques. Although the main idea behind this approach is not new, indeed, it has been applied successfully by various authors to investigate the initial value Stokes problem; a number of nontrivial modifications are needed to adapt the arguments to the time-periodic case. Notably, the system (1) has to be decomposed into a steady-state part and a so-called purely oscillatory part. Without this decomposition, it seems impossible to establish optimal L p estimates. Whereas the estimates for the resulting steady-state problem are well-known and can be found in contemporary literature, the estimates for the purely oscillatory part in the following are new.
It is convenient to formulate  time-periodic problems in a setting of function spaces where the torus T ≔ R∕ Z is used as a time axis. Indeed, via lifting with the quotient map ∶ R → T,  time-periodic functions are canonically identified as functions defined on T and vice versa. For such functions, we introduce the simple decomposition, into a time-independent part u and a part  ⟂ u with vanishing time average over the period. We shall refer to u as the steady-state part and to  ⟂ u as the purely oscillatory part of u. Equipped with the quotient topology, the time-space domain T × R n is a locally compact abelian group, and therefore, has a Fourier transform ℱ T×R n associated to it. Moreover, we may introduce the Schwartz-Bruhat space (T × R n ) and its dual space ′ (T × R n ) of tempered distributions. Consequently, Bessel potential spaces with underlying time-space domain T×R n can be defined as subspaces of ′ (T×R n ) in a completely standard manner. Sobolev spaces are introduced as Bessel potential spaces with integer exponents and Sobolev-Slobodeckiȋ spaces, ie, Sobolev spaces with non-integer exponents, via real interpolation. The same scale of function spaces with respect to the half-space R n + is obtained by restriction. In a setting of these function spaces (see Section 2 for the precise definitions and Remark 2), the main theorem of this article can be formulated as follows: with  ⟂ h n ∈ W 1,q (T; .
which satisfies and ) , with C = C(n, q,  ). If (ũ,p) is another solution to (1) in the class (5), then  ⟂ u =  ⟂ũ , u = ũ + (a 1 x n , … , a n−1 x n , 0) for some vector a ∈ R n−1 , and p =p + d(t) for some function d that depends only on time.
The estimates (6) and (7) of different regularity type for the steady-state u and the purely oscillatory part  ⟂ u of the solution, respectively, demonstrate the necessity of the decomposition. Observe that the purely oscillatory part  ⟂ u of the solution is unique, whereas the steady-state part u is not. The projections  and  ⟂ decompose the time-periodic Stokes problem (1) into a classical steady-state Stokes problem with respect to data ( , g, h) and a time-periodic Stokes problem with respect to purely oscillatory data ( ⟂ ,  ⟂ g,  ⟂ h), respectively. The first estimate (6) is well-known for the former problem, whence the main objective in the following will be to establish existence of a unique solution to the latter that satisfies (7). Remark 1. For properties of solutions to the steady-state Stokes problem we refer the reader to Galdi. 8 The homogeneous Sobolev space . W 2,q (R n + ) used to characterize the solution u in (5), however, is defined slightly different in Galdi 8 compared with the definition (10) employed in this article. In Galdi,8 it is defined as the subspace {u ∈ L 1 loc (R n + ) | ∇ 2 u ∈ L q (R n + )} of locally integrable functions, whereas we introduce it in (10) as a subspace of the distributions Z ′ (R n ). The assertions in Theorem 1, in particular estimate (6), remain valid in both cases. The latter definition has the advantage that the norm ||∇ 2 · || L q (R n + ) appearing on the left-hand side of (6) turns . W 2,q (R n + ) into a Banach space, but the disadvantage that elements of . W 2,q (R n + ) can only be interpreted as functions modulo polynomials. Remark 2. It is possible to avoid the analysis on the torus group T ≔ R∕ Z and instead define the function spaces appearing in Theorem 1 as spaces of  -periodic functions on R. For a Banach space E, let denote the space of smooth  time-periodic E-valued functions. The Bochner-Lebesgue and Bochner-Sobolev spaces of time-periodic functions can then be introduced as follows: Observe that the closures above are taken with respect to a time interval of period  . Time-periodic Sobolev-Slobodeckiȋ spaces can then be defined as real interpolation spaces in the usual way. Via the canonical quotient map ∶ R → T, the spaces C ∞ per (R; E) and C ∞ (T; E) are isometrically isomorphic with respect to the norms || · || W k,q ((0, );E) and || · || W k,q (T;E) , respectively, provided the Haar measure on T is normalized appropriately. It follows that W s,q per (R; E) and W s,q (T; E) are also isometrically isomorphic for all s. In this manner, all the function spaces appearing in Theorem 1 have interpretations as  time-periodic Bochner spaces. Remark 3. Theorem 1 does not characterize the time-periodic Stokes operator as a homeomorphism between the function spaces of the solutions (5) and the function spaces of the data (3 and 4). In particular, there are clearly functions (u, p) in the function spaces (5) for which the condition (4) we require of the purely oscillatory part of the inhomogeneous boundary data is not satisfied for Tr ⟂ u n . The same phenomenon can be observed in the analysis of the initial value Stokes problem in half-spaces (see, for example, Prüss and Simonett 9, Chapter 7.2 ) and seems to manifest itself in a similar manner in the time-periodic case.

