3D incompressible flows with small viscosity around distant obstacles

In this paper, we analyze the behavior of three-dimensional incompressible flows, with small viscosities ν > 0, in the exterior of material obstacles ΩR = Ω0 + (R, 0, 0), where Ω0 belongs to a class of smooth bounded domains and R > 0 is sufficiently large. Applying techniques developed by Kato, we prove an explicit energy estimate which, in particular, indicates the limiting flow, when both ν → 0 and R → ∞, as that one governed by the Euler equations in the whole space. According to this approach, it is natural to contrast our main result to that one already known in the literature for families of viscous flows in expanding domains.


L. Viana
We fix an initial vorticity ω 0 , which is a smooth, divergence-free and compactly supported vector field in R 3 . Since Π R is simply connected, there exists a unique v 0,R ∈ H(Π R ) such that curl v 0,R = ω 0 | Π R (see Proposition 3.4). In addition, let us denote by u 0 the velocity defined on R 3 which is associated to the vorticity ω 0 , as follows: for each x ∈ R 3 , where × represents the cross product of vectors in R 3 . In this context, there exists T * > 0 with the following property: for all T ∈ (0, T * ), we can find a smooth solution u = u(x, t) to the three-dimensional Euler equations in the whole space |u| → 0 as |x| → +∞, (1.4) defined on R 3 × [0, T]. In (1.4), u is understood as the velocity of an ideal incompressible fluid, while p denotes its pressure.
Taking T ∈ (0, T * ) and a small viscosity ν > 0, let us also consider the incompressible Navier-Stokes equations in Π R , with initial data v 0,R , given by (1.5) Above, v ν,R represents the velocity of the particles of a viscous fluid and P ν,R is its pressure. It is well-known that there exists a Leray-Hopf weak solution v ν,R = v ν,R (x, t) to (1.5) (see Definition 4.1 and Theorem 4.2). We emphasize that, since we consider weak solutions to (1.5), there is no dependence of solution's existence time on the viscosity. Under all these notations we have just described, we are ready to state the main result of this paper. Theorem 1.1. As mentioned previously, let ω 0 ∈ (C ∞ c (R 3 )) 3 be a divergence-free vector field in R 3 , and consider the smooth solution u = u(x, t) of (1.4), defined on R 3 × [0, T], with initial data given in (1.3). For ν > 0 and R > 0, let v ν,R be a weak solution of (1.5) in Π R × [0, T), with initial data v 0,R , where v 0,R is the L 2 -orthogonal projection of u 0 | Π R on H(Π R ). Then, there exist C = C(T, Ω 0 , ω 0 ) > 0 and R 0 > 0 such that, for all R > R 0 , we have At this moment, we would like to list some papers where asymptotic behavior of incompressible flows under singular domain perturbation has been considered. Initially, we recall the study of incompressible flows in the presence of small obstacles, presented in [7] and [6]. In [7], it was investigated the asymptotic behavior of 2D incompressible ideal flows in the exterior of a single smooth obstacle that shrinks homothetically to a point. The work developed in [7] allowed to identify the equation satisfied by the limit flow. In fact, if γ is the circula-tion around the obstacle and γ = 0, then the limit velocity verifies the Euler equations in the full-plane, with the same initial vorticity. On the other hand, when γ = 0, the limit equation involves a new forcing term, with an initial vorticity that acquires a pointwise Dirac mass. In a similar analysis for the 2D Navier-Stokes equations, considered in [6], it was proved that, if the circulation is sufficiently small, then the limit equation is the Navier-Stokes equations in the whole space, but an additional pointwise Dirac mass still appears in the vorticity of the limit equation. In [4], the corresponding problem was considered in the three-dimensional case, where it was established that the limit velocity is a solution of the Navier-Stokes equations in the full-space. Later, in [1], the research proceeded with the asymptotic behavior of solutions of the incompressible 2D Euler equations on a bounded domain with a finite number of holes, assuming that the size of one of them vanishes. In that situation, the limit flow was identified as a modified Euler system in the domain without its small hole.
In [5], incompressible flows around a small obstacle, with small viscosity, are considered. Under specific assumptions, it can be seen that solutions of the Navier-Stokes system in exterior domains converge to solutions of the Euler system in the full space when both viscosity and the size of the obstacle vanish. In the proof of this result, it is presented a rate of convergence in terms of the viscosity and the size of the obstacle. In addition, the complementary situation was treated in [9], where 2D Euler and Navier-Stokes systems were analyzed in expanding domains. To be more precise, such asymptotic analysis also pointed out that solutions in large domains converge to the corresponding solution in the full plane.
As we can see, in the context of fluid dynamics, limits of singularly perturbed domain have been extensively studied over the last years. Last but not least, we would like to highlight [10], where Kelliher, Lopes Filho and Nussenzveig Lopes examined, in dimensions 2 and 3, the limiting behavior of incompressible flows with small viscosity inside expanding domains. Based on energy estimates developed by Kato in [8], these three authors identified conditions under which the limit velocity satisfies the Euler system in the whole space when both viscosity vanishes and the domain becomes large. We are supposed to remark that their analysis also exhibits a rate of convergence which takes into account the small viscosity of the fluid and the enlarged boundary domain. The current work intends to be part of the list of papers we have just mentioned. However, our purpose here is closer to [10]. In fact, we study, in dimension 3, the limiting behavior of incompressible flows, with small viscosity, around far obstacles. In this sense, here the boundary domain becomes distant through the translation of Γ 0 = ∂Π 0 by R = (R, 0, 0), while, in [10], the boundary goes far by dilatation. In both cases, there is some effect of distant boundaries in the vanishing viscosity limit. Thus, Theorem 1.1 should be contrasted with the corresponding three-dimensional main result of [10].
The remainder of this paper is organized as follows: in Section 2, we deal with the behavior of smooth solutions of (1.4) at infinity. In Section 3, we set some suitable approximate solutions to the Euler equations and, applying the decay results obtained in Section 2, some indispensable estimates are achieved in Propositions 3.3 and 3.4. Section 4 is devoted to a brief discussion about the Navier-Stokes in exterior domains. At this point, we must emphasize Proposition 4.5, where a well-known relation involving weak solutions to (4.1) is extended to a larger class of test vector fields. In Section 5 we prove Theorem 1.1, our main result. In Section 6, we make further comments about some aspects related to this work. At the end, for the sake of clarity, there is an appendix, where we list domains, differential operators, function spaces and notations related to the PDEs mentioned throughout the development of this work.

