Abstract
We investigate the pointwise behavior of time-periodic Navier–Stokes flows in the whole space. We show that if the time-periodic external force is sufficiently small in an appropriate sense, then there exists a unique time-periodic solution \(\{ u,p \}\) of the Navier–Stokes equation such that \(|u(t,x)|=O(|x|^{1-n})\), \(|\nabla u(t,x)|=O(|x|^{-n})\) and \(|p(t,x)|=O(|x|^{-n})\) uniformly in \(t \in {\mathbb {R}}\) as \(|x| \rightarrow \infty\). Our solution decays more rapidly than the time-periodic Stokes fundamental solution. The proof is based on the representation formula of a solution via the time-periodic Stokes fundamental solution and its properties.
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1 Introduction
We consider the time-periodic problem for the Navier–Stokes equation in \({\mathbb {R}} \times {\mathbb {R}}^n\) with \(n \ge 3\):
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
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
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
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
with the norm
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
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
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
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
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
and
Furthermore, the associated pressure p satisfies
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 \(u_s\):
and time-periodic part \(u_p\):
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
and
If
then we have
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
Derivatives on G are defined by
The differentiable structure on G is inherited from \({\mathbb {R}} \times {\mathbb {R}}^n\) and we can formulate (1)\(_{1,2,4}\) as
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
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,
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
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
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
Then \(\tilde{u}\) is a solution of (8) in an appropriate sense. Lifting (9) by \(\pi\) yields
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
For \((t,x) \in [0,T) \times {\mathbb {R}}^n\), we lift this equation by \(\varPi ^{-1}\) to get
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 \(u_s\):
and time-periodic part \(u_p\):
Furthermore, the associated pressure p is written as
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
with the norm
We define the operator \(K_s\) on Y by
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
and there exists a constant \(C=C(n,\delta )\) such that
Proof
We first recall the basic estimate
(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
Hence we derive
with the estimate
Next, we write
Since \(|x-y|<(|x|+1)/2\) implies \((|x|-1)/2<|y|< (3|x|+1)/2\), we see
We integrate \(I_2\) by parts to get
where
The assumption \(\delta >0\) leads us to
Furthermore, \(|x-y|=(|x|+1)/2\) implies \((|x|-1)/2 \le |y| \le (3|x|+1)/2\) and we thus obtain
The estimates (12) and (13) yield
It follows from (11) and (14) that
Therefore, we conclude
with the estimate
This completes the proof of Lemma 1.
We also define the operator \(K_p\) on Y by
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
and there exists a constant \(C=C(n,\delta ,T)\) such that
If F satisfies (7), then we have
Proof
The periodicity of \(K_p u\) follows from that of \(E_p\) and we may assume \(t \in [0,T)\). We write
According to Proposition 1, we have \(\nabla E_p \in L_{per}^1 ({\mathbb {R}} \times {\mathbb {R}}^n)\) and thus
Furthermore, we use the estimate in Proposition 1 to get
It follows from (16) and (17) that
uniformly in \(t \in [0,T)\), and we derive
with the estimate
Concerning the estimate for derivatives, we write
In addition, we integrate \(I_6\) by parts to obtain
where
We estimate \(I_5\) and \(I_{61}\) in the same way as (16) and (17), respectively, to get
Furthermore, we see that
Hence
and the estimates for \(I_5\) and \(I_6\) lead us to
uniformly in \(t \in [0,T)\). It follows that
with the estimate
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
that is,
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
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
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
According to Lemma 3, we have \(v_k \in Y\) for all \(k=0,1,\ldots\) with the estimate
where \(M_0\) is the constant in the lemma and is independent of k. We assume
to deduce for all \(k \ge 1\) that
We put
In view of Remark 5, we have
so that
Since \(2M_0 M_1 <1\), we see that \(\{ v_k \}\) converges in Y to a function u satisfying
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
where
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
Consequently,
uniformly in \(t \in [0,T)\) and we conclude
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
The pair \(\{ u_s, p_s \}\) is a solution of the stationary Stokes equation
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
Also, we write
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
Consequently, we derive
Similarly, the property \(\partial _t u \in L_{per}^q ({\mathbb {R}} \times {\mathbb {R}}^n)\) (\(1<q<\infty\)) follows from the representation
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Acknowledgements
The research was supported by the Academy of Sciences of the Czech Republic, Institute of Mathematics (RVO: 67985840). The author would like to thank Professor M. Kyed for useful comments.
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Nakatsuka, T. On time-periodic Navier–Stokes flows with fast spatial decay in the whole space. J Elliptic Parabol Equ 4, 51–67 (2018). https://doi.org/10.1007/s41808-018-0011-8
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DOI: https://doi.org/10.1007/s41808-018-0011-8