1 Introduction

We consider the time-periodic problem for the Navier–Stokes equation in \({\mathbb {R}} \times {\mathbb {R}}^n\) with \(n \ge 3\):

$$\begin{aligned} \left\{ \begin{array}{lll} \partial _t u- \Delta u + u \cdot \nabla u + \nabla p = {{\text {div }}}\, F \quad &{}\text {in } \; {\mathbb {R}} \times {\mathbb {R}}^n, \\ {\text {div }}\, u = 0 \quad &{}\text {in } \; {\mathbb {R}} \times {\mathbb {R}}^n , \\ u(\cdot ,x) \rightarrow 0 \quad &{}\text {as }\; |x| \rightarrow \infty , \\ u(t,\cdot ) =u(t+T,\cdot ) \quad &{}\text {for all } \; t \in \; {\mathbb {R}}. \end{array} \right. \end{aligned}$$
(1)

Here \(u=(u_1 (t,x),\cdots , u_n (t,x) )\) and \(p=p(t,x)\) denote, respectively, the unknown velocity and pressure of a viscous incompressible fluid, while \(F=(F_{ij} (t,x) )_{i,j=1}^n\) is a given periodic tensor with \({\text{div }}\, F = \left( \sum _{i=1}^n {\partial }_{x_i} F_{ij} (t,x) \right) _{j=1}^n\) denoting the periodic external force. Furthermore, T denotes a fixed period.

Existence and uniqueness of solutions to (1) are studied in many literatures such as [2, 3, 5, 7, 10, 13,14,15]. The time-periodic problem (1) is studied in various functional settings, and, in particular, it is known that the \(L^2\) theory enables us to construct a solution v with \(\nabla v \in L^2 ((0,T) \times {\mathbb {R}}^n)\) of (1) without restricting the size of the external force, see [7, Theorem 6.3.1] for instance. The asymptotic property of the solution v is, however, still an open problem since the construction of v gives us little information on it. One method for describing the asymptotic behavior of v, under the smallness of the external force in a sense, is to establish the existence of solutions, say w, with desired decay properties to (1) and then apply an appropriate uniqueness theorem so that we conclude \(v=w\). The purpose of this paper is to prove the existence of solutions with pointwise decay properties to (1). Such results were given by Galdi and Sohr [3] and Kang et al. [4]. They considered the Navier–Stokes equation in three-dimensional exterior domains and established the existence of time-periodic solutions satisfying

$$\begin{aligned} |\nabla ^j u(t,x)|=O(|x|^{-j-1}), \quad |\nabla ^j p(t,x)|=O(|x|^{-j-2}) \quad (j=0,1) \end{aligned}$$
(2)

uniformly in time as \(|x| \rightarrow \infty\). Recently, Kyed [9] introduced the notion of the time-periodic Stokes fundamental solution, and its leading term is given by the steady Stokes fundamental solution. The spatial decay rate (2) coincides with that of the Stokes fundamental solution and thus it seems to be natural. It is known that, in the three-dimensional exterior problem, (2) is the best spatial decay rate of periodic solutions expected in general [4], however, we do not know whether it is also optimal in other unbounded domains. Miyakawa [11] and the author [12] constructed a unique solution \(\{ u,p \}\) of the stationary Navier–Stokes equation in \({\mathbb {R}}^n\) such that

$$\begin{aligned} |\nabla ^j u(x)|=O(|x|^{1-n-j}), \quad |\nabla ^j p(x)|=O(|x|^{-n-j}) \quad (j=0,1,\ldots ) \end{aligned}$$
(3)

as \(|x| \rightarrow \infty\). The stationary solution in [11, 12] decays more rapidly than the steady Stokes fundamental solution, and we can expect that the time-periodic problem (1) also admits a solution decaying like (3) since stationary solutions can be regarded as time-periodic ones with arbitrary period.

In this paper, we shall show that there exists a unique solution \(\{ u,p \}\) of (1) such that

$$\begin{aligned} |u(t,x)|=O(|x|^{1-n}), \quad |\nabla u(t,x)|=O(|x|^{-n}), \quad |p(t,x)|=O(|x|^{-n}) \end{aligned}$$

uniformly in \(t \in {\mathbb {R}}\) as \(|x| \rightarrow \infty\), provided that F and \({\text {div }}\, F\) are small in a suitable sense. Our solution has the same spatial decay as (3) and we emphasize that it decays more rapidly than the time-periodic Stokes fundamental solution. Furthermore, the structure of the time-periodic Stokes fundamental solution enables us to write the solution u as the sum of the steady and time-periodic parts. The decay properties of each parts of solutions are also studied in this paper.

The proof relies upon the representation formula of solutions via the time-periodic Stokes fundamental solution. We transform (1) into the integral equation via the fundamental solution, and then we estimate both the steady and time-periodic parts of a solution in order to apply the contraction mapping principle. In the proof, the well known properties of steady Stokes fundamental solution and the results of [1, 9] play an important role.

2 Main results

Before stating our results, we introduce some function spaces. In what follows, we adopt the same symbols for vector and scalar function spaces. Let \(1 \le q \le \infty\) and let X be a Banach space. We denote by \(L_{per}^q ({\mathbb {R}}; X)\) the Banach space of all T-periodic functions \(u : {\mathbb {R}} \rightarrow X\) such that the restriction \(u|_{[0,T)} \in L^q (0,T; X)\). The norm in \(L_{per}^q ({\mathbb {R}}; X)\) is given by \(\Vert u \Vert _{q,X} := \Vert u \Vert _{L^q (0,T; X)}\). In the case \(X=L^q ({\mathbb {R}}^n)\), we write simply \(L_{per}^q ({\mathbb {R}} \times {\mathbb {R}}^n)\) with norm \(\Vert \cdot \Vert _q\). The space \(W_{per}^{1,2,q} ({\mathbb {R}} \times {\mathbb {R}}^n)\) is defined by \(W_{per}^{1,2,q} ({\mathbb {R}} \times {\mathbb {R}}^n) := \{ u \in L_{per}^q ({\mathbb {R}} \times {\mathbb {R}}^n) ; \Vert u \Vert _{1,2,q} < \infty \}\) where \(\Vert u \Vert _{1,2,q} := ( \Vert \partial _t u \Vert _q^q + \sum _{| \alpha | \le 2} \Vert \partial _x^\alpha u \Vert _q^q )^{1/q}\). For \(\mu >0\), we define the Banach space \(X_\mu\) by

$$\begin{aligned} X_\mu := \{ u \in L^\infty ({\mathbb {R}}^n) ; \sup _{x \in {\mathbb {R}}^n} (|x|+1)^\mu |u(x)| < \infty \} \end{aligned}$$

with the norm

$$\begin{aligned} \Vert u \Vert _{X_\mu } := \sup _{x \in {\mathbb {R}}^n} (|x|+1)^\mu |u(x)| . \end{aligned}$$

It is easy to check that \(X_{\mu _1} \subset X_{\mu _2}\) if \(\mu _2 < \mu _1\).

