Paul Deuring

SPATIAL ASYMPTOTICS OF MILD SOLUTIONS TO THE TIME-DEPENDENT OSEEN SYSTEM

Keywords: Mathematics Subject Classification. Primary: 76D07, 35B40; Secondary: 35Q35 Oseen system, mild solution, spatial decay, Oseen semigroup, wake. 1

We consider mild solutions to the 3D time-dependent Oseen system with homogeneous Dirichlet boundary conditions, under weak assumptions on the data. Such solutions are defined via the semigroup generated by the Oseen operator in L q . They turn out to be also L q -weak solutions to the Oseen system. On the basis of known results about spatial asymptotics of the latter type of solutions, we then derive pointwise estimates of the spatial decay of mild solutions. The rate of decay depends in particular on L p -integrability in time of the external force.

This latter system is usually considered as a model for the flow of a viscous incompressible fluid around a rigid body moving with constant velocity and without rotation, with the set Ω corresponding to the rigid body. The functions u :

and U 0 : Ω c → R 3 (initial velocity), as well as the number τ ∈ (0, ∞) (Reynolds number), are given quantities. Mild solutions to (1), (2), which only involve the velocity among the two unknowns velocity and pressure, are introduced via the semigroup generated by the Oseen operator. We refer to (5) for the definition of this operator, and to the proof 2020

1. Introduction. In this article, we consider mild solutions to the 3D time-dependent Oseen system

u -∆ x u + τ ∂ x1 u + ∇ x π = f, div x u = 0 in Ω c × (0, ∞), (1)
where Ω c := R 3 \Ω, with Ω an open, bounded set in R 3 with smooth boundary.

Thus Ω c is an exterior domain, which we suppose to be connected. Equation ( 1) is supplemented by homogeneous Dirichlet boundary conditions on ∂Ω and an initial condition, u(t)|∂Ω = 0 for t ∈ (0, ∞), u(0) = U 0 .

(2) The Oseen system is a linearization of the time-dependent Navier-Stokes system with Oseen term,

u -∆ x u + τ ∂ x1 u + (u • ∇ x )u + ∇ x π = f, div x u = 0 in Ω c × (0, ∞). (3)
of Theorem 3.3 for a discussion of the associated semigroup. A precise definition of a mild solution to (1), ( 2) is given in Theorem 3.5.

In the work at hand, we study the spatial decay of such solutions. It turned out their decay rate is highest if U 0 ∈ L q σ (Ω c ) and f ∈ L 1 0, ∞, L q σ (Ω c ) for some q ∈ (1, 3/2), and if |U 0 (x)| and |f (x, t)| tend to zero sufficiently fast for |x| → ∞.

(See Section 2 for the definition of L q σ (Ω c ). In this situation we obtain that

|∂ α x u(x, t)| ≤ C |x| ν(x) -(3+|α|)/2 (4)
for a. e. t ∈ (0, ∞), a. e. x ∈ B c R0 := R 3 \B R0 , and for α ∈ N 3 0 with |α| ≤ 1, with C being independent of x and t. The parameter R 0 is some fixed number from (0, ∞) with Ω ⊂ B R0 . Concerning the condition |α| ≤ 1, it means that u and the spatial gradient ∇ x u are estimated in [START_REF] Deuring | Spatial decay of time-dependent incompressible Navier-Stokes flows with nonzero velocity at infinity[END_REF]. The function ν appearing in ( 4) is defined by ν(x) := 1 + |x| -x 1 for x ∈ R 3 . Its presence should be interpreted as a mathematical manifestation of the wake extending downstream in the flow behind a rigid body. If f ∈ L p 0, ∞, L q σ (Ω c ) for some p ∈ (1, ∞) and q ∈ (1, 3/2), the rate of decay of |∂ α x u(x, t)| diminishes to -(3 + |α|)/2 + 1/(2 p ). So in particular this rate is linked to L p -integrability in time of f . If the relation U 0 ∈ L q σ (Ω c ) only holds for some q ∈ [3/2, 3), we also obtain a lower rate, and if q ≥ 3, we have to suppose that U 0 = 0 because otherwise our theory is mute. We refer to Theorem 4.6 for a detailed statement of our results. In the remark following this theorem, we explain the condition mentioned above that |U 0 (x)| and |f (x, t)| are to tend to zero sufficiently fast for |x| → ∞.

A link between the rate of spatial decay of ∂ α x u on the one hand and L pintegrability of f with respect to time on the other already appears in [START_REF] Deuring | The 3D time-dependent Oseen system: link between L p -integrability in time and pointwise decay in space[END_REF]Theorem 6.1] and [START_REF] Deuring | L q -weak solutions to the time-dependent Oseen system: decay estimates[END_REF]Theorem 5.2], but with certain L p -norms in time and in space of u additionally influencing the spatial asymptotics of u. The former reference specifies the spatial decay of regular solutions to [START_REF] Deuring | Spatial decay of time-dependent Oseen flows[END_REF], and the latter one extends these results to L q -weak solutions of [START_REF] Deuring | Spatial decay of time-dependent Oseen flows[END_REF]. No specific boundary conditions are imposed in [START_REF] Deuring | The 3D time-dependent Oseen system: link between L p -integrability in time and pointwise decay in space[END_REF] and [START_REF] Deuring | L q -weak solutions to the time-dependent Oseen system: decay estimates[END_REF]. A simplified version of [START_REF] Deuring | L q -weak solutions to the time-dependent Oseen system: decay estimates[END_REF]Theorem 5.2] is stated below as Theorem 2.7, which is the starting point of the work at hand. In [START_REF] Deuring | The 3D time-dependent Oseen system: link between L p -integrability in time and pointwise decay in space[END_REF] and [START_REF] Deuring | L q -weak solutions to the time-dependent Oseen system: decay estimates[END_REF], we applied [START_REF] Deuring | The 3D time-dependent Oseen system: link between L p -integrability in time and pointwise decay in space[END_REF]Theorem 6.1] and [START_REF] Deuring | L q -weak solutions to the time-dependent Oseen system: decay estimates[END_REF]Theorem 5.2], respectively, to some solutions which are known to exist; see [START_REF] Deuring | The 3D time-dependent Oseen system: link between L p -integrability in time and pointwise decay in space[END_REF]Theorem 6.2,6.3] and [START_REF] Deuring | L q -weak solutions to the time-dependent Oseen system: decay estimates[END_REF]Theorem 6.1,6.2,6.3]. However, these examples are either restricted to an L 2 -framework ([7, Theorem 6.2] and [8, Theorem 6.1, 6.2, 6.3]), or they deal with solutions whose lifespan T 0 ∈ (0, ∞) is finite, which are much more regular than required for our theory and are associated with a right-hand side f ∈ L p 0, T 0 , L q (Ω c ) 3 required to satisfy the condition p = q ([7, Theorem 6.3]).

But as the key point of [8, Theorem 6.1, 6.2] we could improve the decay rates obtained in existing literature ( [START_REF] Deuring | Spatial decay of time-dependent Oseen flows[END_REF], [START_REF] Deuring | Pointwise spatial decay of time-dependent Oseen flows: the case of data with noncompact support[END_REF]). The work at hand with its focus on mild solutions shows that we may handle solutions with data of low regularity. In fact, these solutions exist for any t > 0

if U 0 ∈ L q σ (Ω c ) and f ∈ L 1 loc [0, ∞), L q σ (Ω c
) for some q > 1 (Theorem 3.5). In addition, we are able to express our decay bounds exclusively in terms of the data; no norms of the solution are involved. Moreover, in the assumptions

U 0 ∈ L q σ (Ω c ) and f ∈ L p 0, ∞, L q σ (Ω c ) , a large range of parameters q ∈ (1, ∞), p ∈ [1, ∞) is
admitted, and our estimates exhibit how the choice of these parameters influences the spatial asymptotics of the solution. In this respect we recall the example given in (4): if q ∈ (1, 3/2), p = 1, then |∂ α x u(x, t)| tends to zero as O [|x| ν(x)] -(3+|α|)/2 for |x| → ∞. This rate is best possible in the sense that it coincides with standard decay estimates of a fundamental solution to [START_REF] Deuring | Spatial decay of time-dependent Oseen flows[END_REF]; see Lemma 2.4.

Our proof of (4) consists in verifying the assumptions of Theorem 2.7, with two main points. Firstly it must be shown that mild solutions are also L q -weak solutions as considered in that latter theorem. This is not completely obvious due to the low regularity of f ; see the proof of Theorem 3.5. Secondly, certain L p -norms of u with respect to space and time variables must be estimated by the data (Corollary 4. 3 -4.5). This is achieved by means of L p -L q -estimates of the Oseen semigroup. In our context the key feature of these estimates is the rate of temporal decay of spatial L q -norms of this semigroup. Such rates are derived in [START_REF] Kobayashi | On the Oseen equation in three-dimensional exterior domains[END_REF], [START_REF] Enomoto | Local energy decay of solutions to the Oseen equation in the exterior domain[END_REF], [START_REF] Enomoto | On the rate of decay of the Oseen semigroup in exterior domains and its application to Navier-Stokes equation[END_REF], [START_REF] Hishida | Large time behavior of a generalized Oseen evolution operator, with applications to the Navier-Stokes flow past a rotating obstacle[END_REF] and [START_REF] Hishida | Decay estimates of gradient of a generalized Oseen evolution operator arising from time-dependent rigid motions in exterior domains[END_REF]. We will use results established in [START_REF] Kobayashi | On the Oseen equation in three-dimensional exterior domains[END_REF] and [START_REF] Hishida | Decay estimates of gradient of a generalized Oseen evolution operator arising from time-dependent rigid motions in exterior domains[END_REF]; see Theorem 3.3 and 4.1.