Incompressible inviscid flow in the whole space
At the beginning of this section, we would like to have a few words on the local well-posedness for the 3D Euler system in the whole space. As said before, we fix an initial vorticity ω 0 ∈ (C ∞ c (R 3 )) 3 , which is divergence-free, and take the associated velocity u 0 , expressed in (1.3). Under these assumptions, it was proved in [12] that, for sufficiently small times, there exists a smooth solution of (1.4), with u 0 as the initial data. It means that there exists T * > 0, depending on u 0 , such that, for all T ∈ (0, T * ), there exists a unique smooth solution (u, p) of (1.4), defined on Additionally, for each t ∈ [0, T], the vector field ω = curl(u(·, t)) is compactly supported and there exists r > 0 such that what can be seen in [10], for example. It is important to notice that ω and u solve the system and, due to the second PDE in (2.2), it is true that u = curl Ψ, where Ψ is the vector-valued stream function given by As a consequence, for all (x, t) ∈ R 3 × [0, T], u can be recovered from ω throughout the Biot-Savart law which we had already stated in (1.3), for t = 0. In the rest of this section, we will focus our attention on the behavior of the smooth solution (u, p) of (1.4) at infinity. 3 be a compactly supported vector field and consider M > 0 such that supp Φ ⊂B M (0). Then, there exists C > 0 such that, for any x ∈ R 3 \ B 2M (0), we have Additionally, if R 3 Φ(y)dy = 0, then the inequality also holds.
Next, we apply Lemma 2.1 in order to state the decay of u and its derivatives, as |x| → ∞. Proposition 2.2. Consider u and ω as mentioned above and take M > 0 such that supp(ω(·, t)) ⊂ B M (0) for all t ∈ [0, T]. Then, there exists C > 0 such that Proof. During this proof, suppose that ( For the second desired estimate, we use (2.2) and (2.4) in order to obtain Recalling that u and ω are two divergence-free vector fields, we take Hence, applying the inequality (2.6) from Lemma 2.1, we have In the last part of this proof, we will estimate ∇u = [∂ i u j ] 3 i,j=1 . Notice that, if 0 < ε < M and y ∈ B M (0), we clearly get |x − y| ≥ |x| 2 > M > ε for all y ∈B M (0). Consequently, for any 3 × 1 matrix B, we take It completes the proof.
In the next two results, we will specify the decay of the scalar pressure p, given in (1.4), as |x| → ∞. Lemma 2.3. Let (u, p) be the solution of (1.4) and considerȳ ∈ R 3 \ {0}. The following properties hold: Proof. The pointwise estimate for ∇p comes immediately from Proposition 2.2. Let us prove the second part of the result. Let (θ n ) ∞ n=1 be a sequence of positive real numbers which tends to infinity. Since for all positive integers m and n, we conclude that, for each t ∈ [0, T], the sequence (p(θ nȳ , t)) ∞ n=1 converges as n → ∞. Analogously, if (λ n ) ∞ n=1 is another sequence of positive real numbers which tends to infinity, we take for all positive integers m and n, and t ∈ [0, T]. It means that there exists lim θ→∞ p(θȳ, t), as desired.
Next, we will see that Lemma 2.3 allows us to collect some properties of the pressure p.
Proof. Firstly, let us focus on the proof of (a). Without loss of generality, we can assume that z / ∈ Rȳ and |ȳ| ≥ |z|. Take θ > 0 and consider the sphere where v belongs to the tangent plane to S at q. Thus, from Lemma 2.3, we obtain Secondly, we must prove the part (b). Let us set the scalar function This completes the proof of Proposition 2.4.
In the last result of this section, we will make use of Propositions 2.2 and 2.4 in order to describe how the stream vector field Ψ behaves at infinity. Proposition 2.5. Let (u, p) be the solution of (1.4) and Ψ be the associated stream vector field given in Proof. The first inequality in (2.10) is straightforward.
and thus the third estimate in (2.10) also holds.