Let \(\{ E,Q \}\) be the time-periodic Stokes fundamental solution introduced in [9]. The form and properties of the fundamental solution shall be reviewed in the next section. We consider (1) in the form

$$\begin{aligned} u(t,x) = \int _{{\mathbb {R}}^n} \frac{1}{T} \int _0^T \nabla E (t-s,x-y) (F- u \otimes u)(s,y) \,dsdy , \end{aligned}$$
(4)

where \(u \otimes u := (u_i u_j)_{i,j=1}^n\) and \(\nabla\) denotes the gradient with respect to the spatial variable. If F and u decay rapidly at spatial infinity, then (4) can be written as

$$\begin{aligned} u(t,x) = \int _{{\mathbb {R}}^n} \frac{1}{T} \int _0^T E (t-s,x-y) ({\text {div }}\, F- u \cdot \nabla u)(s,y) \,dsdy . \end{aligned}$$
(5)

As we shall see later, we should interpret E in (4) and (5) as the composition \(E \circ \pi\) where \(\pi\) is a map introduced in the next section. In this paper, we use the notation E instead of \(E \circ \pi\) as long as we consider (4) and (5). Also, it shall be seen that the integral equations (4) and (5) are equivalent to (1) in an appropriate sense. The associated pressure p is given by

$$\begin{aligned} p(t,x) = \int _{{\mathbb {R}}^n} \frac{1}{T} \int _0^T Q (t-s,x-y) \cdot ({\text {div }}\, F- u \cdot \nabla u)(s,y) \,dsdy . \end{aligned}$$

The main result of this paper is stated in the following theorem. The leading term of \(\{ E,Q \}\) is the steady Stokes fundamental solution as we shall see in the next section, and our solution in the following theorem decays more rapidly than the time-periodic Stokes fundamental solution.

Theorem 1

Let \(n \ge 3\) and \(0< \delta <1.\) Suppose

$$\begin{aligned} F \in L_{per}^\infty ({\mathbb {R}}; X_{n+\delta } ) \quad and \quad {\text{div }} F \in L_{per}^\infty ({\mathbb {R}}; X_{n+1}). \end{aligned}$$
(6)

If F and \({\text {div }}\,F\) are sufficiently small in \(L_{per}^\infty ({\mathbb {R}}; X_{n+ \delta } )\) and \(L_{per}^\infty ({\mathbb {R}}; X_{n+1})\) respectively, then (4) admits a unique solution u such that

$$\begin{aligned} u \in L_{per}^\infty ({\mathbb {R}}; X_{n-1} ), \quad \nabla u \in L_{per}^\infty ({\mathbb {R}}; X_n) \end{aligned}$$

and

$$\begin{aligned} \partial _t u, \nabla ^2 u \in L_{per}^q ({\mathbb {R}} \times {\mathbb {R}}^n) \quad for\; all \; 1<q<\infty . \end{aligned}$$

Furthermore, the associated pressure p satisfies

$$\begin{aligned} p \in L_{per}^\infty ({\mathbb {R}}; X_n). \end{aligned}$$

Remark 1

The constant \(\delta\) is introduced so that we can apply the inequality (10) below and \(F (t,\cdot )\) is integrable, see also [12, Remark 2.1].

Remark 2

If F is independent of t, then we can verify that u in Theorem 1 is a stationary solution and thus coincides with the one constructed in [11, 12].

Remark 3

It is possible to show that the solution u in Theorem 1 satisfies \(\partial _t u \in L_{per}^\infty ({\mathbb {R}}; X_{n+1})\) provided that \(\partial _t F\) is sufficiently small in \(L_{per}^\infty ({\mathbb {R}}; X_{n+1})\), see Remark 4. However, the pointwise behavior of \(\partial _t E\) is still an open problem and we do not know whether the decay rate of \(\partial _t u\) is faster than that of \(\partial _t E\).

We shall see in the next section that the time-periodic Stokes fundamental solution E is defined as the sum of the steady Stokes fundamental solution \(E_s\) and the time-periodic remainder \(E_p\). Hence every solution u of (4) is written as the sum of the steady part \(u_s\):

$$\begin{aligned} u_s (x) := \int _{{\mathbb {R}}^n} \nabla E_s (x-y) \frac{1}{T} \int _0^T (F- u \otimes u)(s,y) \,dsdy \end{aligned}$$

and time-periodic part \(u_p\):

$$\begin{aligned} u_p (t,x) := \int _{{\mathbb {R}}^n} \frac{1}{T} \int _0^T \nabla E_p (t-s,x-y) (F- u \otimes u)(s,y) \,dsdy . \end{aligned}$$

The next theorem describes the asymptotic behavior of each parts of solutions.

Theorem 2

Let \(n \ge 3\) and \(0<\delta <1.\) Suppose that F satisfies (6). Every solution \(u \in L_{per}^\infty ({\mathbb {R}}; X_{n-1} )\) with \(\nabla u \in L_{per}^\infty ({\mathbb {R}}; X_n)\) of (4) is written as the sum of the steady part \(u_s\) and time-periodic one \(u_p\) such that

$$\begin{aligned} u_s \in X_{n-1}, \quad \nabla u_s \in X_n, \quad \nabla ^2 u_s \in L^q ({\mathbb {R}}^n) \quad for \;all\; 1<q<\infty , \end{aligned}$$

and

$$\begin{aligned}&u_p \in L_{per}^\infty ({\mathbb {R}}; X_{n+\delta } ),\quad \nabla u_p \in L_{per}^\infty ({\mathbb {R}}; X_{n+1}), \\&\partial _t u_p , \nabla ^2 u_p, \in L_{per}^q ({\mathbb {R}} \times {\mathbb {R}}^n) \quad for\; all\; 1<q<\infty . \end{aligned}$$

If

$$\begin{aligned} F \in L_{per}^\infty ({\mathbb {R}}; X_{n+1} ) \quad and \quad {\text{div }} F \in L_{per}^\infty ({\mathbb {R}}; X_{n+2}), \end{aligned}$$
(7)

then we have

$$\begin{aligned} u_p \in L_{per}^\infty ({\mathbb {R}}; X_{n+1}) \quad and \quad {\nabla} u_p \in L_{per}^\infty ({\mathbb {R}}; X_{n+2}). \end{aligned}$$

3 Proof of main theorems

3.1 Time-periodic Stokes fundamental solution

In this subsection, we review the theory for the time-periodic Stokes fundamental solution. For this purpose, we need the following notation. Set \(\mathbb {T}:= {\mathbb {R}}/T\mathbb {Z}\) and \(G:= \mathbb {T} \times {\mathbb {R}}^n\). We define the map \(\pi : {\mathbb {R}} \times {\mathbb {R}}^n \rightarrow G\) by \(\pi (t,x) := ([t],x)\) and let \(\Pi := \pi |_{[0,T) \times {\mathbb {R}}^n }\). The restriction \(\Pi\) is a bijection from \([0,T) \times {\mathbb {R}}^n\) to G, and via \(\Pi\) we identify G with \([0,T) \times {\mathbb {R}}^n\). The Haar measure dg on the locally compact abelian group G, unique up to a normalization factor, is chosen as the product of the Lebesgue measures on \({\mathbb {R}}^n\) and [0, T), and we have

$$\begin{aligned} \int _G u(g) \,dg := \frac{1}{T} \int _0^T \int _{{\mathbb {R}}^n} (u \circ \Pi )(s,y) \,dyds . \end{aligned}$$