We point out that according to [START_REF] Deuring | On Oseen resolvent estimates[END_REF], the velocity part U of a solution (U, Π) to the Oseen resolvent system -∆U + τ ∂ 1 U + λ U + ∇Π = F, div U = 0 in the whole space R 3 does not satisfy the estimate U 2 ≤ C 0 |λ| -1 F 2 with a single constant C 0 > 0 for all F ∈ L 2 (R 3 ) 3 and all λ ∈ C with λ > 0. As a consequence of this negative result, which arises because small values of |λ| are admitted, an analogous resolvent estimate cannot be expected to hold for solutions to the Oseen resolvent problem in Ω c , under whatever boundary conditions. Therefore, in view of [25, Theorem 4.2, point 3.)], it is a safe guess that maximal regularity is not valid for solutions of problem ( 1), [START_REF] Deuring | The Cauchy problem for the homogeneous time-dependent Oseen system in R 3 : spatial decay of the velocity[END_REF]. The negative result in [START_REF] Deuring | On Oseen resolvent estimates[END_REF] is the reason why we discuss some properties of the Oseen operator and its associated semigroup in greater detail; see the proof of Theorem 3.3. We mention that pointwise spatial decay of solutions to the nonlinear problem (3), ( 2) is considered in [START_REF] Knightly | Some decay properties of solutions of the Navier-Stokes equations[END_REF], [START_REF] Mizumachi | On the asymptotic behaviour of incompressible viscous fluid motions past bodies[END_REF], [START_REF] Deuring | Pointwise decay in space and in time for incompressible flow around a rigid body moving with constant velocity[END_REF], [START_REF] Deuring | Spatial decay of time-dependent incompressible Navier-Stokes flows with nonzero velocity at infinity[END_REF], [START_REF] Deuring | Pointwise decay in space and in time for incompressible flow around a rigid body moving with constant velocity[END_REF] and [START_REF] Deuring | Time-dependent incompressible viscous flows around a rigid body: estimates of spatial decay independent of boundary conditions[END_REF].

2.

Notation. Some auxiliary results. The symbol | | denotes the Euclidean norm of R n for any n ∈ N, and the length

α 1 + α 2 + α 3 of a multi-index α ∈ N 3 0 . For R ∈ (0, ∞), x ∈ R 3 , put B R (x) := {y ∈ R 3 : |x -y| < R}. In the case x = 0, we write B R instead of B R (0).
The set Ω ⊂ R 3 and the parameter τ ∈ (0, ∞) introduced in Section 1 will be kept fixed throughout. Recall that Ω is open and bounded, with smooth boundary and connected complement. Further recall that n (Ω) denotes the outward unit normal to Ω. We fix a number

R 0 ∈ (0, ∞) with Ω ⊂ B R0 . For R ∈ (0, ∞), we define Ω R := B R \Ω and Z R,∞ := Ω R × (0, ∞).
We additionally recall that also in Section 1, we introduced the weight function

ν : R 3 → [1, ∞) by setting ν(x) := 1 + |x| -x 1 for x ∈ R 3 .
For I ⊂ R, let χ I stand for the characteristic function of

I on R. If A ⊂ R 3 , we denote by A c the complement R 3 \A of A in R 3 . Put e l := (δ jl ) 1≤j≤3 for 1 ≤ l ≤ 3 (unit vector in R 3 ). If A is some nonempty set and γ : A → R a function, we set |γ| ∞ := sup{|γ(x)| : x ∈ A}. Let p ∈ [1, ∞), m ∈ N. If A ⊂ R 3 is
open, we write p for the norm of the Lebesgue space L p (A), and m,p for the usual norm of the Sobolev space W m,p (A) of order m and exponent p. For an open set B ⊂ R 3 , the spaces L p loc (B) and W m,p loc (B) are defined as the set of all functions V from B into R or C such that V |A ∈ L p (A) and V |A ∈ W 1,p (A), respectively, for any open, bounded set A ⊂ R 3 with A ⊂ B. We put ∇V := (∂ k V j ) 1≤j,k≤3 for V ∈ W 1,1 loc (B) 3 . Let n ∈ N and let V be a normed space, with norm denoted by . Then we will use the same notation for the norm of V n defined by (f 1 , ..., f n ) := 3 : div V = 0}. We write L p σ (A) for the closure of C ∞ 0,σ (A) with respect to the norm of L p (A) 3 . This function space L p σ (A) ("space of solenoidal L p -functions") is equipped with the norm p . Let p ∈ [1, ∞] and B be a Banach space. For any interval J ⊂ R, the notation

n j=1 f j 2 1/2 for (f 1 , ..., f n ) ∈ V n . The space V 3×3 , as concerns its norm, is identified with V 9 . Let A ⊂ R 3 be open and p ∈ (1, ∞). Put C ∞ 0,σ (A) := {V ∈ C ∞ 0 (A)
∈ (0, ∞], A ⊂ R 3 open, p ∈ [1, ∞], q ∈ (1, ∞) and n ∈ {1, 3}.
Then we write q,p;T instead of L p (0,T, L q (A) n ) . For an interval J ⊂ R and a function v : J → W 1,1 loc (A) 3 , the notation ∇ x v stands for the gradient of v with respect to x ∈ A, in the sense that

∇ x v : J → L 1 loc (A) 3×3 , ∇ x v(t)(x) := ∂ x k v j (t) (x) 1≤j,k≤3
for t ∈ J, x ∈ A (spatial gradient of v). Similar conventions are to be valid with respect to the expressions ∆ x v, div x v and ∂ xj v.

For the definition of the Bochner integral, we refer to [26, p. 132-133], or to [15, p. 78-80].

We write C for numerical constants and C(γ 1 , ..., γ n ) for constants depending exclusively on parameters γ 1 , ..., γ n ∈ [0, ∞) for some n ∈ N. However, such a precise bookkeeping will be possible only at some places. Mostly we will use the symbol C for constants whose dependence on parameters is not indicated. Sometimes we write C(γ 1 , ..., γ n ) in order to indicate that the constant in question is influenced in particular but not exclusively by the quantities γ 1 , ..., γ n . However, whenever the symbol C appears, it stands for a constant that does not depend on the quantities in a list introduced by the word "for" and preceding or following the respective inequality. In particular, such a constant never depends on the variable t.

We state an estimate involving the function ν.

Lemma 2.1. The inequality ν(x) ≤ C (1 + |y|) ν(x -y) holds for x, y ∈ R 3 .

Proof. Let x, y ∈ R. If |x| -x 1 ≤ 4 |y|, we get ν(x) ≤ 1 + 4 |y| ≤ 4 (1 + |y|) ν(x -y).
On the other hand, if |x| -x 1 ≥ 4 |y|, hence ν(x) ≥ 4 |y|, we have

ν(x) = 1 + |x -y + y| -(x -y + y) 1 ≤ ν(x -y) + 2 |y| ≤ ν(x -y) + ν(x)/2, so ν(x) ≤ 2 ν(x -y) (1 + |y|).
The Helmholtz-Fujita decomposition of L q (Ω c ) 3 will play an important role in what follows. The ensuing theorem serves to introduce this decomposition, fix the related notation, and indicate which properties of the operators in question will be used.

Theorem 2.2. For q ∈ (1, ∞), there is a linear bounded operator

P q : L q (Ω c ) 3 → L q σ (Ω c ) with P q (V ) = V for V ∈ L q σ (Ω c ). Moreover P q = P q for q ∈ (1, ∞). Proof. See [14, Section III.1], [5, Corollary 2.3].
We state two well-known properties of Bochner integrals in view of clarifying some arguments further below. Let A be another Banach space, A : B → A a linear and bounded operator, J ⊂ R an interval and f : J → B a Bochner integrable mapping. Then A • f : J → A is Bochner integrable, too, and A( J f dx) = J A • f dx, where the integral on the left-hand side is B-valued and the one on the right-hand side A-valued. We define some fundamental solutions. Set H(z, t)

:= (4 π t) -3/2 e -|z| 2 /(4t) for z ∈ R 3 , t ∈ (0, ∞) (heat kernel), Γ jk (z, t) := H(z, t) δ jk + ∞ t ∂ zj ∂ z k H(z, s) ds for z ∈ R 3 , t ∈ (0, ∞), 1 ≤ j, k ≤ 3
(fundamental solution to the time-dependent Stokes system), and

Λ (τ )
jk (z, t) := Γ jk (z -τ t e 1 , t) for z, t, j, k as before. (fundamental solution to the time-dependent Oseen system (1)) We will need the following estimate of Λ (τ ) .

Lemma 2.4 ([7, Corollary 3.3]). Let K > 0. Then for z ∈ B c K , t ∈ (0, ∞), α ∈ N 3 0 with |α| ≤ 2, the estimate |∂ α z Λ (τ ) (z, t)| ≤ C(K, τ ) |z| ν(z) + t) -(3+|α|)/2
holds. The potential functions introduced in the two ensuing lemmas are needed in order to state the decay result from [START_REF] Deuring | L q -weak solutions to the time-dependent Oseen system: decay estimates[END_REF] which we will apply later on (proof of Theorem 4.6).