Approximate inviscid solutions
Firstly, let us recall some notations given in Section 2. Consider ω 0 ∈ (C ∞ c (R 3 )) 3 and let (u, p) be the smooth solution of (1.4) Also, let Ψ be the stream vector field associated to u, given in (2.3).
In this section, we intend to approximate the solution u by an appropriate net (u R ) R>0 of divergence-free vector fields.
where (x, t) ∈ R 3 × [0, T] and C R = −1 4πR R 3 ω 0 (y)dy. Clearly, each u R is a smooth and divergence-free vector field in R 3 , which vanishes in the neighborhood of where the function p ∞ : [0, T] −→ R was obtained in Proposition 2.4, we also have Next, we will prove some important estimates involving (u R ) R>0 , which will allow us to obtain Theorem 1.1.
Proposition 3.3. Under the previous notation, there exist two constants C = C(Ω 0 , T) > 0 and R 0 > 0 such that, for all R > R 0 , we have: Proof. ESTIMATE (a): In the first place, using (3.2) and (3.5), we get for all t ∈ [0, T] and R > 0 sufficiently large. On the other hand, recalling Proposition 2.2, we also obtain for all t ∈ [0, T] and R > 0 sufficiently large. As a result, desired estimate holds.

ESTIMATE (b): From (3.2), we know that
Hence, using Propositions 2.2 and 2.5, and Lemma 3.1, we can argue as in (3.7) and (3.8) in order to prove that for all R > 0 sufficiently large.

ESTIMATE (c):
The proof of the third desired estimate is very similar to the last ones. In fact, it is a consequence of (2.9) and (2.10), given in Propositions 2.4 and 2.5, respectively.

