NAVIER-STOKES-OSEEN FLOWS IN THE EXTERIOR OF A ROTATING AND TRANSLATING OBSTACLE

. In this paper, we investigate Navier-Stokes-Oseen equation de-scribing ﬂows of incompressible viscous ﬂuid passing a translating and rotating obstacle. The existence, uniqueness, and polynomial stability of bounded and almost periodic weak mild solutions to Navier-Stokes-Oseen equation in the solenoidal Lorentz space L 3 σ,w are shown. Moreover, we also prove the unique existence of time-local mild solutions to this equation in the solenoidal Lorentz spaces L 3 ,qσ .

1. Introduction and preliminaries. Consider a rigid body R moving through an incompressible viscous fluid that fills the whole three-dimensional space R 3 exterior to R. We assume that with respect to a frame attached to R, the translational velocity u ∞ and the angular velocity ω of R are both constant vectors. Without loss of generality we may assume that ω = ae 3 , e 3 = (0, 0, 1) T . If the flow is nonslip at the boundary, then the motion of fluid can be described by the following equation:    D t v + (v · ∇)v − ∆v + ∇π = div G, div v = 0 in Ω(t) (t > 0), v(y, t)| ∂Ω(t) = ω × y| ∂Ω(t) , lim |y|→∞ v(y, t) = u ∞ , v(y, 0) = v 0 (y) (1) in the time-dependent exterior domain Ω(t) = O(at)Ω, where Ω is a fixed exterior domain complemented to R at the time t = 0 with smooth boundary, and O(t) denotes the orthogonal matrix Here D t = ∂/∂t, v = v(y, t) is the velocity field of the liquid; π = π(y, t) is the pressure field; and G = G(y, t) is a second-order tensor field.

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To study (1) in the time-independent exterior domain Ω(t), the following change of variables and unknown functions have been introduced π(y, t), F (x, t) = O(at) T G(y, t)O(at).
The Eq. (3) have been investigated by many authors, e.g., Shibata [26,27], Galdi and Silvestre [7,8,9], Geissert, Heck and Hieber [11], and many others (see [2,3,5,10,15,16,18,20,23,24]). Shibata [27] has shown the unique existence of solution to the linearized equation corresponding to Eq. (3) and proved the stability of the stationary solution. Meanwhile, Galdi and Silvestre [9] have shown the existence and uniqueness of the steady solution. In the case of moving or rotating obstacle, the unique existence of time-local mild solution have been proved by Geissert, Heck and Hieber [11]. However, almost authors study the existence and properties of solution to Eq. (3) in the spaces L p σ (see (4) below). The reader is also referred to [2,14,15,24,26,27] for the treatment of the existence and uniqueness of solution to Eq. (3) in various cases.
In this paper we consider the case of rotating and translating obstacle. Thus, the form of Eq. (3) is more general than the equations before. Our purpose is to prove the existence, uniqueness, and polynomial stability of bounded and almost periodic weak mild solutions to the Eq. (3) in the solenoidal Lorentz space L 3 σ,w , and the unique existence of time-local mild solutions to this equation in the solenoidal Lorentz spaces L 3,q σ (see (5) below). When q > 3, the space L 3,q σ is bigger than the space L 3 σ . Thus, the initial value can be chosen in the bigger data class. For the first results, our approach relies on the interpolation spaces combined with the smoothing properties of semigroup corresponding to linearized equation. For the latter result we use the Kato-iteration scheme as in [12,13,17,29].
The keys of our strategy are lying on the duality estimates, the smoothing properties and interpolation functors for semigroup corresponding to linearized equation, and the technique of cut-off function to translate velocity of the liquid on boundary ∂Ω of Eq. (3) to zero. This paper is organized as follows. In Section 2 and 3, we prove the existence, uniqueness, and polynomial stability of bounded and almost periodic weak mild solutions. In Section 4 we investigate the unique existence of time-local mild solutions. The main results are contained in Theorem 2.1, Theorem 3.4 and Theorem 4.1.
We now recall some preliminaries that will be used in the next sections. Throughout this paper, the following spaces will be used We also need the notion of Lorentz space L r,q (Ω), (1 ≤ r ≤ ∞, 1 ≤ q ≤ ∞), defined in as [1,4,19,22,28], and note that L r,r (Ω) = L r (Ω) and for q = ∞ then the space L r,∞ (Ω) is called the weak-L r space and is denoted by L r w (Ω) := L r,∞ (Ω). Note that the Lorentz spaces can be described by using interpolation pairs as follows Some usefulness properties of Lorentz spaces such as embedding property, Hölder type inequality are given in the following lemma.
It is known (see Shibata [26,27]) that −L a,k is a generator of bounded C 0 -semigroup (e −tL a,k ) t≥0 on L r σ (Ω) for each 1 < r < ∞ (note that, generally, this semigroup is not analytic). By interpolation theory, (e −tL a,k ) t>0 is also the bounded C 0semigroup on the space L r,q σ (Ω). Using interpolation theory, we can transfer the L p − L q decay estimates obtained by Shibata in [27,Theorem 3] for (e −tL a,k ) t≥0 on L r σ (Ω) to the L r,q − L p,q decay estimates for that semigroup on the space L r,q σ (Ω) in the following proposition. Proposition 1. Let 1 < r < ∞, 1 ≤ q ≤ ∞ and denote by f r,q the norm in the spaces L r,q (Ω). Then, the following inequalities hold.
In order to prove (11), we take g ∈ C ∞ 0,σ (Ω). Using triangle inequality and (7) we have t . The next step, approximating f by g and sending t → 0 we get (11). The proof of (12) follows the similar manner.
2. Boundedness and polynomial stability of weak mild solutions to Navier-Stokes-Oseen equation. In this section, we will show the unique existence and polynomial stability of bounded weak mild solutions to the Navier-Stokes-Oseen equation. We put Then, the Eq. (3) is rewritten as To transform the boundary condition to the case of the zero vector-field on ∂Ω. We perform as in [25], take a cut-off function ϕ ∈ C ∞ c (R 3 ) such that ϕ ≥ 0, ϕ ≡ 1 on the neighborhood of R = Ω c and supp ϕ ⊂ B(0, r) for some r > 0 and define . Then, equation (13) is equivalent to the following equation with zero-boundary condition: Applying the Helmholtz projection to (15) and considering initial value in the solenoidal Lorentz space L 3 σ,w (Ω), we obtain the following operator equation where L a,k is defined as in (6). By straightforward computations, we have Then, the Eq. (16) has form where It is now convenient to give the definition of weak mild solutions to the above Navier-Stokes-Oseen equation, namely, by a weak mild solution to (17) that means the solution to equation for all ϕ ∈ L 3 2 ,1 σ (Ω) and t ∈ R + . We next come to our first main result on the unique existence and polynomial stability of bounded weak mild solutions to (17) in the following theorem.
σ,w (Ω) 3×3 ). Then, the following assertions hold true. (a) If the norm F ∞, 3 2 ,w , b ω 3,w and z 0 3,w are small enough, Eq. (17) has a unique bounded weak mild solution in a small closed ball of where r is any fixed number satisfying r > 3.

