Large Time Decay of Solutions to a Linear Nonautonomous System in Exterior Domains

In this expository paper, we study Lq-Lr decay estimates of the evolution operator generated by a perturbed Stokes system in n-dimensional exterior domains when the coefficients are time-dependent and can be unbounded at spatial infinity. By following the approach developed by the present author for the physically relevant case where the rigid motion of the obstacle is time-dependent, we clarify that some decay properties of solutions to the same system in whole space Rn together with the energy relation imply the desired estimates in exterior domains provided n≥3.


Introduction
This paper studies the large time decay of solutions to the initial value problem for a linear nonautonomous system arising from fluid dynamics, specifically a viscous incompressible flow past an obstacle. Let Ω be an exterior domain in R n , n ≥ 3, with C 1,1boundary ∂Ω. Complement R n \ Ω is identified with the obstacle (rigid body) immersed in a fluid, and it is assumed to be a compact set in B 1 (0) with nonempty interior. Towards the understanding of the stability or attainability of time-dependent Navier-Stokes flow past the obstacle whose motion could also be time-dependent, an essential step is to deduce some decay properties of a linearized system of form in Ω × (s, ∞), where vector field u = (u 1 (x, t), · · · , u n (x, t)) ∈ R n and scalar function p = p(x, t) ∈ R are unknowns denoting fluid velocity and pressure, while b(x, t) ∈ R n and M(x, t) ∈ R n×n are prescribed functions. When b = 0 and M = O, (1) is just the wellknown Stokes system. We consider perturbed Stokes system (1) subject to homogeneous Dirichlet boundary condition and initial condition u(x, s) = f (x) (3) at initial time s ≥ 0. The adjoint of the solution operator (evolution operator) T(t, s): f → u(t) to (1)-(3) provides solution operator T(t, s) * : g → v(s) to backward problem −∂ s v − ∆v − ∇σ + b(s) · ∇v + M(s) + div b(s) in Ω × [0, t), with I being the n × n identity matrix, where v(y, s) and σ(y, s) are unknowns, subject to v| ∂Ω = 0, v → 0 as |y| → ∞ (5) and final condition v(y, t) = g(y) (6) at final time t > 0. In what follows, let us assume that div b = 0 for simplicity. This condition is actually satisfied for a typical example, (9) below. Initial and final velocities f and g are taken from class L q σ (Ω) of solenoidal L q -vector fields, 1 < q < ∞, with vanishing normal trace at boundary ∂Ω.
Several comments on Assumptions (i)-(iii) above are in order. The well-posedness in Assumption (i), in other words, the generation of the evolution operator, is never obvious; however, this is a different issue from what we address in this paper. When b = η + ω × x and Mu = ω × u with (8), the generation of the evolution operator with (7) for 0 ≤ s < t ≤ T was successfully proved by Hansel and Rhandi [22] for every T ∈ (0, ∞); then, it was verified by [1] that constant C = C(τ * ) in (7) can be taken uniformly in (t, s) with t − s ≤ τ * , and by [2] that smoothing estimate (10) of ∂ t u(t) near the obstacle holds with α = (1 + 1/q)/2. The latter estimate is closely related to the asymptotic behavior of pressure (in a bounded domain near the obstacle) and very crucial in [2] on account of the lack of smoothing action exhibited by analytic semigroups since the evolution operator is not parabolic. The remarkable smoothing rate α = (1 + 1/q)/2 was already found by [6,7,23] for the Stokes and Oseen semigroups with rotating effect, and it was a slight improvement of the rate deduced by Noll ans Saal [24] in another context. Assumption (ii) on the L q -L r estimate (7) for the whole space problem is nontrivial, but it is the starting point of analysis in this paper. When b = η + ω × x and Mu = ω × u with (8), the solution in the whole space can be explicitly described in terms of the heat semigroup in which a change of variable is made, so that Estimates (7) and (11) are in fact available. Energy Relations (12) and (13) in (iii) are reasonable assumptions that play several roles, especially for deduction of (7) j=0 . When b = η + ω × x and Mu = ω × u, (12) and (13) are obvious on account of the skew-symmetry of b · ∇u − Mu without any smallness condition. Concerning case (9) as well, we can easily see (12) and (13), provided that V is small enough in L ∞ (0, ∞; L n,∞ (Ω)), see, for instance, [25], where L n,∞ is the weak-L n space (a Lorentz space). Except for Assumption (iii), we do not have useful higher energy estimates, which play an important role in [4] by Maremonti and Solonnikov for the Stokes semigroup.