PRELIMINARIES
The objective of this section is to formalize the reformulation of (1) in a setting where the time axis is replaced with the torus group T ≔ R∕ Z. This includes definitions of the function spaces appearing in Theorem 1.

Topology, differentiable structure, and Fourier transform
We utilize T × R n as a time-space domain. Equipped with the quotient topology induced by the quotient mapping T × R n becomes a locally compact abelian group. We can identify T × R n with the domain [ 0,  ) × R n via the restriction | [0, )×R n . The Haar measure dg on T × R n is the product of the Lebesgue measure on R n and the Lebesgue measure on [ 0,  ). We normalize dg so that There is a bijective correspondence between points (k, ) ∈ 2  Z × R n and characters ∶ T × R n → C, (t, x) ≔ e ix· +ikt on T × R n . Consequently, we can identify the dual group of T × R n with 2  Z × R n . The compact-open topology on 2  Z × R n reduces to the product of the Euclidean topology on R n and the discrete topology on 2  Z. The Haar measure on 2  Z × R n is therefore the product of the counting measure on 2  Z and the Lebesgue measure on R n . The spaces of smooth functions on T × R n and 2  Z × R n are defined as and The notion of Schwartz spaces can be extended to locally compact abelian groups (see Bruhat 10 and Eiter and Kyed 11 ). The so-called Schwartz-Bruhat space on T × R n is given by Equipped with the semi-norm topology of the family { , , | ( , , ) ∈ N n 0 × N 0 × N n 0 }, (T × R n ) becomes a topological vector space. The corresponding topological dual space ′ (T × R n ) equipped with the weak* topology is referred to as the space of tempered distributions on T × R n . Distributional derivatives for a tempered distribution u are defined by duality as in the classical case. The Schwartz-Bruhat space on 2 with the generic semi-norms̃, inducing the topology. By ℱ T×R n , we denote the Fourier transform associated to the locally compact abelian group T × R n equipped with the Haar measure introduced above: provided the Lebesgue measure d is normalized appropriately. By duality, ℱ T×R n extends to a homeomorphism Via the symbol, the projections  and  ⟂ extend to projections on ′ (T × R n ): At this point, we have introduced ample formalism to reformulate (1) equivalently as a system of partial differential equations in the time-space domain T×R n . Moreover, the Fourier transform ℱ T×R n enables us the investigate the systems in terms of Fourier-multipliers. Because of the lack of a comprehensive L q multiplier theory in the general group setting, we shall utilize a so-called transference principle for this purpose. The transference principle goes back to de Leeuw. 12 The lemma below is due to Edwards and Gaudry. 13 Theorem 2 (de Leeuw, Edwards, and Gaudry). Let G and H be locally compact abelian groups. Moreover, let Φ ∶Ĝ →Ĥ be a continuous homomorphism and q ∈ [1, ∞]. Assume that m ∈ L ∞ (Ĥ; C) is a continuous L q multiplier, ie, there is a constant C such that Proof. See Edwards and Gaudry. 13

Function spaces
Since our proof of the main theorem is based on Fourier multipliers and interpolation theory, we find it convenient to introduce all the relevant Sobolev spaces via Bessel-Potential spaces. Classical (inhomogeneous) Bessel-Potential spaces are defined for s ∈ R and q ∈ [1, ∞) by Classical Sobolev spaces on R n are defined as Bessel-Potential spaces of integer order k ∈ Z and Sobolev spaces on the half-space R n + via restriction: Observe that for negative-order spaces, ie, when k < 0, the Sobolev space W k,q (R n + ) coincides with the dual space ) ′ . In the following, it is essential that the former meaning of W k,q (R n + ) is used. Homogeneous Bessel-Potential spaces are defined in accordance with Triebel 14 by introducing the subspace Because of the lack of regularity of | | s at the origin, the above definition of is clearly a Banach space. As above, we define homogeneous Sobolev spaces on R n as homogeneous Bessel-Potential spaces of integer order k ∈ Z and introduce homogeneous Sobolev spaces on the half-space R n + via restriction: By the Hahn-Banach Theorem, any functional in Z ′ (R n ) can be extended to a tempered distribution in ′ (R n ). If s ≤ 0, the extension of an element in Sobolev-Slobodeckiȋ spaces of both homogeneous and inhomogeneous type are defined via real interpolation in the usual way. For example, the spaces appearing in Theorem 1 are defined by and equipped with the associated interpolation norms. Analogously, Bessel-Potential spaces with underlying time-space domain T × R n are introduced via the Fourier transform ℱ T×R n as } equipped with the canonical norm. Again, we refer to Bessel-Potential spaces of integer order k, l ∈ N as Sobolev spaces: ) .