Proof. (a):
For each z 0 ∈ L 3 σ,w (Ω), denotingẑ 0 (t) = e −tL a,k z 0 . Applying the estimate (7) for p = r = 3, we have and is a contractive mapping. Firstly, we prove that the function T z ∈ C b (R + , L 3 σ,w (Ω)). Indeed, for fixed t > 0 we have By the Hölder type inequality in Lemma 1. Hence, Thus We now use L r,q − L p,q estimate and interpolation technique to prove that We choose real numbers p 1 and p 2 such that 1 < p 1 < 3 2 < p 2 < 3. Consider the sublinear operator A which maps a function ϕ ∈ L p1 σ,w (Ω) 2p j ϕ pj ,w for j = 1, 2.
Combining the above inequality with (19), we obtain Note that we have 3 2 ,w |t 2 − t 1 |. By the functionsẑ 0 and G(z) are continuous on R + , so the function T z is also continuous. Thus, T z ∈ C b (R + , L 3 σ,w (Ω)). Next, we prove that T is a contractive mapping. Indeed, By arguing similarly as above, we obtain By (21), (23) and (22), if ρ, F ∞, 3 2 ,w , b ω 3,w and z 0 3,w are small enough then the mapping T acts from B ρ (ẑ 0 ) into itself and is a contractive mapping. So, Eq. (17) has a unique bounded weak mild solution in a small closed ball of C b (R + , L 3 σ,w (Ω)). (b): Letẑ be bounded weak mild solution of Eq. (17) and z ∈ C b (R + , L 3 σ,w (Ω)) be any other weak mild solutions of Eq. (17) such that z(0) −ẑ(0) 3,w is small enough.
Putting u = z −ẑ we obtain that u satisfies the equation For any fixed r > 3 we set We will prove that if b ω 3,w , z(0) −ẑ(0) 3,w and ẑ ∞,3,w are small enough then the Eq. (24) has a unique solution in a small closed ball of M.
Indeed, for u ∈ M consider the mapping Φ defined as follows Let B ρ be a closed ball in M centered at 0 with radius ρ. We then prove that if ρ, b ω 3,w , z(0) −ẑ(0) 3,w and ẑ ∞,3,w are small enough then the mapping Φ acts from B ρ into itself and is a contraction. Arguing similarly as in the proof of Assertion (a) we have Φu ∈ C b (R + , L 3 σ,w (Ω)). Furthermore, By L r,∞ − L 3,∞ estimate for semigroup e −tL a,k (see (7)) we obtain that For each t > 0, we have We have By the interpolation technique as in proof of Assertion (a), we obtain that Therefore, We also have Combining (25), (26) and (27) we obtain Similar calculations lead to Therefore, if ρ, b ω 3,w , z(0) −ẑ(0) 3,w and ẑ ∞,3,w are small enough then the mapping Φ acts from B ρ into itself and is a contraction. Then the equation Φu = u has unique solution in M. Since u = z −ẑ, we obtain the polynomial stability of the bounded weak mild solutionẑ and the inequality for all t > 0.