Lemma 2.5 ([7, Corollary 3.5]). Let

q ∈ [1, ∞) and V ∈ L q (R 3 ) 3 . Then the integral R 3 |∂ α x Λ (τ ) (x -y, t) V (y)| dy is finite for α ∈ N 3 0 with |α| ≤ 1, x ∈ R 3 , t ∈ (0, ∞). Define the function I (τ ) (V ) : R 3 → R 3 by setting I (τ ) (V )(x, t) := R 3 Λ (τ ) (x -y, t) • V (y) dy for x ∈ R 3 , t ∈ (0, ∞). The derivative ∂ x l I (τ ) (V )(x, t) exists and equals R 3 ∂ x l ∂ σ t Λ (τ ) (x -y, t) • V (
y) dy for x, t as above and for l ∈ {1, 2, 3}. The functions

I (τ ) (V ) and ∂ l I (τ ) (V ) are continuous in R 3 × (0, ∞). If A ⊂ R 3 is measurable and V ∈ L q (A) 3 , the term I (τ ) (V ) is defined in an obvious way via the zero extension of V to R 3 . Lemma 2.6 ([7, Lemma 3.8]). Let q ∈ [1, ∞) and f ∈ L 1 loc [0, ∞), L q (R 3 ) 3 . Then the integral t 0 R 3 |∂ α x Λ (τ ) (x -y, t -σ) • f (y, σ)| dy dσ is finite for a. e. t ∈ (0, ∞), a. e. x ∈ R 3 and for α ∈ N 3 0 with |α| ≤ 1.
Thus we may define

R (τ ) (f )(x, t) := t 0 R 3 Λ (τ ) (x -y, t -σ) • f (y, σ) dy dσ for such t and x. The relation R (τ ) (f )(t) ∈ W 1,1 loc (R 3 ) 3 holds for a. e. t ∈ (0, ∞). If T 0 ∈ (0, ∞], A ⊂ R 3 measurable and f ∈ L 1 loc [0, T 0 ), L q (A) 3 , then R (τ ) (f ) is defined in an obvious way via the zero extension of f to R 3 × (0, ∞).
The next theorem states the decay result from [START_REF] Deuring | L q -weak solutions to the time-dependent Oseen system: decay estimates[END_REF] we mentioned in Section 1. 9 and div x u = 0.

Theorem 2.7. Take S 0 ∈ (0, R 0 ) with Ω ⊂ B S0 . Let q, r 1 , r 2 , r 3 ∈ (1, ∞) and take functions U 0 ∈ L q (Ω c ) 3 , f ∈ L 1 loc [0, ∞), L r1 (Ω c ) 3 and u : (0, ∞) → W 1,1 loc (Ω c ) 3 with u ∈ C 0 [0, ∞), L r2 (Ω c ) 3 , ∇ x u ∈ L 1 loc [0, ∞), L r3 (Ω c )
Further suppose that u satisfies the equation

T0 0 Ω c -γ (t) u(t) • ϑ + γ(t) ∇ x u(t) • ∇ϑ + τ ∂ x1 u(t) • ϑ -f (t) • ϑ dx dt -γ(0) Ω c U 0 • ϑ dx = 0 for γ ∈ C ∞ 0 [0, T 0 ) , ϑ ∈ C ∞ 0,σ (Ω c ).
Assume there are numbers q ∈ (1, ∞) and

υ 1 , υ 2 , υ 3 ∈ [1, ∞] such that the func- tion u|Z R0,∞ belongs to L ∞ 0, ∞, L q (Z R0,∞ ) 3 and to L υ1 0, ∞, L q (Z R0,∞ ) 3 , the restriction ∇ x u|Z R0,∞ is in L υ2 0, ∞, L q (Z R0,∞ ) 9 , and f |Z R0,∞ belongs to L υ3 0, ∞, L q (Z R0,∞ ) 3 . Suppose that the zero flux condition ∂Ω u(t) • n (Ω) do x = 0 holds for t ∈ (0, ∞).
Then there is a zero-measure subset S ∞ of (0, ∞) such that Under suitable assumptions on U 0 , the potential function

|∂ α x u -R (τ ) f |B c S0 × (0, ∞) -I (τ ) (U 0 |B c S0 ) (x, t)| ≤ C ( u|Z R0,∞ q,∞;∞ + u|Z R0,∞ q,υ1;∞ + ∇ x u|Z R0,∞ q,υ2;∞ + f |Z R0,∞ q,υ3;∞ + U 0 q ) |x| ν(x) -(3+|α|)/2+1/(2 min{υ 1 ,υ 2 ,υ 3 }) for α ∈ N 3 0 with |α| ≤ 1, t ∈ (0, ∞)\S ∞ , x ∈ B R0 c \N t ,
I (τ ) (U 0 )(x, t) dimin- ishes as O |x| ν(x) -2 for |x| → ∞.
Here are the details of this result, which will be interesting in the context of our decay estimate of weak solutions presented in Section 4 (Theorem 4.6). It is the highest rate of decay we could find for

I (τ ) (U 0 ). Lemma 2.8. Let q ∈ (3, ∞), U 0 ∈ L q σ (Ω c ) ∩ L 1 σ (Ω c ) 3 with supp(U 0 ) compact. Take R ∈ (0, ∞) with supp(U 0 ) ⊂ B R . Then |∂ α x I (τ ) (U 0 )(x, t)| ≤ C(τ, R) |x| ν(x) -(4+|α|)/2 U 0 1 for x ∈ B c 2R , t ∈ (0, ∞), α ∈ N 3 0 with |α| ≤ 1.
Proof. We apply an approach used by Kozono [21,p. 724] in a different context (temporal decay). By [20, Lemma 2.2], the function U 0 has mean value zero:

Ω c U 0 dx = 0. Take x, t, α as in the lemma. Then we get for y ∈ B R that |x -y| ≥ |x|/2 ≥ R and ν(x) ≤ C (1 + |y|) ν(x -y) (Lemma 2.1), so with Lemma 2.4, ∂ α x ∂ y l Λ (τ ) (x -y, t)| ≤ C(τ, R) |x -y| ν(x -y) + t -(4+|α|)/2 ≤ C(τ, R) |x| ν(x) -(4+|α|)/2 .
Now we find with Lemma 2.5 that

∂ α x I (τ ) (U 0 )(x, t) = B R ∂ α x Λ (τ ) (x -z, t) -∂ α x Λ (τ ) (x, t) U 0 (z) dz = B R 1 0 3 l=1 ∂ α x ∂ y l Λ (τ ) (x -y, t) |y=ϑ z z l dϑ U 0 (z) dz ≤ C(τ, R) |x| ν(x) -(4+|α|)/2 B R |z| |U 0 (z)| dz ≤ C(τ, R) |x| ν(x) -(4+|α|)/2 U 0 (z) 1 .
3. Mild solutions of ( 1), [START_REF] Deuring | The Cauchy problem for the homogeneous time-dependent Oseen system in R 3 : spatial decay of the velocity[END_REF]. We begin by recalling some known results, occasionally discussing a proof if the result in question is slightly modified or is not stated clearly in literature.

Lemma 3.1. Let (B, ) be a Banach space, S : [0, ∞) → B a C 0 -semigroup on B and f ∈ L 1 loc [0, ∞), B . Then t 0 S(t -s) f (s) ds < ∞ for t ∈ (0, ∞). Define u(t) := t 0 S(t -s) f (s) ds for t ∈ (0, ∞). Then u ∈ C 0 [0, ∞), B .
Proof. See [START_REF] Hille | Functional Analysis and Semi-Groups[END_REF]Theorem 3.8.4] and its proof, and the first statement of Theorem 2.3.

Next we introduce the Oseen operator

O q : D(O q ) → L q σ (Ω c ) for q ∈ (1, ∞) by setting D(O q ) := L q σ (Ω c ) ∩ W 1,q 0 (Ω c ) 3 ∩ W 2,q ((Ω c ) 3 , O q (V ) := P q (∆V -τ ∂ 1 V ) for V ∈ D(O q ), (5)
where the operator P q was introduced in Theorem 2.2. We denote the identity mapping on L q σ (Ω c ) by I q . The ensuing theorem gives some details on the resolvent of O q .

Theorem 3.2. Let q ∈ (1, ∞). Then the resolvent set (O q ) of O q is given by

(O q ) = {λ ∈ C : τ 2 λ > -( λ) 2 }. The relation (λ I q -O q ) -1 (F ) ∈ D(O q ) holds for λ ∈ (O q ) and for F ∈ L q σ (Ω c ). Let ϑ 0 ∈ (π/2, π). There is r 0 ∈ (0, ∞) such that {λ ∈ C : |λ| ≥ r 0 , | arg λ| ≤ ϑ 0 } ⊂ (O q ) and |λ| U q + |λ| 1/2 U 1,q + U 2,q ≤ C F q for F ∈ L q σ (Ω c ) (6)
and for λ ∈ C with |λ| ≥ r 0 , | arg λ| ≤ ϑ 0 , where U := (λ I q -O q ) -1 (F ).