As the substitution of analysis of a parametrix of the resolvent in exterior domains for the autonomous case, the key of our approach is how we make use of energy Relations (12) and (13) to deduce (7) j=0 , see Proposition 1. Here and in what follows, by (7) j=0 we denote estimate (7) with j = 0. Case r = q (uniform boundedness) for large t − s > 0 is our main task, yielding the other cases by use of the energy inequality of the differential form. It is reasonable under Assumption (ii) to regard the solution to (1)-(3) as a perturbation from (a modification of) R n -flow by means of a cut-off procedure. The desired uniformly boundedness of the perturbation is discussed by a bootstrap argument and by duality argument with the aid of Assumption (iii) on the energy relations, and the use of duality is why we need to simultaneously study adjoint evolution operator T(t, s) * with T(t, s). Unfortunately, this step does not work when n = 2. If Dirichlet Condition (2) is replaced by another boundary condition, a core part of this step does not follow, even for n ≥ 3. With (7) j=0 at hand, we are able to proceed to the decay estimates of ∂ t u(t) W −1,q (Ω 3 ) and u(t) W 1,q (Ω 3 ) near the obstacle that we call the local energy decay, as in several papers for the autonomous case. Among other papers, the method of local energy decay is traced back to [3] by Iwashita on the Stokes semigroup in the context of mathematical fluid dynamics, and the origin would be even back to studies of hyperbolic equations with dissipation by Shibata. The final step with another cut-off procedure is to derive the decay estimate of ∇u(t) L q (R n \B 3 ) near spatial infinity by the use of Assumption (ii) combined with the local energy decay obtained in the preceding step. The cut-off remainder consists of several terms; among them, two terms are delicate: one is ∂ t u(t), and the other is pressure. What we need is both decay for t − s → ∞ and smoothing rate for t − s → 0 of those terms; see (78). The latter of the temporal derivative is the assumption (10) in (i), and the deduction of (10) was actually one of the main tasks in [2] on case b = η + ω × x and Mu = ω × u with (8). For more general case, as in [2], we have to look into details about the construction of a parametrix of the evolution operator to verify (10). If it is constructed with the use of evolution operators for the whole space problem and for the interior one near the obstacle by a cut-off technique as in [22], smoothing rate (10) of ∂ t u(t) is determined by the one of pressure for the interior problem.
To sum up, we claim that some decay properties of solutions to the same system in whole space R n together with the energy relation imply the desired estimates in exterior domains provided n ≥ 3, and that we need to find (10) through analysis of pressure to justify this statement. Let us close the introductory section with a remark on Case (9). With the results for case V = 0 obtained in [1,2] at hand, it is actually possible to show the stability or attainability of scale-critical Navier-Stokes flow V ∈ L ∞ (0, ∞; L n,∞ (Ω)) by an interpolation technique due to Yamazaki [26] as long as it is small enough; see, for instance, Takahashi [15] on the attainability of steady flow for the purely rotating case. However, we have less information about the asymptotic behavior of disturbance. If we intend to show some decay properties of gradient of the disturbance, we have to know (7) j=1 for Problem (1)-(3) with (9). If we adapt the approach developed in ( [25], Section 4) to case (9) with the aforementioned V that is sufficiently small, we could verify Assumption (ii), but only partially.
In the next section, we precisely formulate the problem and provide the main theorem. Section 3 is devoted to the proof. We close the paper with a conclusion in the final section.

Result
Let us fix the notation. Given a domain D ⊂ R n , q ∈ [1, ∞] and integer k ≥ 0, the standard Lebesgue and Sobolev spaces are denoted by L q (D) and by W k,q (D). We abbreviate norm · q,D = · L q (D) and even · q = · q,Ω , where Ω is the exterior domain with C 1,1 -boundary ∂Ω under consideration. We assumed that R n \ Ω ⊂ B 1 , where B ρ = B ρ (0) denotes the open ball centered at the origin with radius ρ > 0. We set Ω ρ = Ω ∩ B ρ for ρ ≥ 1. By C ∞ 0 (D) we denote the class of all C ∞ functions with compact support in D, and by W k,q 0 (D) the completion of C ∞ 0 (D) in W k,q (D). We set W −1,q (D) = W 1,q 0 (D) * , where 1/q + 1/q = 1 and q ∈ (1, ∞). By ·, · D we denote various duality pairings over domain D. In what follows, we use the same symbols for denoting the scalar and vector functions if there is no confusion. Let X be a Banach space. Then, L(X) stands for the Banach space consisting of all bounded linear operators from X into itself. Let where ν stands for the outer unit normal to ∂Ω. When D = R n , the boundary condition in the sense of a normal trace is absent. The space of the L q -vector fields admits Helmholtz decomposition [27][28][29]; Simader and Sohr [29] established decomposition under condition ∂Ω ∈ C 1 when D = Ω. By P D = P D,q : L q (D) → L q σ (D) we denote the Fujita-Kato projection associated with the decomposition above. We then observe that P D ∈ L(W 1,q (D)) as well as P D ∈ L(L q (D)). Note the duality relation (P D,q ) * = P D,q and L q σ (D) * = L q σ (D), where 1/q + 1/q = 1. We simply write P = P Ω for exterior domain Ω under consideration. We easily see that Suppose that are measurable in x ∈ R n and continuous in t ≥ 0 with the property where | · | denotes the R n -norm. For simplicity, b(t) is assumed to be a solenoidal vector field, that is, div b(t) = 0 in the sense of distributions for each t ≥ 0.