Almost periodic weak mild solutions to Navier-Stokes-Oseen equation.
Assuming that the second-order tensor field F is almost periodic on the whole line, we then show the existence and uniqueness of an almost periodic weak mild solutions for the Navier-Stokes-Oseen equation. Firstly, we recall the following definition of almost periodic functions. Definition 3.1. Let X be a Banach space. A continuous function f : R → X is called almost periodic if for all > 0 there exists a real number L > 0 such that for every a ∈ R, we can find T ∈ [a, a + L ] such that f (t + T ) − f (t) < for all t ∈ R. Remark 1. From definition we easily see that an almost periodic function is bounded and uniformly continuous. Details on almost periodic functions can be found in [21].
To prove the uniqueness of an almost periodic weak mild solution, we need the following lemma (can see [25, Lemma 3.2]).
We then consider the inhomogeneous equation Similarly to the case of semilinear equation, a weak mild solution to (28) that means the solution to the integral equation σ,w (Ω) 3×3 ) is almost periodic. Then, Eq. (28) has a unique almost periodic weak mild solution in C b (R, L 3 σ,w (Ω)), and this solution has form Proof. Indeed, for each ϕ ∈ L 3 2 ,1 On the other hand, (see (20)).

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Thus, u(t) is uniformly bounded on R, and It is straightforward to check that u(t) is continuous on R and that for fixed s ∈ R, u(t) satisfies the equation i.e., u(t) is a weak mild solution to (28).
Since F is almost periodic so for all > 0 there exists a real number L > 0 such that for every a ∈ R, we can find T ∈ [a, a + L ] such that F (t + T ) − F (t) 3 2 ,w < for all t ∈ R. Then, we have Therefore, u(t) is almost periodic weak mild solution of Eq. (28). Next we prove the uniqueness of the almost periodic weak mild solution. To do this, assume that v(t) ∈ C b (R, L 3 σ,w (Ω)) is another almost periodic weak mild solution to (28). Then, σ,w (Ω)), and for any fixed s ∈ R it holds that z(t) = e −(t−s)L a,k z(s) for t ≥ s. By the estimate (7), for r > 3 then we have With T = 0, there exists k ≥ 1 such that k|T | ≥ 1. Thus, z(t) is also almost periodic function in the space C b (R, L r σ,w (Ω)). By the estimate (7), we have lim t→+∞ z(t) r,w = 0. By Lemma 3.2, we have z(t) = 0 on R. So, u ≡ v.
We will now prove the existence and uniqueness of an almost periodic weak mild solution to the Navier-Stokes-Oseen equation on the whole line: where Theorem 3.4. Suppose that the second-order tensor field F ∈ C b (R, L 3 2 σ,w (Ω) 3×3 ) is almost periodic. Then, the following assertions hold true. (a) If the norm F ∞, 3 2 ,w and b ω 3,w are small enough then Eq. (30) has a unique almost periodic weak mild solution in a small closed ball of C b (R, L 3 σ,w (Ω)). (b) The almost periodic weak mild solutionẑ of Eq. (30) is polynomial stability in the sense that for any other weak mild solution z ∈ C b (R + , L 3 σ,w (Ω)) of Eq. (30) such that z(0) −ẑ(0) 3,w is small enough, we have where r is any fixed number satisfying r > 3.
Proof. Since the almost periodic weak mild solutionẑ is bounded, assertion (b) follows directly from the part (b) of Theorem 2.1. Hence, we need only to prove assertion (a).
Consider the following closed set : v is almost periodic and v ∞,3,w ≤ ρ}. Since σ,w (Ω) 3×3 ) and is almost periodic. For each v ∈ B ρ we define the map T as follows σ,w (Ω)) is the unique almost periodic weak mild solution to the equation D t z + L a,k z = P div G(v).
(31) The existence and uniqueness of z are guaranteed by Lemma 3.3. Moreover, by (29) we have z ∞,3,w ≤ M G(v) ∞, 3 2 ,w . In a same way as in (19), we obtain . Therefore, if the norm F ∞, 3 2 ,w and b ω 3,w are small enough then T acts from B ρ into itself.
For v 1 , v 2 ∈ B ρ , by Lemma 3.3 for T v 1 = z 1 and T v 2 = z 2 we obtain So, if ρ, F ∞, 3 2 ,w and b ω 3,w are small enough then the mapping T acts from B ρ into itself and is a contractive mapping. Therefore, Eq. (30) has uniqueness of almost periodic weak mild solution in a small closed ball of C b (R, L 3 σ,w (Ω)). If the function F (t) is periodic then the solution u(t) in the Lemma 3.3 is also periodic and has the same period. Moreover, a periodic function on R + can be extended to become a periodic function on R. Thus without loss of generality, we assume that F is periodic on R. Then, we obtain the following corollary for the unique existence and polynomial stability of periodic weak mild solution to Eq. (30).