Proof. The first claim of that theorem, pertaining to (O q ), holds according to [START_REF] Farwig | On the spectrum of an Oseen-type operator arising from flow around a rotating body[END_REF]Theorem 3.1]. The relation (λ

I q -O q ) -1 (F ) ∈ D(O q ) for λ ∈ (O q ), F ∈ L q σ (Ω c ) is
obvious by the definition of the resolvent. By [19, Lemma 4.5], there is r 0 ∈ (0, ∞)

such that |λ| U q + U 2,q ≤ C F q for λ ∈ C, |λ| ≥ r 0 , | arg λ| ≤ ϑ 0 , F ∈ L q loc (Ω c
), with U defined as above. It follows by interpolation that |λ| 1/2 ∇V q ≤ C F q .

The ensuing theorem deals with the semigroup generated by O q . In particular it presents an L q -L q -estimate (inequality ( 7)), which we take from [START_REF] Kobayashi | On the Oseen equation in three-dimensional exterior domains[END_REF] and [START_REF] Hishida | Decay estimates of gradient of a generalized Oseen evolution operator arising from time-dependent rigid motions in exterior domains[END_REF]. The estimate in the following theorem is "global" in the sense that it gives an upper bound of the Oseen semigroup with respect to L q -norms on the exterior domain Ω c . In Section 4, we will additionally need "local" L p -L q -estimates, that is, upper bounds for L p -norms on Ω R0 instead of Ω c . These latter estimates yield decay rates which are not always available in the global setting. We further note that [START_REF] Hishida | Large time behavior of a generalized Oseen evolution operator, with applications to the Navier-Stokes flow past a rotating obstacle[END_REF] and [START_REF] Hishida | Decay estimates of gradient of a generalized Oseen evolution operator arising from time-dependent rigid motions in exterior domains[END_REF] deal with the case of time-dependent coefficients and rotational terms in the differential equations, a level of generality not needed here.

Theorem 3.3. Let q ∈ (1, ∞). The operator O q generates an analytic semigroup on L q σ (Ω c ). We write e t Oq for its value in t ∈ [0, ∞). Let U ∈ L q σ (Ω c ), and put

u U (t) := e t Oq U for t ∈ [0, ∞). Then u U ∈ C 0 [0, ∞), L q σ (Ω c ) ∩ C ∞ (0, ∞), L q σ (Ω c ) , u U (t) ∈ D(O q ), u U (t) = O q u U (t) for t ∈ (0, ∞), and u U (0) = U. Moreover ∂ α x u U (t) q ≤ C χ (0,1] (t) t -|α|/2 + χ [1, ∞) (t) t -min{1/2, 3/(2q)} |α| U q (7) for t ∈ (0, ∞), α ∈ N 3 0 with |α| ≤ 1. In particular ∇ x u U ∈ L 1 loc [0, ∞), L q (Ω c ) 9 .
Proof. According to Miyakawa [22, Theorem 4.2], the Oseen operator O q generates an analytical semigroup on L q σ (Ω c ). Since Theorem 3. 2 was not yet available in [START_REF] Miyakawa | On nonstationary solutions of the Navier-Stokes equations in an exterior domain[END_REF],

but allows to directly reduce this semigroup property to standard results in [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF], we indicate a proof based on such a reduction, for the convenience of the reader, although the argument in question is in principle well known. Take ϑ 0 ∈ (π/2, π) and choose a number r 0 associated to ϑ 0 as in Theorem 3.2. By that theorem

S 0 := {λ ∈ C : |λ| ≥ r 0 , | arg λ| ≤ ϑ 0 } ⊂ (O q ) and (λ I q -O q ) -1 (F ) q ≤ C |λ| -1 F q for F ∈ L q σ (Ω c ), λ ∈ S 0 . (8)
We may choose a 0 > 0 such that S ϑ0,a0

:= {λ ∈ C\{a 0 } : | arg(λ -a 0 )| ≤ ϑ 0 } ⊂ S 0 .
As a consequence S ϑ0,a0 ∪{a 0 } ⊂ (O q ) and inequality (8) holds for F ∈ L q σ (Ω c ), λ ∈ S ϑ0,a0 ∪ {a 0 }. But for λ ∈ S ϑ0,a0 , we have |λ| ≥ |λ -a 0 | sin(ϑ 0 ), so we may conclude with (6) that (λ I q -O q ) -1 (F ) q ≤ C |λ -a 0 | -1 F q for F and λ as before.

As a consequence the sets {0} and S ϑ0,0 := {λ ∈ C\{0} : | arg(λ)| ≤ ϑ 0 } are contained in the resolvent set (-a 0 I q + O q ) of the operator -a 0 I q + O q , and λ -(-

a 0 I q + O q ) -1 (F ) q ≤ C |λ| -1 F q for F ∈ L q σ (Ω c
) and λ ∈ S ϑ0,0 . Now it follows that -a 0 I q + O q generates an analytic semigroup on L q σ (Ω c ) ([24, Theorem 1.7.7, 2.5.2]). We write e t (-a0 Iq+Oq) for its value in t ∈ [0, ∞), so that the mapping t → e t (-a0 Iq+Oq) t ∈ [0, ∞) is the restriction of an analytic semigroup to [0, ∞).

Then the mapping t → e t a0 e t (-a0 Iq+Oq) t ∈ [0, ∞) is also such a restriction, and the operator O q is the infinitesimal generator of this semigroup. In view of the uniqueness result in [24, Theorem 1.2.6] and the notation introduced in Theorem 3.3, we have e t Oq = e t a0 e t (-a0 Iq+Oq) for t ∈ [0, ∞). The function u U introduced in Theorem 3.3 thus belongs to Concerning inequality [START_REF] Deuring | The 3D time-dependent Oseen system: link between L p -integrability in time and pointwise decay in space[END_REF], we indicate that it is valid in the case t ≤ 1 according to [19, (6.38)], whereas the case t ≥ 1 is covered by [19, (1.2)] if α = 0, [19, (1.3)] if |α| = 1, q ≤ 3, and [17, (2.23)] if |α| = 1, q > 3. Actually, in the case t ≥ 1, |α| = 1, q > 3, another reference would be [19, (1.6)], but that latter inequality is proved only implicitly in [START_REF] Kobayashi | On the Oseen equation in three-dimensional exterior domains[END_REF].

C 0 [0, ∞), L q σ (Ω c ) and to C ∞ (0, ∞), L q σ (Ω c ) , with u U (t) ∈ D(O q ), u U (t) = O q u U (t) for t ∈ (0, ∞),
Theorem 3.4. Let q ∈ (1, ∞) and f ∈ L 1 loc [0, ∞), L q σ (Ω c ) . Define u f (t) := t 0 e (t-s) Oq f (s) ds for t ∈ [0, ∞)
(see Lemma 3.1), with the preceding integral denoting an L q σ (Ω c )-valued and an

L q (Ω c ) 3 -valued Bochner integral. Then the integral t 0 ∂ x l e (t-s) Oq f (s) q ds ex- ists and u f (t) ∈ W 1,q 0 (Ω c ) 3 for a. e. t > 0. Moreover div x u f = 0, ∇ x u f ∈ L 1 loc [0, ∞), L q (Ω c ) 3 , and
∂ x l u f (t) = t 0 ∂ x l e (t-s) Oq f (s) ds for 1 ≤ l ≤ 3 and a. e. t ∈ (0, ∞). (9)
Proof. Of course, it does not make any difference whether the integral in the definition of u f is considered as an L q σ (Ω c )-valued or an L q (Ω c ) 3 -valued Bochner integral, due to Theorem 2.3 and the fact that the canonical imbedding of L p σ (Ω c ) into L q (Ω c ) 3 is linear and bounded. Let T ∈ (0, ∞), α ∈ N 3 0 with |α| ≤ 1, and put Z T := Ω c × (0, T ). Then by [START_REF] Deuring | The 3D time-dependent Oseen system: link between L p -integrability in time and pointwise decay in space[END_REF],

T 0 t 0 ∂ α x e (t-s) Oq f (s) q ds dt ≤ C T 0 t 0 χ (0,1) (t -s) (t -s) -|α|/2 + χ [1,∞) (t -s) χ (0,T ) (t -s) f (s) q ds dt ≤ C R χ (0,1) (r) r -|α|/2 + χ [1,∞) (r) χ (0,T ) (r) dr f |Z T q,1;T ≤ C(T ) f |Z T q,1;T , (10)
where we used Young's inequality in the second estimate. Inequality [START_REF] Deuring | On Oseen resolvent estimates[END_REF] yields in particular that t 0 ∂ α x e (t-s) Oq f (s) q ds < ∞ for a. e. t ∈ (0, ∞) and for α ∈ N 3 0 with |α| ≤ 1. Take any such t. Let l ∈ {1, 2, 3} and ψ ∈ C ∞ 0 (Ω) 3 . By the second statement in Theorem 2.3 and because e (t-s) Oq f (s) ∈ D(O q ) for s ∈ (0, t), we get that

Ω c ∂ l ψ(x) • u f (x, t) dx = - Ω c ψ(x) • t 0 ∂ x l e (t-s) Oq f (s) ds (x) dx.
Thus the weak derivative ∂ x l u f (t) exists and equation ( 9) holds. Since the integral t 0 ∂ α x e (t-s) Oq f (s) q ds is finite by the choice of t, we thus have u f (t) ∈ W 1,q (Ω c ) 3 . Equation ( 9) and inequality [START_REF] Deuring | On Oseen resolvent estimates[END_REF] imply that the function

∇ x u f belongs to L 1 loc [0, ∞), L q (Ω c ) 3 .
In order to show that u f (t) ∈ W 1,q 0 (Ω c ) 3 and div x u f (t) = 0, again take t ∈ (0, ∞)

with t 0 ∂ α x e (t-s) Oq f (s) q ds < ∞ for α ∈ N 3 0 , |α| ≤ 1. We have e (t-s) Oq f (s) ∈ D(O q ) ⊂ W 1,q 0 (Ω c ) 3 ∩ L q σ (Ω c
) for s ∈ (0, t), in particular div x e (t-s) Oq f (s) = 0 by a density argument in L q σ (Ω c ). It follows with (9) that div x u f (t) = 0. By the choice of t and because e (t-s) Oq f (s) ∈ W 1,q 0 (Ω c ) 3 we may conclude that the integral t 0 e (t-s) Oq f (s) ds exists also as a W 1,q 0 (Ω c ) 3 -valued Bochner integral. For 3 , the operator V → Ω c ψ V dx is linear and bounded as a mapping on L q σ (Ω c ) and on W 1,q 0 (Ω c ) 3 . This observation and Theorem 2.3 imply that the integral t 0 e (t-s) Oq f (s) ds yields the same function both as L q σ (Ω c )-and W 1,q 0 (Ω c ) 3 -valued Bochner integral. As a consequence we have u f (t) ∈ W 1,q 0 (Ω c ) 3 .

