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On the spatially asymptotic structure of time-periodic solutions to the Navier–Stokes equations

The asymptotic behavior of weak time-periodic solutions to the Navier-Stokes equations with a drift term in the three-dimensional whole space is investigated. The velocity field is decomposed into a time-independent and a remaining part, and separate asymptotic expansions are derived for both parts and their gradients. One observes that the behavior at spatial infinity is determined by the corresponding Oseen fundamental solutions.


Introduction
We study the behavior for |x| → ∞ of time-periodic solutions to the Navier-Stokes equations lim |x|→∞ u(t, x) = 0 for t ∈ T, (1.1) which model the flow of a viscous incompressible fluid. Here f : T × R 3 → R 3 is an external force, and u : T × R 3 → R 3 and p : T × R 3 → R denote velocity and pressure fields of the fluid flow. The torus group T := R/T Z serves as time axis and encodes that all involved functions are time periodic with prescribed period T > 0. In this paper, we consider the case λ = 0, which models a non-vanishing inflow velocity λ e 1 at infinity. Asymptotic properties in the case λ = 0 are different and shall not be treated here. For λ = 0 the pointwise decay of time-periodic solutions to (1.1) was studied by Galdi and Sohr [10] and by Galdi and Kyed [12]. By [12] a weak solution u to (1.1) satisfies where Γ λ 0 is the fundamental solution to the steady-state Oseen system 3) and the remainder term satisfies |R(t, x)| ≤ C|x| −3/2+ε . In particular, (1.2) shows that the asymptotic behavior of the velocity field u is, in general, determined by the steadystate Oseen fundamental solution Γ λ 0 . Moreover, (1.2) coincides with the anisotropic expansion of weak solutions to the corresponding steady-state problem, which is due to Finn [8,9], Babenko [1] and Galdi [11] and may be seen as a special case of the time-periodic setting.
The main theorem of this paper, Theorem 4.3 below, extends the results from [12] in several ways. Firstly, we improve the pointwise estimate of R(t, x) in such a way that it reflects the anisotropic structure of the solution. Secondly, we derive an asymptotic expansion for ∇u by establishing pointwise estimates of ∇R(t, x). Thirdly, we decompose u into its time mean over one period Pu and a time-periodic remainder P ⊥ u = u−Pu, for which we derive separate asymptotic expansions. We shall observe that the asymptotic properties of the steady-state part Pu are governed by the steady-state fundamental solution Γ λ 0 , while those of the purely periodic part P ⊥ u are determined by Γ λ ⊥ , the (faster decaying) purely periodic part of the fundamental solution Γ λ to the time-periodic Oseen system In particular, this shows that the purely periodic part P ⊥ u decays faster than the steadystate part Pu as |x| → ∞. This paper is structured as follows. After introducing the basic notation in Section 2, we recall the fundamental solution to the time-periodic Oseen equations and collect related results in Section 3. In Section 4 we present and prove our main theorems.

Notation
In general, we denote points in T × R 3 by (t, x) and call t ∈ T time variable and x ∈ R 3 spatial variable, respectively. For a sufficiently regular function u : T×R 3 → R 3 we write ∂ j u := ∂ x j u, and we set ∆u := ∂ j ∂ j u and div u := ∂ j u j . As in this definition, we use Einstein's summation convention frequently. If U : By L q (Ω) and W k,q (Ω) we denote classical Lebesgue and Sobolev spaces, and we set Observe that G := T × R 3 is a locally compact abelian group and that its dual group can be identified with G = Z × R 3 , the elements of which we denote by (k, ξ) ∈ Z × R 3 . We equip the group T with the normalized Haar measure given by and G with the corresponding product measure. Moreover, F G denotes the Fourier is the space of tempered distributions on G, which was introduced by Bruhat [2]; see also [3]. Moreover, for f : Pf is time independent, we call Pf the steady-state part and P ⊥ f the purely periodic part of f . A straightforward calculation shows where δ Z is the delta distribution on Z.
By the letter C we denote generic positive constants. In order to specify the dependence of C on quantities a, b, . . ., we write C(a, b, . . .).