Corollary 1.
Suppose that the second-order tensor field F ∈ C b (R, L 3 2 σ,w (Ω) 3×3 ) is periodic. Then, the following assertions hold true. (a) If the norm F ∞, 3 2 ,w and b ω 3,w are small enough then Eq. (30) has an uniquely periodic weak mild solution in a small closed ball of C b (R, L 3 σ,w (Ω)). (b) The periodic weak mild solutionẑ of Eq. (30) is polynomial stability in the sense that for any other weak mild solution z ∈ C b (R + , L 3 σ,w (Ω)) of Eq. (30) such that z(0) −ẑ(0) 3,w is small enough, we have where r is any fixed number satisfying r > 3.
The under viewpoint, a stationary solution is considered as periodic solution with arbitrary period. On the other hand, by Corollary 1 for each the periodic function F (t) then Eq. (30) having unique solution u(t) is periodic and has the same period. So, if the function F (t) is time-independent then Eq. (30) has an uniquely weak stationary solution and this solution is polynomial stability.
where r is any fixed number satisfying r > 3.

4.
Time-local mild solutions to Navier-Stokes-Oseen equation. In order to easily track, we restate Navier-Stokes-Oseen equation (see (17)) under the following form where In this section we prove that Eq. (32) has time-local mild solution in the solenoidal Lorentz space L 3,q σ , 1 < q < ∞, that means z(t) is solution of the integral equation on [0, T ], with T > 0 is small enough. Here, we use Kato iteration scheme combining with interpolation theory and duality estimates to prove the existence of time-local mild solution.
Theorem 4.1. Assume that f ∈ C(R + , L 3,q σ (Ω)). Then, for each r > 3 there exists In the functional space C(R + , L 3,q σ (Ω)), we set the following iterative sequence z 1 (t) = e −tL a,k z 0 , where T > 0 will be defined later. We firstly show that K j and K j are bounded for all j.
For ϕ ∈ C ∞ 0,σ (R 3 ) 3×3 , using Hölder type inequality and embedding property in Lemma 1.1 we have where B(·, ·) is the Beta functions. So, and in which (11) and e −tL a,k is C 0 -semigroup, for every λ > 0 there exists T 0 > 0 such that Pick up fixed λ < min{ 1 24C2 , 1 8A2 } and then choose T ≤ T * , in which The next, using (35), (34) and the induction method we obtain the following estimates K j ≤ 4λ and K j ≤ 2(λ + The next step, we prove that {z j (t)} is Cauchy sequence in the Banach space M T . In which, The same argument as above, we obtain if pick up fixed λ < min{ 1 24C2 , 1 32A2 , 1 8 } and then choose T such that So, {z j (t)} is Cauchy sequence in the Banach space M T . Hence, there exists z(t) ∈ M T such that z j (t) converges to z(t).
The same argument as Q j , we obtain This yields to Therefore, {t  Finally, the uniqueness of time-local mild solution follows as in [13] from Gronwall's inequality. So, z(t) is unique solution of Eq. (33) on [0, T ].