ψ ∈ C ∞ 0 (Ω c )
In the ensuing theorem, we collect some of our previous results. They allow us to introduce the notion of "mild solution ". The theorem then states that such a solution satisfies [START_REF] Deuring | Spatial decay of time-dependent Oseen flows[END_REF] in the sense of an L q -weak solutions as formulated in [START_REF] Deuring | Oseen resolvent estimates with small resolvent parameter[END_REF], for a right-hand side f of low regularity.

Theorem 3.5. Let q ∈ (1, ∞), U 0 ∈ L q σ (Ω c ) and f ∈ L 1 loc [0, ∞), L q σ (Ω c ) . Let
the functions u U0 and u f be defined as in Theorem 3.3 and 3.4, respectively, and put u := u U0 + u f . This function u is called a "mild solution" to [START_REF] Deuring | Spatial decay of time-dependent Oseen flows[END_REF]. It satisfies the relations

u ∈ C 0 [0, ∞), L q σ (Ω c ) , u(0) = U 0 , u(t) ∈ W 1,q 0 (Ω c ) 3 , div x u(t) = 0 for t ∈ (0, ∞),
and

∇ x u ∈ L 1 loc [0, ∞), L q (Ω c ) 9 .
Moreover the function u fulfills equation [START_REF] Deuring | Oseen resolvent estimates with small resolvent parameter[END_REF].

Proof. By Theorem 3.3 we know that

u U0 ∈ C 0 [0, ∞), L q σ (Ω c ) , u U0 (0) = U 0 and u U0 (t) ∈ D(O q ). Let γ ∈ C ∞ 0 [0, ∞) and ϑ ∈ C ∞ 0,σ (Ω c ). Choose some T ∈ (0, ∞) with supp(γ) ⊂ [0, T ].
The properties of u U0 listed above imply that the function

t → Ω c u U0 (t) • ϑ dx t ∈ [0, ∞) belongs to C 0 [0, ∞) and to C ∞ (0, ∞) , with ∂ t Ω c u U0 (t) • ϑ dx = Ω c u U0 (t) • ϑ dx for t ∈ (0, ∞), and Ω c u U0 (t) • ϑ dx |t=0 = Ω c U 0 • ϑ dx. Thus γ( ) Ω c u U0 ( ) • ϑ dx → γ(0) Ω c U 0 • ϑ dx for ↓ 0. From the preceding relations we get ∞ 0 γ (t) Ω c u U0 (t) • ϑ dx dt = lim ↓0 T γ (t) Ω c u U0 (t) • ϑ dx dt = -lim ↓0 T γ(t) Ω c u U0 (t) • ϑ dx dt -γ(0) Ω c U 0 • ϑ dx. ( 11
)
Since by [START_REF] Deuring | The 3D time-dependent Oseen system: link between L p -integrability in time and pointwise decay in space[END_REF],

T 0 Ω c |∇ x u U0 (t) • ∇ϑ| dx dt < ∞, and because u U0 (t) ∈ W 2,q (Ω c ) 3 for t > 0, we have ∞ 0 γ(t) Ω c ∇ x u U0 (t) • ∇ϑ dx dt = -lim ↓0 T γ(t) Ω c ∆ x u U0 (t) • ϑ dx ds. But ϑ ∈ C ∞ 0,σ (Ω c
), so P q (ϑ) = ϑ by Theorem 2.2. Therefore due to the equation

P q = P q (Theorem 2.2), we get ∞ 0 γ(t) Ω c ∇ x u U0 (t) • ∇ϑ dx dt = -lim ↓0 T γ(t) Ω c P q ∆ x u U0 (t) • ϑ dx ds.
(12) By a similar reasoning we find that

∞ 0 γ(t) Ω c τ ∂ x1 u U0 (t)•ϑ dx dt = lim ↓0 T γ(t) Ω c P q τ ∂ x1 u U0 (t) •ϑ dx ds. (13)
Since (u U0 ) (t) = O q u U0 (t) for t > 0, we may conclude from ( 11) -( 13) that equation ( 5) is valid with f = 0 and with u U0 in the role of u. Note that in the preceding argument, the integral T 0 γ(t) Ω c P q ∆ x u U0 (t) • ϑ dx ds, which does not exist in general, does not arise.

By Lemma 3.1 we know that 9 . Take γ, ϑ and T as above. By the previous relation and the second claim in Theorem 2.3, we get

u f ∈ C 0 [0, ∞), L q σ (Ω c ) . Moreover Theorem 3.4 yields that u f (t) ∈ W 1,q 0 (Ω c ) 3 , div x u f (t) = 0 for a. e. t ∈ (0, ∞), and u f ∈ L 1 loc [0, ∞), L q (Ω c )
∞ 0 γ (t) Ω c u f (t) • ϑ dx dt = T 0 t 0 Ω c γ (t) ϑ(x) • e (t-s) Oq f (s) (x) dx ds dt. (14)
Due to inequality [START_REF] Deuring | The 3D time-dependent Oseen system: link between L p -integrability in time and pointwise decay in space[END_REF], we have e (t-s) Oq f (s) q ≤ C(T ) f (s) q for t ∈ (0, T ), s ∈ (0, t). Moreover the function s → f (s) q s ∈ (0, T ) is integrable, the function γ is bounded, and ϑ is bounded with compact support. As a consequence the integral

T 0 t 0 Ω c |γ (t) ϑ(x)• e (t-s) Oq f (s) (x)
| dx ds dt exists. Thus we may apply Fubini's and Lebesgue's theorem on the right-hand side of ( 14), to obtain

∞ 0 γ (t) Ω c u f (t) • ϑ dx dt = lim ↓0 T 0 T s+ Ω c γ (t) ϑ(x) • e (t-s) Oq f (s) (x) dx dt ds. (15)
By Theorem 3.3 we know that for s ∈ (0, ∞),

the function t → e (t-s) Oq f (s) t ∈ [s, ∞) belongs to C 0 [s, ∞), L q σ (Ω c ) and to C ∞ (s, ∞), L q σ (Ω c ) . Hence, for s ∈ (0, ∞), the function K ϑ,s (t) := Ω c ϑ(x) • e (t-s) Oq f (s) (x) dx t ∈ [s, ∞) belongs to C 0 [s, ∞) ∩ C ∞ (s, ∞) , with K ϑ,s (t) := Ω c ϑ(x) • ∂ t e (t-s) Oq f (s) (x) dx.
Thus the right-hand side in (15) may be transformed by an integration by parts into the term lim

↓0 - T 0 T s+ γ(t) K ϑ,s (t) dt ds - t 0 γ(s + ) Ω c ϑ(x) • e Oq f (s) (x) dx ds .
But with Hölder's inequality and (

)

Ω c ϑ(x) • e Oq f (s) (x) dx| ≤ C |γ| ∞ ϑ q f (s) q ,
with the function s → f (s) q s ∈ (0, T ) being integrable, as already mentioned before. Moreover, by the continuity of the function r → e r Oq V r ∈ [0, ∞) , and because this function takes the value

V if r = 0, for V ∈ L q σ (Ω c ) (Theorem 3.3), we obtain that γ(s + ) Ω c ϑ(x) • e Oq f (s) (x) dx → γ(s) Ω c ϑ(x) • f (x, s) dx ( ↓ 0). Therefore T 0 γ(s+ ) Ω c ϑ(x)• e Oq f (s) (x) dx ds → T 0 γ(s) Ω c ϑ(x)•f (x, s) dx ds ( ↓ 0)
by Lebesgue's theorem. From [START_REF] Hille | Functional Analysis and Semi-Groups[END_REF], the transformation of the right-hand side of (15) presented above and the preceding relation, we deduce that

∞ 0 γ (t) Ω c u f (t) • ϑ dx dt = - T 0 γ(t) Ω c f (t) • ϑ dx dt -lim ↓0 T 0 T s+ γ(t) Ω c ϑ(x) • ∂ t e (t-s) Oq f (s) (x) dx dt ds. (16)
Since ∇ x u f ∈ L 1 loc [0, ∞), L q (Ω c ) 9 by Theorem 3.4, and because of the second statement in Theorem 2.3, we get

