On the Green’s formula for a Stokes type problem

A time-periodic Stokes problem is studied in the domain with cylindrical outlets to infinity. Using the Fourier series the problem is reduced to a sequence of elliptic problem. For each of these elliptic boundary value problems a generalized Green’s formula is constructed. The analogous Green’s formula for the steady Stokes problem was obtained in [1].


Formulation of the problem
Let ⊂ R 3 be a domain with cylindrical outlets to infinity, i.e., outside the ball B R = x ∈ R 3 : |x| R the domain coincides with a system of J semi-infinite cylinders j + of a constant cross section ω j . Let j + ∩ k + = ∅, j = k and let the boundary ∂ be smooth. We consider in the time-periodic Stokes problem We assume that the external force f = (f 1 , f 2 , f 3 ) t is 2π -time-periodic function. Problem (1)-(4) could be decomposed into a sequence of elliptic problems. Indeed, we can look for the solution to problem (1) Inserting series (5), (6) into equations and boundary conditions we get for coefficients v ck , v sk , p ck , p sk series of the systems of elliptic problems Here f c0 /(2π), f ck /π, f sk /π, k = 0, 2, . . . , are Fourier coefficients of the function f = f(x, t).
In this paper we derive so-called generalized Green's formula for problem (7). The analogous Green's formula for the steady Stokes problem was obtained in [1]. The obtained below results are important for the construction of correct asymptotic conditions at infinity which describe real time-periodic physical processes (for example bloodstream).

The asymptotics of the solution to problem (7)
3 ) be the local coordinate system related to the cylinder j + such that the axis x j 3 is directed along cylinder axis. We consider problem (7) in a weighted Sobolev space W l β ( ) which is a closure of C ∞ 0 ( ) (a class of infinitely differentiable functions with compact supports in ) with respect to the norm If β > 0, elements of this space exponentially vanish as x j 3 tends to infinity, and they may exponentially grow, if β < 0 .
Consider problem (7) in the cylinder j + . Using the methods of the book [2] and arguing in the same way as in [1] we obtain four special solutions of the homogeneous problem (7): where the pair of functions (ϕ According to Theorem 3.1.4 in [2] the sum of linear combinations of these solutions gives the main term (up to an exponentially vanishing term) of the asymptotic decom-position of the "growing" at infinity solution. Let χ j (x) be a smooth cut-off function such that supp(χ j ) ⊆ j + and χ j (x) = 1 if x j 3 > L for j = 1, . . . , J .

Generalized Green's formula
Let u k = (v ck , p ck , v sk , p sk ), U k = (V ck , P ck , V sk , P sk ) ∈ C ∞ 0 ( ). Integrating twice by parts in one gets the standard Green's formula (see [3]) here ( , ) stands for a scalar product in L 2 ( ). Denoting by q(u k , U k ) the left-hand side of the above formula we get q(u, U) = q(U, u) = 0 for any u ∈ D l β W ( ) and U ∈ D l −β W ( ). Let S be an operator of problem (7) and S * be an operator of the problem It is clear that S * is an adjoint operator to S with respect to the Green's formula (12). Note that S is not self-adjoint operator. Homogeneous problem (13) in the cylinder j + has four special solutions where functions ϕ j k and ψ j k are defined by formula (10). We denote by D l ±β W ( ) the subset of functions u k ∈ D l −β W ( ) having expansion (11) and by D l ±β W ( ) * the subset of D l −β W ( ) consisting of functions having an expansions where U jh ,k , h ∈ {0, 1} , ∈ {c, s}, are defined by (14) and (15), Using the fact that functions (8), (9) and (14), (15) are exact solutions to homogeneous problems (7) and (13), respectively, we get, after cumbersome computation, that Inserting representations (11) and (16) into q(u k , U k ) we get that a number of terms in q(u k , U k ) vanishes and, finally, we find Let us calculate the term q(χ j u j 0 ck , χ j U j 1 ck ). We note, firstly, that the cut-off function χ j restricts all considerations to the cylinder j + , secondly, that S(χ j u jh ,k ) and S * (χ j u jh ,k ) have compact supports. Applying the Green's formula (12) in the domain L = {x ∈ : ifx ∈ j + then x j 3 < L, j = 1, . . . , J } we get an additional integral over the cross-section ω j . Let n = (0, 0, 1) t be the outward normal to ∂ L on ω j and ∂ 3 = ∂ \ ∂x j 3 . Taking into account (8), (9) and (14), (15)