Then initial value Problem (1)-(3) and backward adjoint problem (4)-(6) (with div b = 0) are formulated, respectively, as It is easily seen that By (14) and by u · v ∈ L 1 (D), passing to the limit as R → ∞ justifies (18) and, therefore, At this point, we need the maximality in the sense that This means the solvability of the associated elliptic problem, which implies that λ(t) + L D,b,M (t) * is injective. Under this condition, (18) leads to duality relation Hence, [23] by Shibata along with the Mozzi-Chasles transform ( [30], Chapter VIII) (to reduce the problem to the particular case when the translational and angular velocities are parallel each other).
As in [2,22], auxiliary spaces for D ∈ {Ω, R n } play a role to describe the regularity of solutions, since domain D q (L D (t)) varies as t goes on. Note that for every t ≥ 0, and that the homogeneous Dirichlet condition at ∂Ω is not involved in the space Z q (Ω).
We further make Assumptions (A3)-(A5): in L(L q σ (Ω)) and that the map is continuous for every f ∈ L q σ (Ω). Moreover, we have the following properties: 1. Let and in L q σ (Ω) (condition q > n is consistent with Lemma 3); 2.
Assumption (A5) is related to the solution operator to backward adjoint problem (17).
We next make the following assumption on the evolution operators for the whole space problem.
In the same manner as for the exterior problem mentioned above, we observe We need the following estimates of T R n (t, s) * as well as T R n (t, s).
We are now in a position to give our main theorem.

Proof of Theorem 1
We start with a quite elementary but useful lemma on optimal growth rate of the integral (see (46) below) under a condition on the decay of the square integral.

Proof. From (45), it follows that
for all σ > 0, which leads to (46) since The proof is complete.
Our proof of (43) with j = 0 is based on the following lemma with the aid of energy Relations (39)-(42).
The following proposition provides us with (43) j=0 for all t > s ≥ 0 except the case r = ∞. Note that κ(0) = (n/q − n/r)/2. Proof. Given f ∈ C ∞ 0,σ (Ω), we set u(t) = T(t, s) f and denote by p(t) the pressure associated with u(t). Let us take T R n (t, s) f with T R n (t, s) being the evolution operator in the whole space R n given by (A6) and single out associated pressure p R n (t), such that We regard (u, p) as a perturbation from (a modification of) the R n -flow, to be precise, and φ ∈ C ∞ 0 (B 3 ) is a fixed cut-off function satisfying φ = 1 on B 2 . Here, B is called the Bogovskii operator [30,[32][33][34] in the domain G := B 3 \ B 1 that provides a particular solution (among many solutions) of the boundary value problem for the equation of continuity subject to homogeneous Dirichlet boundary condition, and enjoys optimal regularity estimates (1 < q < ∞, k = 0, 1, 2, · · · ) Let 2 < r < ∞, and let us show (47) (with r 0 = r). Since we have the desired estimate for u(t) above by (36), together with (53), our task is to find uniform boundedness for v(t).
Then, by virtue of energy Relation (42) along with (35) and by the decay estimate obtained in the previous step, we find t−s−1 for t − s > 2. As for the former integral of (62), we use (64) to furnish The latter integral of (62) must be comparable with that. In fact, due to Lemma 1 with σ = (t − s − 2)/2, it follows from (64) that In this way we accomplish (47) for all r 0 ∈ (n, ∞) when n ≥ 4, while we still need to repeat the procedure once more when n = 3. This procedure in 3D is indeed possible for every r < ∞ by the same manner as before with the aid of (48) for 6/5 < q ≤ r ≤ 2. We completed the proof of (43) j=0 for all t > s ≥ 0 and 2 ≤ q ≤ r < ∞. Uniform boundedness (49) for the adjoint is similarly proven by use of (26), (A7) for T R n (t, s) * and (41). Here, the Duhamel formula can be justified merely in weak form by (18) and (24), but it is enough for a duality argument. We thus employ Lemma 2 to obtain (43) j=0 for all t > s ≥ 0 and 1 < q ≤ r ≤ 2. Remaining case q < 2 < r is easily filled on account of semigroup property (22) to conclude (43) j=0 for all t > s ≥ 0 and 1 < q ≤ r < ∞.