Sobolev spaces on the time-space domain T × R n + are defined via restriction of the elements in the spaces above. In order to introduce homogeneous spaces, we let As above, we may consider H r,q (T; . H s,q (R n )) as a subspace of ′ (T×R n ) by extension. Finally, Sobolev-Slobodeckiȋ spaces on the domain T × R n are defined via real interpolation. For example, In this way, all the function spaces appearing in Theorem 1 attain rigorous definitions. It is easy to verify that these definitions coincide with a classical interpretation as Bochner spaces of vector-valued functions defined on the torus T.

Interpolation
Although the function spaces appearing in Theorem 1 can all be defined in terms of classical interpolation, our proof of the theorem relies on a somewhat more refined scale of interpolation spaces. More specifically, it is based on anisotropic Besov spaces with underlying time-space domain T×R n , which we shall show coincide with the function spaces obtained by real interpolation of the Bessel-Potential spaces introduced above. Although this task is mainly technical and does not require any significantly new ideas, indeed, we shall mimic the proofs of similar results for classical isotropic Besov spaces; these spaces and their interpolation properties are not part of contemporary literature, and we shall therefore carry out the identification here (even in slightly more generality than actually needed for the proof of the main theorem). To this end, we fix an m ∈ N and introduce the parabolic length scale The anisotropic Besov spaces defined below pertain to time-periodic parabolic problems of order 2m. In our analysis of the Stokes problem, we thus put m = 1. For simplicity, we omit m in the notation for the function spaces below.
Definition 1 (Anisotropic Besov and Bessel-potential spaces). Let ∈ C ∞ 0 (R × R n ) be as in Lemma 1, s ∈ R and p, q ∈ [1, ∞). We define anisotropic Besov spaces and anisotropic Bessel-Potential spaces Observe that B s pq,⟂ (T×R n ) and H s p,⟂ (T×R n ) are defined as subspaces of the purely oscillatory distributions  ⟂ ′ (T×R n ) rather than ′ (T × R n ). Recalling (9), it is easy to verify that || · || B s pq,⟂ and || · || H s p,⟂ are therefore norms (rather than mere semi-norms) and B s pq,⟂ (T × R n ) and H s p,⟂ (T × R n ) Banach spaces. As in the case of classical (isotropic) spaces, real interpolation of anisotropic Bessel-potential spaces yields anisotropic Besov spaces: Proof. For l ∈ N 0 and r ∈ R let We claim that r l is an L p (R; L p (R n )) multiplier, which we verify by showing that r l meets the condition of Marcinkiewicz's multiplier theorem (see, for example, Grafakos 15, Corollary 6.2.5 ). For this purpose, we utilize only and that g( , ) ≔ | , | −r is parabolically (−r)-homogeneous, that is, From (17), we immediately obtain | r l | ∞ ≤ C| | ∞ 2 −lr , with C independent on l. By (18), we further observe that Choosing ≔ | , | and recalling (17), we thus deduce || with C independent on l. It follows from Marcinkiewicz's multiplier theorem that r l is an L p (R; L p (R n )) multiplier. Consequently, the transference principle (Theorem 2) implies that r l| 2  Z×R n is an L p (T; L p (R n )) multiplier with . Consider a decomposition f = f 0 + f 1 with 0 ∈ H s 0 p,⟂ (T × R n ) and 1 ∈ H s 1 p,⟂ (T × R n ). We deduce .
We now employ the K-method (see, for example, Bergh and Löfström 16, Chapter 3.1 ) to characterize the interpolation . Taking infimum over all decompositions f 0 and f 1 in the inequality above, we find that where the last inequality above is valid since s 0 ≠ s 1 .