Derivatives on G are defined by

$$\begin{aligned} \partial _t \partial _{x_i} u = \left( \partial _t \partial _{x_i} (u \circ \pi ) \right) \circ \Pi ^{-1}. \end{aligned}$$

The differentiable structure on G is inherited from \({\mathbb {R}} \times {\mathbb {R}}^n\) and we can formulate (1)\(_{1,2,4}\) as

$$\begin{aligned} \left\{ \begin{array}{lll} \partial _t \tilde{u}- \Delta \tilde{u} + \tilde{u} \cdot \nabla \tilde{u} + \nabla \tilde{p} = {\text {div }}\, \tilde{F} \quad &{}\text {in } \;G, \\ {\text {div }}\, {\tilde{u}} = 0 \quad &{}\text {in }\; G. \end{array} \right. \end{aligned}$$
(8)

We can verify that if u is a solution of (1)\(_{1,2,4}\), then \(\tilde{u} = u \circ \Pi ^{-1}\) is a solution of (8) with \(\tilde{F} = F \circ \Pi ^{-1}\). Conversely, if \(\tilde{u}\) is a solution of (8), then \(u = \tilde{u} \circ \pi\) is a solution of (1)\(_{1,2,4}\) with \(F= \tilde{F} \circ \pi\). For more details on the analysis on G, see [7, 8].

According to Kyed [9], the time-periodic Stokes fundamental solution \(\{ E, Q \}\) is given by

$$\begin{aligned} E = E_s \otimes 1_{\mathbb {T}} + E_p, \quad Q = Q_s \otimes \delta _{\mathbb {T}}, \end{aligned}$$

where \(1_{\mathbb {T}}\) is the constant distribution 1 and \(\delta _{\mathbb {T}}\) the Dirac delta distribution on \(\mathbb {T}\). Here \(\{ E_s, Q_s \}\) is the steady Stokes fundamental solution, that is,

$$\begin{aligned}&E_s= \left( E_{s,ij} \right) _{i,j=1}^n, \qquad E_{s, {ij}}(x) = \frac{1}{2 \omega _n} \left( \frac{\delta _{ij}}{n-2} |x|^{2-n}+ \frac{x_i x_j}{|x|^n}\right) ,\\&Q_s= \left( Q_{s,i} \right) _{i=1}^n, \qquad Q_{s,i} (x) = \frac{x_i}{\omega _n |x|^n}, \end{aligned}$$

with \(\omega _n\) denoting the surface area of the unit sphere in \({\mathbb {R}}^n\). The time-periodic remainder \(E_p\) is defined as a tempered distribution on G via the Fourier transform, see Kyed [9] and Eiter and Kyed [1] for its precise form. Properties of \(E_s\) are well-known and those of \(E_p\) are studied in [1, 9]. The properties of \(E_p\) are stated within the functional framework on G in [1, 9], however, they are still valid even if we replace \(E_p\) by \(E_p \circ \pi\), since \(L^q (G)\) and \(L_{per}^q ({\mathbb {R}} \times {\mathbb {R}}^n)\) are isometrically homeomorphic and so are \(W^{1,2,q} (G)\) and \(W_{per}^{1,2,q} ({\mathbb {R}} \times {\mathbb {R}}^n)\). For the sake of convenience, we shall simply write \(E_p\) instead of \(E_p \circ \pi\) below. Also, we denote by \(C=C(\cdot , \cdots , \cdot )\) various constants depending only on the quantities in parentheses.

Proposition 1

[1, 9] The time-periodic remainder \(E_p\) satisfies

$$\begin{aligned}&E_p \in L_{per}^q ({\mathbb {R}} \times {\mathbb {R}}^n) \quad for\; all \; q \in \left( 1,\frac{n+2}{n} \right) ,\\&\nabla E_p \in L_{per}^q ({\mathbb {R}} \times {\mathbb {R}}^n) \quad for\; all \; q \in \left[ 1,\frac{n+2}{n+1} \right) , \\&\Vert \nabla ^j E_p (\cdot ,x) \Vert _{L^q (0,T)} \le \frac{C}{|x|^{n+j}} \quad for\; all \; j=0,1,\ldots \ and \; q \in [ 1,\infty ), \end{aligned}$$

with \(C=C(n,q,T).\) Furthermore, for \(f \in L_{per}^q ({\mathbb {R}} \times {\mathbb {R}}^n)\) there exists a constant \(C=C(n,q,T)\) such that

$$\begin{aligned} \left\| \int _0^T \int _{{\mathbb {R}}^n} E_p (\cdot -s,\cdot -y) f(s,y) \,dyds \right\| _{1,2,q} \le C \Vert f \Vert _q \quad for \;1< q < \infty . \end{aligned}$$

The time-periodic Stokes fundamental solution is a tempered distribution on G and thus we should consider the convolution integral on G. However, we can apply it in the classical setting as stated in [9, Remark 1.2]. To see this, we note that in (5) the periodicity of E implies that of u and we may assume \(t \in [0,T)\). Let \(f \in G\) with \(\Pi ^{-1} (f)=(t,x)\) and let \(\tilde{u}\) be a solution of

$$\begin{aligned} \tilde{u} (f) = \int _G E (f-g) ({\text {div }} \tilde{F}- \tilde{u} \cdot \nabla \tilde{u})(g) \,dg. \end{aligned}$$
(9)

Then \(\tilde{u}\) is a solution of (8) in an appropriate sense. Lifting (9) by \(\pi\) yields

$$\begin{aligned} (\tilde{u} \circ \pi )(t,x) = \int _{{\mathbb {R}}^n} \frac{1}{T} \int _0^T (E \circ \pi ) (t-s,x-y) ({\text {div }}\, \tilde{F}- \tilde{u} \cdot \nabla \tilde{u}) (\pi (s,y)) \,dsdy , \end{aligned}$$

which is (5) with \(E=E \circ \pi\), \(u = \tilde{u} \circ \pi\) and \(F= \tilde{F} \circ \pi\). Conversely, let u be a solution of (5) and we write

$$\begin{aligned} u(t,x)&= \int _{{\mathbb {R}}^n} \frac{1}{T} \int _0^t (E \circ \pi ) (t-s,x-y) ({\text {div }}\, F- u \cdot \nabla u)(s,y) \,dsdy \\&\quad + \int _{{\mathbb {R}}^n} \frac{1}{T} \int _t^T (E \circ \pi ) (t-s+T,x-y) ({\text {div }}\, F- u \cdot \nabla u)(s,y) \,dsdy. \end{aligned}$$

For \((t,x) \in [0,T) \times {\mathbb {R}}^n\), we lift this equation by \(\varPi ^{-1}\) to get

$$\begin{aligned} (u \circ \varPi ^{-1}) (f) =\int _G E (f-g) ({\text {div }}\, F- u \cdot \nabla u)(\varPi ^{-1} (g)) \,dg, \end{aligned}$$

which is (9) with \(\tilde{u} = u \circ \varPi ^{-1}\) and \(\tilde{F} = F \circ \varPi ^{-1}\). Here we have used \(\varPi ^{-1} (f-g) = (t-s,x-y)\) for \(0 \le s \le t\) and \(\varPi ^{-1} (f-g) = (t-s+T,x-y)\) for \(t<s<T\). Therefore, there is a natural correspondence between the integral equations (5) and (9), and we readily see that a function u satisfying (5) is a solution of (1) in a suitable sense. If F and u decay rapidly at spatial infinity, we can obtain (4) by integrating (5) by parts and hence it suffices to consider (4) for our purpose.