The proof of this proposition would be a novelty among other arguments in [1,2]. However, the first step of the proof above does not work well when n = 2; in fact, (61) cannot be bounded for r > 2.
For the proof of (43) j=1 , it suffices to show for all (t, s) with t − s > 1, r ∈ (1, ∞) and f ∈ L r σ (Ω) since we combine (65) with (43) j=0 to obtain the desired estimates. As in several papers for the autonomous case mentioned in Section 1, it is standard to split (65) into ∇T(t, s) f r,Ω 3 and ∇T(t, s) f r,R n \B 3 . The former is given by Proposition 2 below, for which the following lemma is needed. Lemma 3. Let n < q < ∞, where 1/n + 1/n = 1.
Proof. The second assertion follows from (23) because (66) implies Pg ∈ Z q (Ω). Let us consider Neumann problem It then suffices to show which implies (66) since Pg = g + ∇w. We take the same cut-off function φ ∈ C ∞ 0 (B 3 ) as in the proof of Proposition 1, and choose a solution w satisfying Ω 3 w dx = 0, so that where the last inequality is due to [28,29]. Then, φw obeys where ν denotes the outer unit normal to ∂Ω 3 . On the other hand, For Riesz transform R, we know |x|Rh q,R n ≤ C |x|h q,R n from Muckenhoupt theory for singular integrals, as long as n < q < ∞; in fact, for such q, weight |x| q belongs to Muckenhoupt class A q (R n ); see Farwig and Sohr ([35], Section 2), and Torchinsky ([36], Chapter IX) for details. We thus obtain for n < q < ∞. We collect (68)-(70) to conclude (67).
For the proof of (71), it suffices to show it for u(t) = T(t, s) f with f ∈ C ∞ 0,σ (Ω). For such f , function u(t) given by (52) satisfies (71) because of (36). Let us consider v(t) = u(t) − u(t) obeying (55). Since both f and F(t) vanish outside Ω 3 , one can apply (74) to those vector fields. Then, we immediately see that T(t, s) f satisfies the desired estimate. Combining (58) with the above observation, we also find for τ ∈ (s, t), which concludes (71).
(Ω) and single out pressure p(t) associated with u(t) subject to Ω 3 p dx = 0. Let φ ∈ C ∞ 0 (B 3 ) be the cutoff function as in the proof of Proposition 1, and B the Bogovskii operator in domain G = B 3 \ B 1 . Then, for each m ∈ (0, ∞), there is a constant C = C(m, q, n, Ω) > 0, such that for all f ∈ L q σ (Ω), whenever (b, M) ≤ m.
Let us complete the proof of Theorem 1 by showing estimates near spatial infinity.

Conclusions
The stability or attainability of physically relevant basic (Navier-Stokes) flow V is a significant issue in mathematical fluid dynamics. Estimate (7) for the associated linearized flow describes linearized stability with a definite decay rate of disturbance, and it is always a crucial step toward nonlinear stability, where nonlinearity is regarded as a small perturbation as long as the initial disturbance is small enough. The exterior problem is even more interesting since the rigid motion (translation or rotation) of the obstacle is involved in stability analysis. Spectral analysis of the linearized operator through resolvent problem is quite useful to deduce (7) when basic flow V is steady. In this paper, we considered the situation where both the motion of the obstacle and basic flow V were time-dependent (see (9) as a typical example), for which stability analysis is much less developed because spectral analysis does not work well. The novelty is to provide the essence of the new approach proposed by the present author [1,2] to show linearized stability (7) even for nonautonomous systems in exterior domains under reasonable assumptions on regularity of the evolution operator, including smoothing estimate (10) of the temporal derivative, linearized stability for the whole space problem, and energy Relations (12) and (13) with dissipation. Theorem 1, together with standard analysis of the nonlinear problem, tells us roughly that linearized stability for the whole space problem and the energy structure lead to nonlinear stability in exterior domains. Emphasis is on how we utilize the energy relation to find this conclusion. In addition to example (9) with V = 0 discussed in [1,2], we could apply Theorem 1 to Case (9), with V decaying faster than scale-critical rate O(|x| −1 ) at spatial infinity. This is indeed the case when the better spatial decay structure of V with wake behind the translating obstacle is available. As mentioned at the end of the introduction, we still have to await further analysis to apply our theory to the more important case of (9) with V, which decays at the scale-critical rate uniformly in t. This is left as future work.