∞ 0 γ(t) Ω c τ ∂ x1 u f (t) • ϑ dx dt = T 0 γ(t) t 0 Ω c ϑ(x) • τ ∂ x1 e (t-s) Oq f (s) (x) dx ds dt. (17)
By [START_REF] Deuring | The 3D time-dependent Oseen system: link between L p -integrability in time and pointwise decay in space[END_REF], ∇ x e r Oq f (s) q ≤ C(T ) r -1/2 f (s) q for r, s ∈ (0, T ). Thus with Hölder's inequality,

T 0 t 0 Ω c |γ(t)ϑ(x) • τ ∂ x1 e (t-s) Oq f (s) (x)| dx ds dt ≤ C(T ) |γ| ∞ ϑ q T 0 t 0 (t -s) -1/2 f (s) q ds dt. But T 0 t 0 (t -s) -1/2 f (s) q ds dt = T 0 T s (t -s) -1/2 dt f (s) q ds ≤ C(T ) T 0 f (s) q ds.
Since the function s → f (s) q s ∈ (0, T ) is integrable, it follows that the integral

T 0 t 0 Ω c |γ(t)ϑ(x) • τ ∂ x1 e (t-s) Oq f (s) (x)
| dx ds dt is finite. Thus from ( 17) and Fubini's and Lebesgue's theorem,

∞ 0 γ(t) Ω c τ ∂ x1 u f (t) • ϑ dx dt = lim ↓0 T 0 T s+ Ω c γ(t) ϑ(x) • τ ∂ x1 e (t-s) Oq f (s) (x) dx dt ds. (18)
The same reasoning yields that

∞ 0 γ(t) Ω c ∇ x u f (t) • ∇ϑ dx dt = lim ↓0 T 0 T s+ Ω c γ(t) ∇ϑ(x) • ∇ x e (t-s) Oq f (s) (x) dx dt ds. (19)
On the other hand, we have

e (t-s) Oq f (s) ∈ W 2,q (Ω c ) 3 for s ∈ (0, ∞), t ∈ (s, ∞),
hence we obtain

Ω c ∇ϑ(x) • ∇ x e (t-s) Oq f (s) (x) dx = - Ω c ϑ(x) • ∆ x e (t-s) Oq f (s) (x) dx
for s, t as before. So we may combine ( 18) and ( 19) to get

∞ 0 γ(t) Ω c ∇ x u f (t) • ∇ϑ + τ ∂ x1 u f (t) • ϑ dx dt = lim ↓0 T 0 T s+ Ω c γ(t) ϑ(x) • -∆ x + τ ∂ x1 e (t-s) Oq f (s) (x) dx dt ds. (20)
At this point we proceed as in the first part of the proof, using the equations P q (ϑ) = ϑ and P q = P q provided by Theorem 2.2. Due to them, we may rewrite [START_REF] Kozono | L 1 -solutions of the Navier-Stokes equations in exterior domains[END_REF] as

∞ 0 γ(t) Ω c ∇ x u f (t) • ∇ϑ + τ ∂ x1 u f (t) • ϑ dx dt = -lim ↓0 T 0 T s+ Ω c γ(t) ϑ(x) • O q e (t-s) Oq f (s) (x) dx dt ds. (21)
But (∂ t -O q ) e (t-s) Oq f (s) = 0 for s ∈ (0, ∞), t ∈ (s, ∞) by Theorem 3.3, so it follows from ( 16) and ( 21) that equation ( 5) holds with U 0 = 0 and u f in the role of u. Since u = u U0 + u f , equation ( 5) holds as stated in the theorem.

4. Spatial decay of mild solutions. We use L p -L q -estimates of the Oseen semigroup e t Oq in order to deduce rates of spatial decay of mild solutions to (1), [START_REF] Deuring | The Cauchy problem for the homogeneous time-dependent Oseen system in R 3 : spatial decay of the velocity[END_REF].

Our main tools are Theorem 3.3 as well as the following theorem which reproduces results from [START_REF] Kobayashi | On the Oseen equation in three-dimensional exterior domains[END_REF] and [START_REF] Hishida | Decay estimates of gradient of a generalized Oseen evolution operator arising from time-dependent rigid motions in exterior domains[END_REF]. Recall that the parameter R 0 ∈ (0, ∞) and the set Z R0,∞ were introduced at the beginning of Section 2, and the functions u U and u f in Theorem 3.3 and 3.4, respectively, for

U ∈ L q σ (Ω c ) and f ∈ L 1 loc [0, ∞), L q σ (Ω c ) .
Theorem 4.1 ([19, (6.18)], [17, (6.4)]). Let q ∈ (1, ∞). Then

u V |Ω R0 1,q ≤ C t -3/(2 q) V q for V ∈ L q σ (Ω c ), t ∈ [1, ∞),
with the constant C being independent of t and V .

Corollary 4.2. Let q ∈ (1, ∞). Then

∂ α x u V |Ω R0 q ≤ C χ (0,1) (t) t -|α|/2 + χ [1,∞) (t) t -3/(2 q) ) V q for V ∈ L q σ (Ω c
), a. e. t ∈ (0, ∞) and α ∈ N 3 0 with |α| ≤ 1. Proof. Theorem 3.3 (t ≤ 1) and 4.1 (t > 1).

In the ensuing three corollaries, we apply Theorem 3.3 and the preceding corollary in order to estimate the functions u f and u U0 .

Corollary 4.3. Let q ∈ (1, ∞). If q < 3/2, take p 1 ∈ [1, ∞) and set p := p 1 .

In the case q ≥ 3/2, let p 1 ∈ 1, (1

-3/(2q)) -1 . Then 1 ≥ 3/(2q) > 1 -1/p 1 ≥ 0. Fix some p 2 ∈ (1, ∞) with 3/(2q) > 1/p 2 > 1 -1/p 1 . For example, choose p 2 := 2 3/(2q) + 1 -1/p 1 -1 . Then 1 ≥ 1/p 1 > 1/p 1 + 1/p 2 -1 > 0. Define p := (1/p 1 +1/p 2 -1) -1 . Then p 1 , p 2 , p ∈ [1, ∞) with 1/p = 1/p 1 +1/p 2 -1, 3 p 2 /(2q) > 1 and p ∈ [p 1 , ∞). Moreover u f |Z R0,∞ q,p;∞ ≤ C f q,p1;∞ and u f (t)|Ω R0 q ≤ C f q,p1;∞ for f ∈ L p1 0, ∞, L q σ (Ω c ) , t ∈ (0, ∞), with C independent of t and f . Proof. For any f ∈ L 1 loc [0, ∞), L q σ (Ω c ) , t ∈ (0, ∞),
u f (t)|Ω R0 q ≤ C t 0 g(t -s) f (s) q ds, (22)
with g(r) := χ (0,1) (r) + χ [1,∞) (r) r -3/(2q) for r ∈ R. In the case q < 3/2, put p 2 := 1. Then we may conclude for any choice of q that p 1 , p 2 , p ∈ [1, ∞), 1/p = 1/p 1 + 1/p 2 -1 and 3 p 2 /(2q) > 1. The latter inequality yields that g ∈ L p2 (R).

At this point we see that inequality [START_REF] Miyakawa | On nonstationary solutions of the Navier-Stokes equations in an exterior domain[END_REF] and Young's inequality imply the estimate u f |Z R0,∞ q,p;∞ ≤ C f q,p1;∞ for f ∈ L p1 0, ∞, L q σ (Ω c ) . Since p 1 < 1 -3/(2q) -1 in the case q ≥ 3/2, and 3/(2q) > 1 if q < 3/2, we have in any case that 3 p 1 /(2q) > 1. Therefore ( 22) and Hölder's inequality yield the estimate of u f (t)|Ω R0 q stated in the corollary.

Corollary 4.4. Let the numbers q, p 1 , p 2 be given in one of the following three ways:

Either q ∈ [1, 3/2), p 1 ∈ [1, ∞), p := p 1 , or q ∈ [3/2, 3), p 1 ∈ 1, (1 -3/(2q)) -1
, and p 2 and p are defined by p 2 :=

2 3/(2q) + max{1/2, 1 -1/p 1 } -1 , p := (1/p 1 + 1/p 2 -1) -1 , or q ∈ [3, ∞), p 1 ∈ 1, (1 -3/(2q)) -1 , p 2 := 2 3/(2q) + 1 -1/p 1 -1 and again p := (1/p 1 + 1/p 2 -1) -1 .
In any case, p belongs to [p 1 , ∞). If q < 3, the inequality

(∇ x u f )|Z R0,∞ q,p;∞ ≤ C f q,p1;∞ holds for f ∈ L p1 0, ∞, L q σ (Ω c
) . In the case q ≥ 3, the estimate

(∇ x u f |Z R0,∞ ) q,p;∞ ≤ C ( f q,p1;∞ + f q,p;∞ )
is valid for functions f belonging to L υ 0, ∞, L q σ (Ω c ) for both υ = p 1 and υ = p.