In view of the form of E, we note again that every solution u of (4) can be written as the sum of the steady part \(u_s\):

$$\begin{aligned} u_s (x) = \int _{{\mathbb {R}}^n} \nabla E_s (x-y) \frac{1}{T} \int _0^T (F- u \otimes u)(s,y) \,dsdy \end{aligned}$$

and time-periodic part \(u_p\):

$$\begin{aligned} u_p (t,x) = \int _{{\mathbb {R}}^n} \frac{1}{T} \int _0^T \nabla E_p (t-s,x-y) (F- u \otimes u)(s,y) \,dsdy . \end{aligned}$$

Furthermore, the associated pressure p is written as

$$\begin{aligned} p(t,x) = \int _{{\mathbb {R}}^n} Q_s (x-y) \cdot ({\text {div }}\, F- u \cdot \nabla u)(t,y) \,dy. \end{aligned}$$

3.2 Proof of main results

With properties of the fundamental solution in hand, we estimate the steady and time-periodic parts. Let Y be a Banach space defined by

$$\begin{aligned} Y := \left\{ u \in L_{per}^\infty ({\mathbb {R}}; X_{n-1} ); \nabla u \in L_{per}^\infty ({\mathbb {R}}; X_n ) \right\} \end{aligned}$$

with the norm

$$\begin{aligned} \Vert u \Vert _Y := \Vert u \Vert _{\infty ,X_{n-1}} + \Vert \nabla u \Vert _{\infty ,X_n}. \end{aligned}$$

We define the operator \(K_s\) on Y by

$$\begin{aligned} (K_s u)(x) := \int _{{\mathbb {R}}^n} \nabla E_s (x-y) \frac{1}{T} \int _0^T (F- u \otimes u)(s,y) \,dsdy . \end{aligned}$$

Estimates for \(K_s u\) are studied in the next lemma.

Lemma 1

Let \(0< \delta <1\) and suppose F satisfies (6). For \(u \in Y\) we have

$$\begin{aligned} K_s u \in X_{n-1} \quad and \quad \nabla (K_s u) \in X_n , \end{aligned}$$

and there exists a constant \(C=C(n,\delta )\) such that

$$\begin{aligned} \Vert K_s u \Vert _{X_{n-1}} + \Vert \nabla (K_s u) \Vert _{X_n} \le C \left( \Vert F \Vert _{\infty , X_{n+ \delta }} + \Vert {\text {div }}\, F \Vert _{\infty , X_{n+1}} + \Vert u \Vert _Y^2 \right) . \end{aligned}$$

Proof

We first recall the basic estimate

$$\begin{aligned} \int _{{\mathbb {R}}^n} \frac{dy}{|x-y|^{n-1} (|y|+1)^{\mu }} \le C(|x|+1)^{1-n} \quad \text {if } \; \mu >n \end{aligned}$$
(10)

(see [12, Lemma 3.1]). Since \(| \nabla ^j E_s (x-y) | \le C|x-y|^{2-n-j}\) (\(j=0,1,\ldots\)), it follows from (10) that

$$\begin{aligned} \left| (K_s u) (x) \right|&\le \int _{{\mathbb {R}}^n} \frac{C}{|x-y|^{n-1}} \left\{ \frac{\Vert F \Vert _{\infty , X_{n+ \delta }}}{(|y|+1)^{n+ \delta }} + \frac{\Vert u \Vert _{\infty , X_{n-1}}^2}{(|y|+1)^{2n-2}} \right\} \,dy \\&\le C \left( \Vert F \Vert _{\infty , X_{n+\delta }} + \Vert u \Vert _{\infty , X_{n-1}}^2 \right) (|x|+1)^{1-n}. \end{aligned}$$

Hence we derive

$$\begin{aligned} K_s u \in X_{n-1} \end{aligned}$$

with the estimate

$$\begin{aligned} \Vert K_s u \Vert _{X_{n-1}} \le C \left( \Vert F \Vert _{\infty , X_{n+\delta }} + \Vert u \Vert _Y^2 \right) . \end{aligned}$$

Next, we write

$$\begin{aligned} \nabla (K_s u) (x)&= \int _{{\mathbb {R}}^n} \nabla E_s (x-y) \frac{1}{T} \int _0^T \left( {\text {div }} (F - u \otimes u ) \right) (s,y) \,dsdy \\&= \int _{|x-y|<\frac{|x|+1}{2}} \nabla E_s (x-y) \frac{1}{T} \int _0^T \left( {\text {div }} (F - u \otimes u ) \right) (s,y) \,dsdy \\&\quad + \int _{|x-y|>\frac{|x|+1}{2}} \nabla E_s (x-y) \frac{1}{T} \int _0^T \left( {\text {div }} (F - u \otimes u ) \right) (s,y) \,dsdy \\&=: I_1 + I_2. \end{aligned}$$

Since \(|x-y|<(|x|+1)/2\) implies \((|x|-1)/2<|y|< (3|x|+1)/2\), we see

$$\begin{aligned} | I_1|&\le \int _{|x-y|<\frac{|x|+1}{2}} \frac{C}{|x-y|^{n-1}} \left\{ \frac{\Vert {\text {div }}\, F \Vert _{\infty ,X_{n+1}}}{(|y|+1)^{n+1}} + \frac{\Vert u \Vert _{\infty ,X_{n-1}} \Vert \nabla u \Vert _{\infty ,X_n}}{(|y|+1)^{2n-1}} \right\} \,dy \nonumber \\&\le \frac{C \left( \Vert {\text {div }}\, F \Vert _{\infty ,X_{n+1}} + \Vert u \Vert _{\infty ,X_{n-1}} \Vert \nabla u \Vert _{\infty ,X_n} \right) }{(|x|+1)^{n+1}} \int _{|x-y|<\frac{|x|+1}{2}} \frac{dy}{|x-y|^{n-1}} \nonumber \\&\le C \left( \Vert {\text {div }}\, F \Vert _{\infty ,X_{n+1}} + \Vert u \Vert _{\infty ,X_{n-1}} \Vert \nabla u \Vert _{\infty ,X_n} \right) (|x|+1)^{-n}. \end{aligned}$$
(11)