ESTIMATE (d):
Let us prove the last estimate. Since the inviscid velocity u is a smooth vector field, Proposition 2.2 assures that ∇u R L ∞ ([0,T];L 2 (Π R )) ≤ C for R > 0 is sufficiently large. Finally, for all i ∈ {1, 2, 3}, it is clear that for all x ∈ R 3 and R > 0. It implies that u R L ∞ ([0,T];L ∞ (Π R )) and ∇u R L ∞ ([0,T];L ∞ (Π R )) are uniformly bounded with respect to R > 0. This ends the proof.
Next, we prove the last result of this section, which yields a suitable convergence related to the initial data. Proposition 3.4. As before, let ω 0 ∈ (C ∞ c (R 3 )) 3 be a divergence-free vector field and consider the initial velocity u 0 as in (3.1). Then, for each R > 0, there exists a unique v 0,R ∈ H(Π R ) such that curl v 0,R = ω 0 | Π R . In addition, there exist C > 0 and R 0 > 0 such that for all R > R 0 , whereṽ 0,R vanishes on Ω R and equals v 0,R on Π R .
Proof. For each R > 0, the existence of exactly one vector field v 0,R ∈ H(Π R ) satisfying curl v 0,R = ω 0 | Π R was given in [4], where the authors have used the Leray-Helmholtz-Weyl orthogonal decomposition as well as the simple connectedness of Π R . In order to prove (3.9), we observe that for all w ∈ H(Π R ). In particular, taking w(x) = u R (x, 0), where x ∈ R 3 , and applying Proposition 3.3, we obtain for all R > 0 sufficiently large. Besides, taking R > 2M 0 , we observe thatΩ R ∩ supp ω 0 = ∅. As a consequence, for any x ∈ Ω R and y ∈ supp ω 0 , we have and (3.9) holds.

Leray-Hopf solutions in exterior domains
Throughout this section, let Π = R 3 \ Ω ⊂ R 3 be a smooth exterior domain, which means that Ω is a smooth compact set in R 3 . Given T > 0 and v 0 ∈ H(Π), we consider the Navier-Stokes system where v = v(x, t) is the velocity field evaluated at the point x ∈ Π and at the time t ∈ [0, T], P = P(x, t) is the related scalar pressure field, and ν > 0 is the kinematic viscosity. is said to be a weak solution of (4. The next result assures the existence of a weak solution to (4.1) satisfying a very important additional estimate, which is called a Leray-Hopf solution of (4.1).

Theorem 4.2.
Given T > 0 and v 0 ∈ H(Π), the system (4.1) there exists a weak solution v : for all t ∈ [0, T] Proof. The proof of this result can be found in [2].
Later, we will apply the relation (4.4) replacing Φ by each approximate inviscid solutions u R , where R > 0. For this reason, we are supposed to prove that (4.4) remains valid when Φ decays sufficiently fast at infinity, but is not compactly supported.
Under these notations, we are ready to present the next two results.

PART (a):
It follows immediately from Lebesgue's dominated convergence theorem.

PART (b):
Let i ∈ {1, 2, 3}. If a ∈ (3, +∞), the desired convergence is a consequence of On the other hand, the estimate ∂ i η s L ∞ (Π) ≤ C s gives us the complete conclusion.

PART (e):
For the last estimate, we assume that α > 3/2. Once again, applying (4.5), we have The following result is the last one of this section. We emphasize that its content brings the information that (4.4) holds for a larger class of test functions. (c) There exists C 1 > 0 such that Additionally, considerΦ := curl(Ψ) + F and suppose that there exists C 2 > 0 such that Then, a weak solution v of (4.1), with initial data v 0 ∈ H(Π), satisfies Proof. As in the proof of Lemma 2.10, consider Π = R 3 \ Ω and take d > 0 such that Ω ⊂ B d (0). For each s ≥ d, define Φ s ∈ D(Π × [0, T)) given by Φ s := curl(η sΨ ) + F = ∇η s ×Ψ + η s curlΨ + F.

Proof of Theorem 1.1
This section is devoted to the main result of this paper. Let us recall its hypotheses: • ω 0 is a smooth, compactly supported and divergence-free vector field in R 3 ; • (u, p) is the smooth solution of (1.4), defined in R 3 × (0, T), with initial data u 0 , as in (1.3).
From now on, we will focus on the verification of (5.1). To do so, the main arguments used below take into account those presented in [8]. Let us fix t ∈ [0, T]. From (4.3), whereṽ 0,R equals v 0,R on Π R and vanishes otherwise.