Proof. Suppose that q ∈ [3/2, 3). Then 3/(2q) ∈ (1/2, 1]. On the other hand, the assumption Next assume that q ≥ 3. Then the conditions on p 1 and the definition of p 2 imply that 3/(2q) > 1/p 2 > 1 -1/p 1 , so Corollary 4.3 yields that p 2 ∈ (1, ∞), 1/p 1 > 1/p 1 + 1/p 2 -1 > 0 and 3 p 2 /(2q) > 1. In particular p is again well defined and p ∈ [p 1 , ∞).

p 1 ∈ 1, (1 -3/(2q)) -1 implies 3/(2q) > 1 -1/p 1 . Therefore we have 3/(2q) > max{1/2, 1 -1/p 1 } > 0, so 3/(2q) > 1/p 2 > max{1/2, 1 -1/p 1 }, hence 1 > 1/p 2 > 1/2, that is, p 2 ∈ (1, 2). Moreover 3/(2q) > 1/p 2 > 1 -1/p 1 ,
If q < 3/2, choose p 2 = 1. Then (1/p 1 + 1/p 2 -1) -1 = p by the choice of p, and 3 p 2 /(2q) > 1 because 3/(2q) > 1 in the case under consideration.

Altogether we have for any choice of q that p 1 , p 2 , p ∈ [1, ∞), 3 p 2 /(2q) > 1 and 1/p = (1/p 1 + 1/p 2 -1) -1 . If q < 3, we additionally have p 2 ∈ [1, 2). Theorem 3.4 and 2.3 yield that ∂

x l u f (t) |Ω R0 = t 0 ∂ x l e (t-s) Oq f (s) |Ω R0 ds for f ∈ L 1 loc [0, ∞), L q σ (Ω c ) , 1 ≤ l ≤ 3
∇ x u f (t) |Ω R0 q ≤ C t 0 g(t -s) f (s) q ds (23)
for f, t as before, with g(r) := χ (0,1) (r) r -1/2 + χ [1,∞) (r) r -3/(2q) for r ∈ R. Since p 2 ∈ [1, 2) and 3 p 2 /(2q) > 1 in the case q < 3, we obtain g ∈ L p2 (R) in that case. Recalling that p 1 , p 2 , p ∈ [1, ∞) and p = (1/p 1 + 1/p 2 -1) -1 , we may thus conclude from ( 23) and Young's inequality that in the case q < 3, the estimate

(∇ x u f )|Z R0,∞ q,p;∞ ≤ C f q,p1;∞ is valid for f ∈ L p1 0, ∞, L q σ (Ω c ) .
Now suppose that q ≥ 3. Since in this case the relations p = (1/p 1 + 1/p 2 -1) -1 and 3 p 2 /(2q) > 1 are valid, too, we may apply Young's inequality once more, to obtain [START_REF] Mizumachi | On the asymptotic behaviour of incompressible viscous fluid motions past bodies[END_REF] and the two preceding estimates, we may conclude that (∇ x u f )|Z R0,∞ q,p;∞ ≤ C ( f q,p1;∞ + f q,p;∞ ).

∞ 0 R χ [1,∞) (t -s) (t -s) -3/(2q) f (s) q ds p dt 1/p ≤ C f q,p1;∞ . Also by Young's inequality we get ∞ 0 R χ (0,1) (t -s) (t -s) -1/2 f (s) q ds p dt 1/p ≤ C f q,p;∞ . Due to
Corollary 4.5. Let q ∈ (1, ∞). If q < 3/2, take p ∈ [1, ∞), else let p ∈ (2 q/3, ∞).

Then the inequalities

u U0 |Z R0,∞ q,p;∞ ≤ C U 0 q and u U0 (t)|Ω R0 q ≤ C U 0 q hold for U 0 ∈ L q σ (Ω c ), t ∈ (0, ∞). Let q ∈ (1, 3). If q < 3/2, take p ∈ [1, 2), else let p ∈ (2 q/3, 2). Then (∇ x u U0 )|Z R0,∞ q,p;∞ ≤ C U 0 q for U 0 ∈ L q σ (Ω c ).
Proof. Use Corollary 4.2.

Now we are in a position to establish our decay result for mild solutions to (1), (2). Theorem 4.6. Fix some S 0 ∈ (0, R 0 ) with Ω ⊂ B S0 . Choose parameters q, p 0 , p 1 , p 2 , γ in the following way:

Either take q ∈ (1, 3/2), p 1 ∈ [1, ∞) and set p 0 := 1, γ := p 1 , or let q ∈ [3/2, 3), p 0 ∈ 2q/3, 2 and p 1 ∈ 1, (1 -3/(2q)) -1 , and define

p 2 := 2 3/(2q) + max{1/2, 1 -1/p 1 } -1 , γ := (1/p 1 + 1/p 2 -1) -1 ,
or choose q ∈ [3, ∞) and p 1 ∈ 1, (1 -3/(2q)) -1 , and then define p 2 := 2 3/(2q) + 1 -1/p 1 -1 and again γ := (1/p 1 + 1/p 2 -1) -1 .

According to Corollary 4.4, the parameter γ is well defined in all three cases and belongs to [p 1 , ∞).

In the case q < 3, let U 0 ∈ L q σ (Ω c ) and f ∈ L p1 0, ∞, L q σ (Ω c ) , and define u := u U0 + u f . If q ≥ 3, take f ∈ L p1 0, ∞, L q σ (Ω c ) ∩ L γ 0, ∞, L q σ (Ω c ) and set u := u f .

Then there is a zero-measure subset S ∞ of (0, ∞) and for any t ∈ (0, ∞)\S ∞ a zero-measure subset N t of B R0 c such that

|∂ α x u -R (τ ) f |B c S0 × (0, ∞) -I (τ ) (U 0 |B c S0 ) (x, t)|
≤ C U 0 q |x| ν(x) -(3+|α|)/2+1/(2p 0 ) + |x| ν(x) -(3+|α|)/2+1/(2 γ ) f q,p1;∞ [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF] if q < 3, and

|∂ α x u -R (τ ) f |B c S0 × (0, ∞) (x, t)| ≤ C |x| ν(x) -(3+|α|)/2+1/(2 γ ) ( f q,p1;∞ + f q,γ;∞ ) (25)
else, for t ∈ (0, ∞)\S ∞ , x ∈ B R0 c \N t , α ∈ N 3 0 with |α| ≤ 1. The constant in these inequalities is independent of U 0 , f, x and t. -(5/2+|α|)/2 -with a slightly larger exponent in the case of ∂ α x R (τ ) f |B c S0 × (0, ∞) if |α| = 1 -is considered in [START_REF] Deuring | L q -weak solutions to the time-dependent Oseen system: decay estimates[END_REF]Theorem 4.13,4.14]. These remarks explain why we stated in Section 1 that inequality (4) holds if U 0 (x) and f (x, t) decay sufficiently fast for |x| → ∞.

It should be noted that the sum R (τ ) f |B c S0 × (0, ∞) + I (τ ) (U 0 |B c S0 ) solves (1) in the whole space R 3 × (0, ∞). So the left-hand side in (5) may be interpreted as the perturbation generated by the presence of the rigid object, in the region far from that object.

Proof of Theorem 4.6. Suppose that q < 3. Then we have u = u U0 +u f . Corollary 4.5 yields the estimate u U0 |Z R0,∞ q,∞;∞ + u U0 |Z R0,∞ q,p0;∞ + (∇ x u U0 )|Z R0,∞ q,p0;∞ ≤ C U 0 q . Moreover, by Corollary 4.3 and 4.4,

u f |Z R0,∞ q,∞;∞ + u f |Z R0,∞ q,γ;∞ + (∇ x u f )|Z R0,∞ q,γ;∞ ≤ C f q,p1;∞ .
The assumptions in Theorem 4.6, the preceding inequalities and Theorem 3.5 show that the conditions on U 0 , f and u in Theorem 2.7 are fulfilled if this latter theorem is applied separately to the cases f = 0 and U 0 = 0. By making use of this theorem in this way, we may conclude that inequality [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF] is valid in the case q < 3 under consideration. Note that since γ ≥ p 1 (Corollary 4.4), the rate of decay of ∂ α x u f provided by Theorem 2.7 is -(3 + |α|)/2 + 1/(2 γ ).

In the case q ≥ 3, we obtain (25) by the same reasoning, but only Corollary 4.3 and 4.4 are relevant because of our definition u := u f if q ≥ 3.

L

  p (J,B) stands for the norm of L p (J, B). Let a, b ∈ R ∪ {∞} with a < b. Then we write L p (a, b, B) instead of L p (a, b), B . The expression L p loc [a, b), B denotes the space of all functions v : (a, b) → B such that v|(a, T ) ∈ L p (a, T, B) for any T ∈ (a, b). This space is to be distinguished from the space L p loc a, b, B , defined in the usual way. Let T

Theorem 2 . 3 .

 23 Let B be a Banach space and f : R → B a Bochner integrable function. Then R f (s + h) -f (s) B ds → 0 for h → 0, where B denotes the norm of B.

Proof. See [ 15 ,

 15 Theorem 3.8.3], [26, p. 134, Corollary 2], [15, Theorem 3.7.12].

  The asymptotics of the functions |∂ αx R (τ ) f |B c S0 ×(0, ∞) | and |∂ α x I (τ ) (U 0 |B c S0 )| (Lemma 2.6 and 2.5) are separate problems, only depending on the behaviour of f and U 0 , respectively. If both these latter functions have compact support and are L 1 , then the former two functions are bounded byC |x| ν(x) -(3+|α|)/2 for x ∈ B c R0 , t > 0; see [7, Lemma 4.1, 4.2] In this situation the functions |∂ α x I (τ ) (U 0 |B c S0 )| and |∂ α x R (τ ) f |B c S0 × (0, ∞)| decrease faster than the right-hand side of (24), except if q < 3/2, p 1 = 1 in Theorem 4.6. Then the two convergence rates coincide. Under the assumptions of Lemma 2.8, the term |∂ α x I (τ ) (U 0 |B c S0 )| even goes to zero as O |x| ν(x) -(4+|α|)/2 for |x| → ∞. For conditions on f and U 0 leading to the decay bound C |x| ν(x) -(2+|α|)/2 for |∂ α x I (τ ) (U 0 |B c S0 )| and |∂ α x R (τ ) f |B c S0 ×(0, ∞) |, we refer to [3, Theorem 3.1] and [2, Theorem 1.1], respectively. An asymptotic behaviour of f and U 0 entailing the bound C |x| ν(x)

  and a. e. t ∈ (0, ∞), for any choice of q; see the beginning of the proof of Corollary 4.3 as concerns the role of Theorem 2.3. Hence by Corollary 4.2,

Acknowledgments. We would like to thank Prof. Toshiaki Hishida (Nagoya University, Japan) for his comments and indications, which were of great help to us in writing this article.