We integrate \(I_2\) by parts to get

$$\begin{aligned} | I_2 | \le I_{21} + I_{22}, \end{aligned}$$

where

$$\begin{aligned} I_{21}&:= \int _{|x-y|>\frac{|x|+1}{2}} \left| \nabla ^2 E_s (x-y) \right| \frac{1}{T} \int _0^T \left| (F- u \otimes u)(s,y) \right| \,ds dy, \\ I_{22}&:= \int _{|x-y|=\frac{|x|+1}{2}} \left| \nabla E_s (x-y) \right| \frac{1}{T} \int _0^T \left| (F- u \otimes u)(s,y) \right| \,ds dS_y . \\ \end{aligned}$$

The assumption \(\delta >0\) leads us to

$$\begin{aligned} I_{21}&\le \int _{|x-y|>\frac{|x|+1}{2}} \frac{C}{|x-y|^n} \left\{ \frac{\Vert F \Vert _{\infty ,X_{n+ \delta }}}{(|y|+1)^{n+ \delta }} + \frac{\Vert u \Vert _{\infty ,X_{n-1}}^2}{(|y|+1)^{2n-2}} \right\} \, dy \nonumber \\&\le \frac{C \left( \Vert F \Vert _{\infty ,X_{n+ \delta }} + \Vert u \Vert _{\infty ,X_{n-1}}^2 \right) }{(|x|+1)^n} \int _{{\mathbb {R}}^n} \frac{dy}{(|y|+1)^{n+ \delta }} \nonumber \\&\le C \left( \Vert F \Vert _{\infty ,X_{n+ \delta }} + \Vert u \Vert _{\infty ,X_{n-1}}^2 \right) (|x|+1)^{-n}. \end{aligned}$$
(12)

Furthermore, \(|x-y|=(|x|+1)/2\) implies \((|x|-1)/2 \le |y| \le (3|x|+1)/2\) and we thus obtain

$$\begin{aligned} I_{22}&\le \int _{|x-y|=\frac{|x|+1}{2}} \frac{C}{|x-y|^{n-1}} \left\{ \frac{\Vert F \Vert _{\infty ,X_{n+ \delta }}}{(|y|+1)^{n+ \delta }} + \frac{\Vert u \Vert _{\infty ,X_{n-1}}^2}{(|y|+1)^{2n-2}} \right\} \, dS_y \nonumber \\&\le C \left( \Vert F \Vert _{\infty ,X_{n+ \delta }} + \Vert u \Vert _{\infty ,X_{n-1}}^2 \right) (|x|+1)^{1-n} (|x|+1)^{-n - \delta } \int _{|x-y|=\frac{|x|+1}{2}} \, dS_y \nonumber \\&\le C \left( \Vert F \Vert _{\infty ,X_{n+ \delta }} + \Vert u \Vert _{\infty ,X_{n-1}}^2 \right) (|x|+1)^{-n - \delta }. \end{aligned}$$
(13)

The estimates (12) and (13) yield

$$\begin{aligned} |I_2|&\le C \left( \Vert F \Vert _{\infty ,X_{n+ \delta }} + \Vert u \Vert _{\infty ,X_{n-1}}^2 \right) \left\{ (|x|+1)^{-n} + (|x|+1)^{-n - \delta } \right\} \nonumber \\&\le C \left( \Vert F \Vert _{\infty ,X_{n+ \delta }} + \Vert u \Vert _{\infty ,X_{n-1}}^2 \right) (|x|+1)^{-n} . \end{aligned}$$
(14)

It follows from (11) and (14) that

$$\begin{aligned} |\nabla (K_s u) (x)| \le C \Big ( \Vert F \Vert _{\infty ,X_{n+ \delta }} &+ \Vert {\text {div }}\, F \Vert _{\infty ,X_{n+1}} + \Vert u \Vert _{\infty ,X_{n-1}}^2 \\&\qquad + \Vert u \Vert _{\infty ,X_{n-1}} \Vert \nabla u \Vert _{\infty ,X_n} \Big ) (|x|+1)^{-n}. \end{aligned}$$

Therefore, we conclude

$$\begin{aligned} \nabla (K_s u) \in X_n \end{aligned}$$

with the estimate

$$\begin{aligned} \Vert \nabla (K_s u) \Vert _{X_n} \le C \left( \Vert F \Vert _{\infty ,X_{n+ \delta }} + \Vert {\text {div }}\, F \Vert _{\infty ,X_{n+1}} + \Vert u \Vert _Y^2 \right) . \end{aligned}$$

This completes the proof of Lemma 1.

We also define the operator \(K_p\) on Y by

$$\begin{aligned} (K_p u)(t,x) := \int _{{\mathbb {R}}^n} \frac{1}{T} \int _0^T \nabla E_p (t-s,x-y) (F- u \otimes u)(s,y) \,dsdy, \end{aligned}$$

and we use Proposition 1 to obtain the following estimates.

Lemma 2

Let \(0< \delta <1\) and suppose F satisfies (6). For \(u \in Y\) we have

$$\begin{aligned} K_p u \in L_{per}^\infty ({\mathbb {R}}; X_{n+\delta } ) \quad and \quad \nabla (K_p u) \in L_{per}^\infty ({\mathbb {R}}; X_{n+1}), \end{aligned}$$

and there exists a constant \(C=C(n,\delta ,T)\) such that

$$\begin{aligned} \Vert K_p u \Vert _{\infty , X_{n+\delta }} + \Vert \nabla (K_p u) \Vert _{\infty , X_{n+1}} \le C \left( \Vert F \Vert _{\infty ,X_{n+ \delta }} + \Vert {\text {div }}\, F \Vert _{\infty ,X_{n+1}} + \Vert u \Vert _Y^2 \right) . \end{aligned}$$

If F satisfies (7), then we have

$$\begin{aligned} K_p u \in L_{per}^\infty ({\mathbb {R}}; X_{n+1}) \quad and \quad \nabla (K_p u) \in L_{per}^\infty ({\mathbb {R}}; X_{n+2}). \end{aligned}$$
(15)

Proof

The periodicity of \(K_p u\) follows from that of \(E_p\) and we may assume \(t \in [0,T)\). We write

$$\begin{aligned} (K_p u) (t,x)&= \int _{|x-y|<\frac{|x|+1}{2}} \frac{1}{T} \int _0^T \nabla E_p (t-s,x-y) \left( F- u \otimes u \right) (s,y) \,dsdy \\&\quad + \int _{|x-y|>\frac{|x|+1}{2}} \frac{1}{T} \int _0^T \nabla E_p (t-s,x-y) \left( F- u \otimes u \right) (s,y) \,dsdy \\&=: I_3 + I_4. \end{aligned}$$