L. Viana
Next, due to (2.7) and (2.10), we can apply Proposition 4.5, with v = v ν,R andΦ = u R , in order to obtain Besides, recalling that the kinetic energy E(t) = 1 2 R 3 |u(x, t)| 2 dx is a conserved quantity in time, we also take As a result, from (5.2), (5.3) and (5.4) , we conclude that Firstly, using Proposition 3.3(d), we easily get Likewise, A 6 was completely analyzed in Proposition 3.3(a). Thus, in the rest of this proof, we will estimate each A i , with i ∈ {1, 2, 3, 4}, in terms of R > 0 and ν > 0. To see that |A 1 | ≤ C R 2 , we just apply Propositions 3.3(a) and 3.4. In fact, observe that and recall (3.9). Next, using the relation and Proposition 3.3(c), we obtain Besides, using Young's inequality and Proposition 3.3(d), we can easily check that At last, the estimate comes from Theorem 4.2 and Proposition 3.3(a),(b),(d).

L. Viana
Therefore, there exist K > 0 and L > 0, independent of time t, such that for any t ∈ [0, T]. Thus, the integral form of Gronwall's inequality allows us to achieve the estimate (5.1) .
6 Some additional comments 1. In order to obtain Proposition 3.4, the circulation of the velocity on the boundary Γ R is not required, since each Π R is a 3D simply connected domain.
2. In this paper, we deal with three-dimensional incompressible flows, with small viscosity, around distant obstacles. Perhaps, the analogous two-dimensional case can also be studied.
To be more precise, let U 0 ⊂ R 2 be a smooth bounded domain, which is also connected and simply connected. For each R ≥ 0, let us set Let us consider ω 0 ∈ C ∞ c (R 2 ) and γ ∈ R, which are both independent of R > 0. Set curl y 0,R = ω 0 , in V R , y 0,R ·n = 0, on C R , |y 0,R (x)| → 0, as |x| → +∞, C R u 0,R · ds = γ. Thus, it would be very nice to understand the asymptotic behavior of the family of 2D incompressible flows, with small viscosity, around distant obstacles, governed by It means that we have fixed ω 0 as an initial vorticity and γ as the circulation of the initial flow around each obstacle U R . We should notice that y 0,R ∈ L 2,∞ (Π R ), but y 0,R / ∈ L 2 (Π R ). In this case, it seems to us that Kato's argument can not be applied following the same structure of the proof of Theorem 1.1. However, another approach may work if we take mild solutions of (6.1), obtained [11]. 3. In order to simplify calculations and estimates, we have considered material obstacles of the form Ω R = Ω 0 + R = Ω 0 + R1, where 1 = (1, 0, 0). However, our main result remains valid if we replace Ω R by Ω 0 + R z, for each R > 0, where z is another unit vector in R 3 .
4. We believe that Theorem 1.1 is related to that one obtained in [10], in the threedimensional case. Let Ω 0 ⊂ R 3 be a smooth and simply connected bounded domain Ω 0 ⊂ R 3 . In that work, the authors started with a smooth initial vorticity ω 0 ∈ C ∞ c (R 3 ), which is divergence-free and, for each L > 0, they took u L 0 as the unique divergence-free vector field in Ω L = LΩ 0 that is tangent to ∂Ω L and has a curl equal to ω 0 in Ω L . At this point, they studied the limiting behavior of a family where u = u(x, t) is a smooth solution to (1.4), with initial data u 0 , given in (1.3). Above, we observe that u L 0 − u 0 L 2 (Ω L ) ≤ CL −1/2 , which allows us to contrast the current main result of this paper with (6.3). In both cases, there are results for 3D incompressible flows, with small viscosity, in domains with distant boundaries. Thus, it seems to us that the rates of convergence R −1 and L −1/2 on the right side of (1.6) and (6.3), respectively, suggest that the boundary Γ R = ∂Π R , in (1.5), produces a bigger effect in the vanishing viscosity limit, when compared to the same effect produced by ∂Ω L , in (6.2).

Operators
For a smooth vector field F = (F 1 , F 2 , F 3 ) in R 3 , we denote:

Function spaces
Given an open set O ⊂ R 3