Using the same technique as above, this time utilizing the multiplier || p , and similarly, We thus obtain We now employ the J-method (see, for example, Bergh and Löfström 16, . By the last two estimates above, we see that with convergence in the space H

PROOF OF MAIN THEOREM
Utilizing the formalism introduced in Section 2, we can equivalently reformulate (1) in a setting where the time axis R is replaced with the torus T ≔ R∕ Z. In this setting, the periodicity condition is no longer needed, and we obtain the equivalent problem In order to investigate (19), we employ the projections  and  ⟂ to decompose the problem into a steady-state and a so-called purely oscillatory problem. More specifically, we observe that (u, p) is a solution to (19) if and only if (v, Π) ≔ (u, p) is a solution to the steady-state problem and The steady-state problem (20) is a classical Stokes problem, for which a comprehensive theory is available. We therefore focus on the purely oscillatory problem (21), which only differs from (1) by having purely oscillatory data. We start with the case of nonhomogeneous boundary values.
Proof. We shall employ the Fourier transform ℱ T×R n−1 to transform (24) into a system of ODEs. For this purpose, we denote by (k, ) ∈ 2  Z × R n−1 the coordinates in the dual group of T × R n−1 . Letting u ≔ u ′ ≔ (u 1 , … , u n−1 ), w ≔ u n , u ∶ = ℱ T×R n−1 [v], andŵ ∶ = ℱ T×R n−1 [u n ] in (24), we obtain an equivalent formulation of the system as a family of ODEs. More precisely, (24) is equivalent to the following ODE being satisfied for each (fixed) (k, ) ∈ 2  Z × R n−1 : To solve the ODE, we first consider the case k ≠ 0. Taking divergence on both sides in (24) 1 and utilizing that divu = 0, we find that Δp = 0 and thus −| | 2p (x n ) + 2 x np (x n ) = 0 in R + . Consequently, for some functions , ∶ 2  Z × R n−1 → C. Utilizing (26) 3 and the boundary conditions (26) 4 , we deduce By (27) to (30), a solution to (26) is identified in the case k ≠ 0. Since H is purely oscillatory, we haveĤ ′ (0, ) = H n (0, ) = 0, whence (up, un, p) ≔ (0, 0, 0) solves (26) in the case k = 0. We thus obtain a formula for the solution to (24): Formally at least, (v, w, p) as defined above is a solution to (24). It remains to show that this solution is well-defined in the class (23) for data in the class (22). We start by considering data which is not a trivial assertion since it entails the construction of an approximating sequence that converges simultaneously in Sobolev spaces of positive order and in homogeneous Sobolev spaces of negative order. Nevertheless, it can be shown by a standard "cut-off" and mollifier technique; see Triebel. 14 Proof of Theorems 2.3.3 and 5.1.5 Consequently,  ⟂ Z(T × R n−1 ) n is dense in the class (22). Clearly, for purely oscillatory data H, the solution given by (31) is also purely oscillatory. If we can therefore show (25) for arbitrary H ∈  ⟂ Z(T × R n−1 ) n , the claim of the proposition will follow by a density argument. We first examine the pressure term p (more specifically ∇p). The terms in (30) have different order of regularity, so we decompose q 0 = q 1 + q 2 by and introduce the operators and For m ∈ N 0 , we observe for any x n > 0 that the symbol → (| |x n ) m e −| |x n is an L q (R n−1 ) multiplier. Specifically, one verifies that sup whence it follows from the Marcinkiewicz multiplier theorem (see, for example, Grafakos, 15, Corollary 6.2.5 ) that the Fourier multiplier operator with symbol → (| |x n ) m e −| |x n is a bounded operator on L q (R n−1 ) with operator norm independent on x n , that is, We can thus estimate || (H)|| L ∞ (R + ;L q (T;L q (R n−1 ))) ≤ C ||ℱ −1 Employing again the Marcinkiewicz multiplier theorem, we find that the symbol M is an L q (R; L q (R n−1 )) multiplier. An application of the transference principle (Theorem 2) therefore implies that the restriction M | 2  Z×R n−1 is an L q (T; L q (R n−1 )) multiplier. We thus conclude || (H)|| L ∞ (R + ;L q (T;L q (R n−1 ))) ≤ C ||H|| L q (T;H 2,q (R n−1 ))∩H 1,q (T;L q (R n−1 )) .