According to Proposition 1, we have \(\nabla E_p \in L_{per}^1 ({\mathbb {R}} \times {\mathbb {R}}^n)\) and thus

$$\begin{aligned} |I_3|&\le \int _{|x-y|<\frac{|x|+1}{2}} \left\{ \frac{\Vert F \Vert _{\infty , X_{n+\delta }}}{(|y|+1)^{n+\delta }} + \frac{\Vert u \Vert _{\infty ,X_{n-1}}^2}{(|y|+1)^{2n-2}} \right\} \frac{1}{T} \int _0^T \left| \nabla E_p (t-s,x-y) \right| \,dsdy \nonumber \\&\le \frac{C \left( \Vert F \Vert _{\infty ,X_{n+\delta }} + \Vert u \Vert _{\infty ,X_{n-1}}^2 \right) }{(|x|+1)^{n+\delta }} \int _{{\mathbb {R}}^n} \int _0^T \left| \nabla E_p (t-s,x-y) \right| \,dsdy \nonumber \\&\le C \left( \Vert F \Vert _{\infty ,X_{n+\delta }} + \Vert u \Vert _{\infty ,X_{n-1}}^2 \right) (|x|+1)^{-n-\delta }. \end{aligned}$$
(16)

Furthermore, we use the estimate in Proposition 1 to get

$$\begin{aligned} |I_4|&\le \int _{|x-y|>\frac{|x|+1}{2}} \left\{ \frac{\Vert F \Vert _{\infty ,X_{n+\delta }}}{(|y|+1)^{n+\delta }} + \frac{\Vert u \Vert _{\infty ,X_{n-1}}^2}{(|y|+1)^{2n-2}} \right\} \frac{1}{T} \int _0^T \left| \nabla E_p (t-s,x-y) \right| \,dsdy \nonumber \\&\le C \left( \Vert F \Vert _{\infty ,X_{n+\delta }} + \Vert u \Vert _{\infty ,X_{n-1}}^2 \right) \int _{|x-y|>\frac{|x|+1}{2}} \frac{dy}{|x-y|^{n+1} (|y|+1)^{n+\delta }} \nonumber \\&\le \frac{C \left( \Vert F \Vert _{\infty ,X_{n+\delta }} + \Vert u \Vert _{\infty ,X_{n-1}}^2 \right) }{(|x|+1)^{n+1}} \int _{{\mathbb {R}}^n} \frac{dy}{(|y|+1)^{n+\delta }} \nonumber \\&\le C \left( \Vert F \Vert _{\infty ,X_{n+\delta }} + \Vert u \Vert _{\infty ,X_{n-1}}^2 \right) (|x|+1)^{-n-1}. \end{aligned}$$
(17)

It follows from (16) and (17) that

$$\begin{aligned} \left| (K_p u) (t,x) \right| \le C \left( \Vert F \Vert _{\infty, X_{n+\delta }} + \Vert u \Vert _{\infty, X_{n-1}}^2 \right) (|x|+1)^{-n-\delta } \end{aligned}$$

uniformly in \(t \in [0,T)\), and we derive

$$\begin{aligned} K_p u \in L_{per}^\infty ({\mathbb {R}}; X_{n+\delta }) \end{aligned}$$

with the estimate

$$\begin{aligned} \Vert K_p u \Vert _{\infty , X_{n+\delta }} \le C \left( \Vert F \Vert _{\infty ,X_{n+\delta }} + \Vert u \Vert _Y^2 \right) . \end{aligned}$$

Concerning the estimate for derivatives, we write

$$\begin{aligned} \nabla (K_p u) (t,x)&= \int _{{\mathbb {R}}^n} \frac{1}{T} \int _0^T \nabla E_p (t-s,x-y) \left( {\text {div }} (F - u \otimes u ) \right) (s,y) \,dsdy \\&= \int _{|x-y|<\frac{|x|+1}{2}} \frac{1}{T} \int _0^T \nabla E_p (t-s,x-y) \left( {\text {div }} (F - u \otimes u ) \right) (s,y) \,dsdy \\&\quad + \int _{|x-y|>\frac{|x|+1}{2}} \frac{1}{T} \int _0^T \nabla E_p (t-s,x-y) \left( {\text {div }} (F - u \otimes u ) \right) (s,y) \,dsdy \\&=: I_5 + I_6. \end{aligned}$$

In addition, we integrate \(I_6\) by parts to obtain

$$\begin{aligned} |I_6| \le I_{61} + I_{62}, \end{aligned}$$

where

$$\begin{aligned} I_{61}&:= \int _{|x-y|>\frac{|x|+1}{2}} \frac{1}{T} \int _0^T \left| \nabla ^2 E_p (t-s,x-y) \right| \left| (F- u \otimes u)(s,y) \right| \,ds dy, \\ I_{62}&:= \int _{|x-y|=\frac{|x|+1}{2}} \frac{1}{T} \int _0^T \left| \nabla E_p (t-s,x-y) \right| \left| (F- u \otimes u) (s,y) \right| \,ds dS_y . \end{aligned}$$

We estimate \(I_5\) and \(I_{61}\) in the same way as (16) and (17), respectively, to get

$$\begin{aligned} |I_5|&\le C \left( \Vert {\text {div }}\, F \Vert _{\infty ,X_{n+1}} + \Vert u \Vert _{\infty ,X_{n-1}} \Vert \nabla u \Vert _{\infty ,X_n} \right) (|x|+1)^{-n-1}, \\ I_{61}&\le C \left( \Vert F \Vert _{\infty ,X_{n+\delta }} + \Vert u \Vert _{\infty ,X_{n-1}}^2 \right) (|x|+1)^{-n-2}. \end{aligned}$$

Furthermore, we see that

$$\begin{aligned} I_{62}&\le \int _{|x-y|=\frac{|x|+1}{2}} \left\{ \frac{\Vert F \Vert _{\infty ,X_{n+\delta }}}{(|y|+1)^{n+\delta }} + \frac{\Vert u \Vert _{\infty ,X_{n-1}}^2}{(|y|+1)^{2n-2}} \right\} \frac{1}{T} \int _0^T \left| \nabla E_p (t-s,x-y) \right| \,dsdS_y \\&\le C \left( \Vert F \Vert _{\infty ,X_{n+\delta }} + \Vert u \Vert _{\infty ,X_{n-1}}^2 \right) \int _{|x-y|=\frac{|x|+1}{2}} \frac{dS_y}{|x-y|^{n+1} (|y|+1)^{n+\delta }} \\&\le C \left( \Vert F \Vert _{\infty ,X_{n+\delta }} + \Vert u \Vert _{\infty ,X_{n-1}}^2 \right) (|x|+1)^{-n -\delta -2}. \end{aligned}$$