The time-periodic fundamental solution
In this section, we consider a fundamental solution Γ λ to the time-periodic problem (1.4) such that the velocity field is given by u = Γ λ * f , where the convolution is taken with respect to the group G = T × R 3 . Such a fundamental solution was recently introduced in [12,4] and is given by Here the symbol 1 T denotes the constant 1 distribution on T, and The function Γ λ 0 is the fundamental solution to the steady-state Oseen problem (1.3); see [11,Section VII.3]. Its anisotropic behavior is reflected by the pointwise estimates see [5,Lemma 3.2]. The examination of convolutions of Γ λ 0 with functions satisfying similar estimates was carried out by Farwig [5,6] in dimension n = 3, and later by Kračmar, Novotný and Pokorný [13] in the general n-dimensional case. The following theorem collects some of their results.
Then there exists C = C(A, B, λ) > 0 with the following properties: Proof. These are special cases of [13, Theorems 3.1 and 3.2].
In order to derive a similar result to control convolutions with the purely periodic part Γ λ ⊥ , we recall the following theorem established in [4]. Theorem 3.2. The purely periodic velocity fundamental solution Γ λ ⊥ satisfies ∀q ∈ 1, Proof. See [4, Theorem 1.1].
From these properties we conclude the following theorem.
The next lemma can be used to conclude asymptotic expansions in the linear case, where the velocity field is given by u = Γ λ * f .
The following auxiliary result treats convolutions of functions with anisotropic decay.
Proof. This is a consequence of the calculations in [13, Section 2].

Main results
We consider weak solutions to (1.1) in the following sense.
iii. the identity (R 3 )) 3 . Therefore, this class seems to be a natural outset for further investigation. Nevertheless, at first glance, instead of ii. one would expect the condition u ∈ L ∞ (T; L 2 (Ω)) 3 , which naturally appears for weak solutions to the Navier-Stokes initial-value problem. However, this property cannot be expected for general time-periodic data f . As was shown by Kyed [14,Theorem 5.2.4], for smooth data f ∈ C ∞ 0 (T × R 3 ) 3 one has u ∈ L ∞ (T; L 2 (R 3 )) 3 if and only if T×R 3 f d(x, t) = 0. An analogous property was established by Finn [7] for the corresponding steady-state problem.
As our main result, we establish the following asymptotic expansions. (4.1) Then such that there exists C > 0 such that for all t ∈ T and |x| ≥ 4 it holds
(4.10) Remark 4.4. As explained in [12], assumption (4.1) merely appears for technical reasons. It ensures additional local regularity but does not improve spatial decay of the solution.
One main observation is that the asymptotic behavior of u and ∇u for |x| → ∞ is governed by the time-periodic Oseen fundamental solution Γ λ . In particular, the purely periodic part P ⊥ u decays faster than the steady-state part Pu. As a direct consequence of Theorem 4.3, we obtain the following pointwise estimates, which we shall derive as intermediate results on the way to a proof of Theorem 4.3.
Theorem 4.5. Under the assumptions of Theorem 4.5 there is C > 0 such that for all t ∈ T and x ∈ R 3 the function u satisfies (4.14) In order to prove these theorems, we recall the following regularity result.
and there is a pressure function p ∈ C ∞ (T × R 3 ) such that (1.1) is satisfied pointwise.
We also need a uniqueness statement for solutions to the linear problem (1.4).
These lemmas enable us to derive the following representation formulas.

Proposition 4.8. Let u be a weak solution as in Theorem 4.3. Then
for all α ∈ N 3 0 with |α| ≤ 1. In particular, v := Pu and w := P ⊥ u satisfy Proof. From Lemma 4.6 we conclude u · ∇u ∈ L q (T × R 3 ) for all q ∈ (1, ∞). Therefore, U := Γ λ * (f − u · ∇u) is well defined as a classical convolution integral, and we have ∂ j U = ∂ j Γ λ * (f − u · ∇u) for j = 1, 2, 3 by the dominated convergence theorem. Since both U and u satisfy the time-periodic Oseen system (1.4) for suitable pressure functions p, Lemma 4.7 implies P ⊥ u = P ⊥ U and that Pu−PU is a polynomial in each component. With Young's inequality we obtain PU ∈ L 6 (R 3 ) since Γ λ 0 ∈ L 12/5 (R 3 ) by [12,Lemma 5.4]. Hence, Pu − PU ∈ L 6 (R 3 ). This leads to Pu = PU and thus u = U , which yields (4.15). The remaining formulas now follow from together with the identity u · ∇u = div(u ⊗ u) and integration by parts.
Based on these formulas, we can now prove Theorem 4.5 and Theorem 4.3.