This estimate shall serve as an interpolation endpoint. To obtain the opposite endpoint, we again employ (35) to estimate sup ||x n (H)|| L q (T;L q (R n−1 )) ≤ C ||q 1 || L q (T;L q (R n−1 )) , which implies || (H)|| L 1,∞ (R + ;L q (T;L q (R n−1 ))) = Recalling (32), we estimate Again, one can utilize the Marcinkiewicz multiplier theorem to show that both M 1 and M 2 are L q (R; L q (R n−1 )) multipliers and subsequently, obtain from the transference principle (Theorem 2) that their restrictions to 2  Z × R n−1 are L q (T; L q (R n−1 )) multipliers. Consequently, we find that . (37) By (36) and (37), the operator extends uniquely to a bounded operator These extensions rely on the fact that Z(T × R n−1 ) is dense in the function spaces on the left-hand side above. We once more refer to Triebel, 14 Proof of Theorems 2.3.3 and 5.1.5 for a verification of this fact. Using the projection  ⟂ on the left-hand side in (38), we obtain scales of the anisotropic Bessel-Potential spaces introduced in (16). Consequently, is a bounded operator: ∶ H 2 q,⟂ (T × R n−1 ) n → L ∞ ( R + ; L q (T; L q (R n−1 )) ) n , Utilizing Lemma 2, we find that * ) .
*The interpolation identity can also be established with the help of the reiteration theorem, which eliminates the need to introduce Besov spaces. We thank for reviewer for pointing this out to us.
Recalling (39), we conclude that extends uniquely to a bounded operator We now recall (34) and examine the operator ℬ. Utilizing (35) with m = 0, we obtain We again employ (35) to estimate .
. H −1,q (R n−1 )) . It follows that ℬ extends to a bounded operator Real interpolation thus implies that ℬ extends to a bounded operator We now return to the solution formulas (31) and consider H ∈  ⟂ Z(T × R n−1 ) n . In this case, an application of (35) ensures that p is well-defined as an element in the function space L q ( T; .
In a similar manner, it can be shown that (v, w) is well-defined as an element in the space L q ( T; H 2,q (R n + ) ) . To this end, one may consider the symbol and verify that sup It follows that the Fourier multiplier operator corresponding to the symbol m is a bounded operator on L q (R; L q (R n−1 )) with operator norm independent on x n . An application of the transference principle (Theorem 2) therefore implies that the operator corresponding to the symbol m | 2  Z×R n−1 is a bounded operator on L q (T; L q (R n−1 )) with operator norm independent on x n , that is, With both (35) and (44) at our disposal, it is now straightforward to verify that u ≔ (v, w) is well-defined as element in the space L q ( T; H 2,q (R n + ) ) ∩ H 1,q ( T; L q (R n + ) ) . By construction, this choice of (u, p) is a solution to (24). Moreover, since  ⟂ H = H, also  ⟂ u = u. This means that u is a purely oscillatory solution in the aforementioned function space to the time-periodic heat equation in the half-space By Kyed and Sauer, 18, Theorem 2.1 (see also Kyed and Sauer,19, Theorem 1.4 ), it is known that this problem has a unique purely oscillatory solution in the space L q ( T; H 2,q (R n + ) ) , which satisfies ||u|| H 1,q (T;L q (R n + ))∩L q (T;H 2,q (R n ) .
In the next step, we consider the resolution of the fully nonhomogeneous system (21), that is, (1) with purely oscillatory data, and establish L q estimates. This step concludes the main result of the article.
A solution to the purely oscillatory Stokes system, is obtained via the solution formulas ] .
By the symmetry of G, the vector fieldw obtained by odd reflection with respect to x n of the n'th component of w, , is also a solution to (49) corresponding to the same pressure term . This means that w andw both solve the same time-periodic heat equation in the whole-space T × R n . By Kyed and Sauer, 18, Theorem 2.1 w =w. It follows that Tr T×R n−1 [w n ] = 0. Consequently, H ≔ h − Tr T×R n−1 [w] belongs to the space (22) (see for example Kyed and Sauer 19 for a rigorous definition of the trace operator in this setting). Let (U, ) be the corresponding solution from Proposition 1. It follows that (u, p) ≔ (U + w + v, + ) is a solution to (1) in the class (47) satisfying (48).
It remains to show uniqueness, which follows from a standard duality argument. To this end, assume that (ũ,p) is a solution in the class (47) to the homogeneous Stokes problem ⎧ ⎪ ⎨ ⎪ ⎩ tũ − Δũ + ∇p = 0 in T × R n + , divũ = 0 i nT × R n + , u = 0 o n T × R n + .
Since this identity holds for all ∈ C ∞ 0 (T × R n + ) n , it follows thatũ = 0. In turn, we deduce ∇p = 0, whencep ∈  ⟂ L q (T), that is,p depends only on time.