Hence

$$\begin{aligned} | I_6| \le C \left( \Vert F \Vert _{\infty ,X_{n+\delta }} + \Vert u \Vert _{\infty ,X_{n-1}}^2 \right) (|x|+1)^{-n-2} \end{aligned}$$

and the estimates for \(I_5\) and \(I_6\) lead us to

$$\begin{aligned} | \nabla (K_p u) (t,x)| \le C \Big ( \Vert F \Vert _{\infty ,X_{n+ \delta }} &+ \Vert {\text {div }}\, F \Vert _{\infty ,X_{n+1}} + \Vert u \Vert _{\infty ,X_{n-1}}^2 \\&\qquad + \Vert u \Vert _{\infty ,X_{n-1}} \Vert \nabla u \Vert _{\infty ,X_n} \Big ) (|x|+1)^{-n-1} \end{aligned}$$

uniformly in \(t \in [0,T)\). It follows that

$$\begin{aligned} \nabla (K_p u) \in L_{per}^\infty ({\mathbb {R}}; X_{n+1}) \end{aligned}$$

with the estimate

$$\begin{aligned} \Vert \nabla (K_p u) \Vert _{\infty , X_{n+1}} \le C \left( \Vert F \Vert _{\infty ,X_{n+ \delta }} + \Vert {\text {div }}\, F \Vert _{\infty ,X_{n+1}} + \Vert u \Vert _Y^2 \right) . \end{aligned}$$

We can easily verify (15) by using (7), instead of (6), in the estimates for \(I_3\) and \(I_5\).

Remark 4

Let \(0< \gamma \le 1\) and set \(\tilde{Y} := \{ u \in Y ; \partial _t u \in L_{per}^\infty ({\mathbb {R}}; X_{n+\gamma }) \}\). Assuming that \(\partial _t F \in L_{per}^\infty ({\mathbb {R}}; X_{n+\gamma })\), we can observe that \(\partial _t (K_p u) \in L_{per}^\infty ({\mathbb {R}}; X_{n+\gamma })\) (\(u \in \tilde{Y}\)) with the estimate \(\Vert \partial _t (K_p u) \Vert _{\infty ,X_{n+\gamma }} \le C ( \Vert \partial _t F \Vert _{\infty ,X_{n+\gamma }} +\Vert u \Vert _{\infty ,X_{n-1}} \Vert \partial _t u \Vert _{\infty ,X_{n+\gamma }})\). This observation, together with suitable modifications of the proof of Theorem 1 below, yields Remark 3.

Set

$$\begin{aligned} K := K_s + K_p , \end{aligned}$$

that is,

$$\begin{aligned} (Ku)(t,x) = \int _{{\mathbb {R}}^n} \frac{1}{T} \int _0^T \nabla E (t-s,x-y) (F- u \otimes u)(s,y) \,dsdy . \end{aligned}$$

Since the embedding \(L_{per}^\infty ({\mathbb {R}}; X_{\mu _1}) \subset L_{per}^\infty ({\mathbb {R}}; X_{\mu _2})\) for \(\mu _2 < \mu _1\) is continuous, Lemmas 1 and 2 yield the following lemma.

Lemma 3

Let \(0< \delta <1\) and suppose F satisfies (6). The operator K maps Y to itself and there exists a constant \(C=C(n,\delta ,T)\) such that

$$\begin{aligned} \Vert Ku \Vert _Y \le C \left( \Vert F \Vert _{\infty ,X_{n+ \delta }} + \Vert {\text {div }}\, F \Vert _{\infty ,X_{n+1}} + \Vert u \Vert _Y^2 \right) \quad (u \in Y). \end{aligned}$$

Remark 5

The constant C in Lemma 3 is determined as follows. By Lemmas 1 and 2, there exists a constant \(C_1 = C_1 (n,\delta ,T)\) such that the estimate for Ku above holds. On the other hand, in the proof of Theorem 1 below, we also need the estimate

$$\begin{aligned} \left\| \int _{{\mathbb {R}}^n} \int _0^T \nabla E (\cdot -s,\cdot -y) (u \otimes v)(s,y) \,dsdy \right\| _Y \le C_2 \Vert u \Vert _Y \Vert v \Vert _Y \quad (u,v \in Y), \end{aligned}$$
(18)

which follows immediately from the proofs of Lemmas 1 and 2. Here \(C_2\) is a constant depending only on n and T. It is not clear from the proofs of Lemmas 1 and 2 whether \(C_1\) is larger than \(C_2\). The proof of Theorem 1 shall require the condition \(C \ge \max \{ C_1, C_2\}\), and hence we take \(C = \max \{ C_1, C_2\}\).

Now we follow the standard argument via the contraction mapping principle to construct a solution with desired decay properties of (4). We prove only Theorem 1, since Theorem 2 follows from Lemmas 1 and 2 together with the proof of Theorem 1 below. Indeed, the argument in the third paragraph of the proof of Theorem 1 is applicable to arbitrary solutions \(u \in Y\) of (4) and the pointwise decay properties stated in Theorem 2 follow immediately from Lemmas 1 and 2.

Proof of Theorem 1

We employ the successive approximation

$$\begin{aligned} v_0 (t,x)&= \int _{{\mathbb {R}}^n} \frac{1}{T} \int _0^T \nabla E (t-s,x-y) F(s,y) \,dsdy , \\ v_{k+1} (t,x)&= \int _{{\mathbb {R}}^n} \frac{1}{T} \int _0^T \nabla E (t-s,x-y) (F- v_k \otimes v_k)(s,y) \,dsdy . \end{aligned}$$

According to Lemma 3, we have \(v_k \in Y\) for all \(k=0,1,\ldots\) with the estimate

$$\begin{aligned} \Vert v_{k+1} \Vert _Y \le M_0 \left( \Vert F \Vert _{\infty ,X_{n+ \delta }} + \Vert {\text {div }}\, F \Vert _{\infty ,X_{n+1}} + \Vert v_k \Vert _Y^2 \right) , \end{aligned}$$

where \(M_0\) is the constant in the lemma and is independent of k. We assume

$$\begin{aligned} \Vert F \Vert _{\infty ,X_{n+ \delta }} + \Vert {\text {div }}\, F \Vert _{\infty ,X_{n+1}} < \frac{1}{4M_0^2} \end{aligned}$$

to deduce for all \(k \ge 1\) that

$$\begin{aligned} \Vert v_k \Vert _Y&\le M_1 := \frac{1-\sqrt{1- 4M_0^2 ( \Vert F \Vert _{\infty ,X_{n+ \delta }} + \Vert {\text {div }}\, F \Vert _{\infty ,X_{n+1}}} ) }{2M_0} \\&< \frac{1}{2M_0}. \end{aligned}$$

We put

$$\begin{aligned} w_k :&= v_{k+1} - v_k \\&= - \int _{{\mathbb {R}}^n} \frac{1}{T} \int _0^T \nabla E (t-s,x-y) (w_{k-1} \otimes v_k + v_{k-1} \otimes w_{k-1} )(s,y) \,dsdy . \end{aligned}$$

In view of Remark 5, we have

$$\begin{aligned} \Vert w_k \Vert _Y \le M_0 \left( \Vert v_k \Vert _Y + \Vert v_{k-1} \Vert _Y \right) \Vert w_{k-1} \Vert _Y \le 2M_0 M_1 \Vert w_{k-1} \Vert _Y , \end{aligned}$$

so that

$$\begin{aligned} \Vert w_k \Vert _Y \le (2M_0 M_1)^k \Vert w_0 \Vert _Y . \end{aligned}$$

Since \(2M_0 M_1 <1\), we see that \(\{ v_k \}\) converges in Y to a function u satisfying

$$\begin{aligned} u(t,x) = \int _{{\mathbb {R}}^n} \frac{1}{T} \int _0^T \nabla E (t-s,x-y) (F- u \otimes u)(s,y) \,dsdy. \end{aligned}$$

Noting that this solution \(u \in Y\) of (4) satisfies the estimate \(\Vert u \Vert _Y \le C ( \Vert F \Vert _{\infty ,X_{n+ \delta }} + \Vert {\text {div }}\, F \Vert _{\infty ,X_{n+1}} )\) with \(C=C(n,\delta ,T)\) and that F and \({\text {div }}\, F\) are sufficiently small in \(L_{per}^\infty ({\mathbb {R}}; X_{n+ \delta } )\) and \(L_{per}^\infty ({\mathbb {R}}; X_{n+1})\) respectively, we can easily verify that u is unique in the class of small solutions in Y by applying the estimate (18).

Next, we prove the decay property of the associated pressure p. By the integration by parts, we get

$$\begin{aligned} p(t,x)&= \int _{{\mathbb {R}}^n} Q_s (x-y) \cdot ({\text {div }}\, F- u \cdot \nabla u)(t,y) \,dy \\&\le I_7 + I_8 + I_9, \end{aligned}$$

where

$$\begin{aligned} I_7&:= \int _{|x-y|<\frac{|x|+1}{2}} \left| Q_s (x-y) \right| \left| \left( {\text {div }}\, F - u \cdot \nabla u \right) (t,y) \right| \,dy ,\\ I_8&:= \int _{|x-y|>\frac{|x|+1}{2}} | \nabla Q_s (x-y) | \left| (F - u \otimes u)(t,y) \right| \,dy , \\ I_9&:= \int _{|x-y|=\frac{|x|+1}{2}} | Q_s (x-y)| \left| (F - u \otimes u)(t,y) \right| \,dS_y . \end{aligned}$$

It is clear that p is T-periodic and we may assume \(t \in [0,T)\). Recalling the estimate \(| \nabla ^j Q_s (x-y) | \le C| x-y |^{1-n-j}\) \((j=0,1,\ldots )\), we calculate \(I_7\), \(I_8\) and \(I_9\) in the same way as (11), (12) and (13), respectively, to deduce

$$\begin{aligned} I_7&\le C \left( \Vert {\text {div }}\, F \Vert _{\infty ,X_{n+1}} + \Vert u \Vert _{\infty ,X_{n-1}} \Vert \nabla u \Vert _{\infty ,X_n} \right) (|x|+1)^{-n}, \\ I_8&\le C \left( \Vert F \Vert _{\infty ,X_{n+ \delta }} + \Vert u \Vert _{\infty ,X_{n-1}}^2 \right) (|x|+1)^{-n}, \\ I_9&\le C \left( \Vert F \Vert _{\infty ,X_{n+ \delta }} + \Vert u \Vert _{\infty ,X_{n-1}}^2 \right) (|x|+1)^{-n-\delta }. \end{aligned}$$

Consequently,

$$\begin{aligned} |p(t,x)| \le C \left( \Vert F \Vert _{\infty ,X_{n+ \delta }} + \Vert {\text {div }}\, F \Vert _{\infty ,X_{n+1}} + \Vert u \Vert _Y^2 \right) (|x|+1)^{-n} \end{aligned}$$

uniformly in \(t \in [0,T)\) and we conclude

$$\begin{aligned} p \in L_{per}^\infty ({\mathbb {R}}; X_n). \end{aligned}$$

Finally, let \(u_s\) and \(u_p\) be the steady and time-periodic parts of the solution u obtained above. By Lemma 1, we have \(u_s \in X_{n-1}\) and \(\nabla u_s \in X_n\). Calculations similar to those of the estimate for the associated pressure p above yield

$$\begin{aligned} p_s (x) := \int _{{\mathbb {R}}^n} Q_s (x-y) \cdot \frac{1}{T} \int _0^T ({\text {div }}\, F- u \cdot \nabla u)(s,y) \,dsdy \in X_n. \end{aligned}$$

The pair \(\{ u_s, p_s \}\) is a solution of the stationary Stokes equation

$$\begin{aligned} \left\{ \begin{array}{lll} - \Delta u_s + \nabla p_s = \frac{1}{T} \int _0^T ({\text {div }}\, F- u \cdot \nabla u)(s,\cdot ) \,ds \quad &{}\text {in } \;{\mathbb {R}}^n, \\ {\text {div }}\, u_s = 0 \quad &{}\text {in } \; {\mathbb {R}}^n , \end{array} \right. \end{aligned}$$

in the sense of distributions. By the class of \(\{ u_s, p_s \}\) and \(\int _0^T ({\text {div }}\, F- u \cdot \nabla u)(s,\cdot ) \,ds \in X_{n+1}\), we can apply the theory for the existence and uniqueness of strong solutions to the stationary Stokes equation ([6, Proposition 2.9]) to deduce that

$$\begin{aligned} \nabla ^2 u_s \in L^q ({\mathbb {R}}^n) \quad {\text {for all }} \;1<q<\infty . \end{aligned}$$

Also, we write

$$\begin{aligned} \nabla ^2 u_p (t,x) = \nabla ^2 \int _{{\mathbb {R}}^n} \frac{1}{T} \int _0^T E_p (t-s,x-y) ({\text {div }}\, F- u \cdot \nabla u)(s,y) \,dsdy , \end{aligned}$$

and the estimate for convolution in Proposition 1, together with \({\text {div }}\, F- u \cdot \nabla u \in L_{per}^\infty ({\mathbb {R}}; X_{n+1})\), implies

$$\begin{aligned} \nabla ^2 u_p \in L_{per}^q ({\mathbb {R}} \times {\mathbb {R}}^n) \quad {\text {for all }} \;1<q<\infty . \end{aligned}$$

Consequently, we derive

$$\begin{aligned} \nabla ^2 u = \nabla ^2 u_s + \nabla ^2 u_p \in L_{per}^q ({\mathbb {R}} \times {\mathbb {R}}^n) \quad {\text {for all}}\;1<q<\infty . \end{aligned}$$

Similarly, the property \(\partial _t u \in L_{per}^q ({\mathbb {R}} \times {\mathbb {R}}^n)\) (\(1<q<\infty\)) follows from the representation

$$\begin{aligned} \partial _t u_p (t,x) = \partial _t \int _{{\mathbb {R}}^n} \frac{1}{T} \int _0^T E_p (t-s,x-y) ({\text {div }}\, F- u \cdot \nabla u)(s,y) \,dsdy \end{aligned}$$

and Proposition 1. The proof of Theorem 1 is complete.