The Ethiopian Linguistic Area

Ronny Meyer (Inalco/LLacan, Paris)

DEFINITION OF LINGUISTIC AREA

Languages may resemble each other due to four major reasons: (i) a common ancestor language, (ii) typological universals, (iii) borrowings due to language contact, and (iv) chance. The last reason, chance, can be neglected because it is always present to a limited extent. However, it is not always easy to clearly distinguish between the remaining three reasons. Common ancestry is usually established through cognates, most frequently lexical items but also paradigmatic sets of bound morphemes, which show that the concerned languages most probably developed out of a single proto-language. With regard to Ethiosemitic and Cushitic, see for instance [START_REF] Kogan | Common origin of Ethiopian Semitic: the lexical dimension[END_REF]2015), [START_REF] Hudson | Highland East Cushitic dictionary[END_REF] and [START_REF] Sasse | An etymological dictionary of Burji[END_REF] for lexical cognates, and [START_REF] Hetzron | Ethiopian Semitic: studies in classification[END_REF]1975a) for the use of morphology for genetic classification. Typological universals, in contrast, are based on the comparison of genetically divergent languages which share similar grammatical structures and possess constituents with a similar functional range (see e.g. [START_REF] Comrie | Language universals and linguistic typology: syntax and morphology[END_REF]). 1Long lasting and intense contact between speakers of different languages, genetically related or not, usually results in widespread bi-and multilingualism and may also trigger various layers of multidirectional borrowing processes of lexical items and grammatical structures so that eventually the involved languages closely resemble each other -although it often remains unclear from which particular language or language group the concerned feature originates. Even if borrowing processes are most commonly found the lexicon (here particularly nouns and discourse particles), all parts of a language can be affected by contact-induced linguistic change (see e.g. [START_REF] Weinreich | Languages in contact: findings and problems[END_REF][START_REF] Thomason | Language contact, creolization, and genetic linguistics[END_REF][START_REF] Thomason | Language contact: an introduction[END_REF]. If languages in a geographically defined area share one or more features, which cannot be explained through genetic ancestry or linguistic universals, then they might form a linguistic area or convergence zone. In an ideal situation, a linguistic area includes genetically unrelated languages that also have sister languages which are spoken outside the area. In this case, languages in the linguistic area can be compared with genetically related languages outside it to detect contact-induced changes. With regard to the Ethiopian Linguistic Area, for instance, a prominent feature is the SOV word order. It could be detected by comparing ancient with modern Ethiosemitic languages (i.e. Geez and Amharic) as well as with Semitic languages spoken in Asia, like Arabic, and Cushitic languages (e.g. Oromo), as in (1).

( 1 'The man saw that big horse.'

The neutral word in Proto-Semitic was VSO (Huehnergard 2019: 69), as shown in (1a) for Arabic and in (1b) for ancient Ethiosemitic Geez. However, The modern Ethiosemitic languages changed the word order to SOV, as in the Amharic example (1c), most probably through the influence of Cushitic languages, which are of the SOV type, as Oromo in (1d). Thus, the geographically related languages Amharic and Oromo are more similar to each other than the genetically closely related languages Amharic and Geez, as well as Amharic and Arabic. This scenario suggests that Proto-Ethiosemitic might still have been of the Proto-Semitic VSO type but later changed through language contact to SOV.

The word order change on the clausal level also affected the order of constituents on the phrasal level, although here the picture is still more divergent. Accordingly, the order MODIFIER-HEAD predominates in Ethiosemitic, Highland East Cushitic, and Omotic. Its reversal, MODIFIER-HEAD, also occurs in Omotic languages (e.g. Aari, Sezo, Anfillo), but particularly in Lowland East Cushitic. Uncommon mixed patterns are also found, such as ADJECTIVE-NOUN/NOUN-GENITIVE in Bayso (cf. Bisang 2006: 90-91) and Tigre/Tigrinya (which also have GENITIVE-NOUN) [START_REF] Tosco | A parsing view on inconsistent word order: articles in Tigre and its relatives[END_REF] 'Kebede started to wear (various pieces of) new clothes' (Leslau 1995: 334)

THE ETHIOPIAN LINGUISTIC AREA

The Ethiopian Linguistic Area is found at the Horn of Africa in present day Ethiopia and Eritrea -sometimes it is also considered to include Djibouti and Somalia (e.g. [START_REF] Bender | Northeast Africa: a case study in genetic and areal linguistics[END_REF]).2 It comprises languages from Nilo-Saharan and Afroasiatic phyla (i.e. Ethiosemitic, Cushitic and Omotic), and is an often citied as a typical example for an African linguistic area (e.g. [START_REF] Thomason | Language contact, creolization, and genetic linguistics[END_REF]. The Ethiopian Linguistic Area has arisen through intense contact between speakers of Semitic and Cushitic languages, which to a lesser extent also affected neighboring Omotic and Nilo-Saharan languages, i.e. it is mainly based on Cushitic-Ethiosemitic comparisons. As the Ethiosemitic languages have a number of particularities not found in Asian Semitic languages, it is commonly assumed that lasting and intense linguistic contacts with Cushitic languages triggered changes that made Ethiosemitic and Cushitic linguistically more similar to each other.

Early studies which particularly deal with Cushitic influence on Ethiosemitic include [START_REF] Praetorius | Hamitische Bestandteile im Äthiopischen[END_REF][START_REF] Praetorius | Kuschitische Bestandteile im Äthiopischen[END_REF], [START_REF] Moreno | L'azione del cuscitico sul sistema morfologico delle lingue semitiche dell'Etiopia[END_REF], and [START_REF] Leslau | The influence of Cushitic on the Semitic languages of Ethiopia: a problem of substratum[END_REF][START_REF] Leslau | The influence of Sidamo on the Ethiopic languages of Gurage[END_REF]1959). Subsequently, Greenberg (1959: 24) proposed the existence of a linguistic area in (former) Ethiopia and Somali, based on the following features: (i) complex consonant system with glottalized sounds, (ii) absence of tone, (iii) predominance of closed syllables, (iv) phrasal order of determiner-determined, and (v) shared lexical idiomatic expressions. However, most of these features are too general to define the Ethiopian linguistic area and partly not true (especially concerning tone). However, at the end of the 1960s, they reflect the then state-of-art, in which only some of the Ethiosemitic and Omotic languages were known to a certain extent, while the research on Omotic -the term itself was coined only years later (see Lamberti 1991) -and Ethiopian Nilo-Saharan languages was still in its infancy.

It was [START_REF] Ferguson | The Ethiopian language area[END_REF][START_REF] Ferguson | The Ethiopian Language Area[END_REF] who established the concept of the Ethiopian Linguistic Area through twenty-six phonological and grammatical features, which are shown in Table 1.

Table 1 Ferguson's (1976: 69, 75) Most Ethiosemitic languages and many Cushitic and Omotic languages exhibit several of these features, but they are less frequently found in Nilo-Saharan languages. For instance, unmarked SOV word order -see example (1) above -is common for Omotic languages and also found in a few Nilo-Saharan languages of the area, like Nara, Kunama and Nubian, while elsewhere in Nilo-Saharan SVO dominates (Güldemann 2010: 574;Bender 2003: 33). Furthermore, Nilo-Saharan languages in close proximity to Ethiosemitic and Cushitic languages may also exhibit the features (1) and ( 4).

Besides SOV word order, feature (23) in 'He did not take the canoe.'

The various clausal status markers seems to have grammaticalized from a construction consisting of a lexical verb followed by an auxiliary or a copula constructions in declarative clauses (see [START_REF] Tosco | The northern Highland East Cushitic verb in an areal perspective[END_REF][START_REF] Crass | Some remarks on the compound suffix conjugation in Highland East Cushitic languages[END_REF][START_REF] Dimmendaal | The grammaticalization of mood and modality in Omotic: a typological perspective[END_REF]Meyer 2014b: §3.5).4

Most of Ferguson's features, have been criticized in the literature (for further details see [START_REF] Zaborski | Ethiopian language subareas[END_REF][START_REF] Zaborski | Ethiopian language macroarea. Sprawozdania z Posiedzeń Komisji Naukowych Oddziału[END_REF]2010a;2010b;[START_REF] Tosco | Is there an "Ethiopian language area[END_REF]2008;[START_REF] Bender | Northeast Africa: a case study in genetic and areal linguistics[END_REF][START_REF] Bisang | Linguistic areas, language contact and typology: some implications from the case of Ethiopia as a linguistic area[END_REF][START_REF] Crass | Ethiopia[END_REF]. Some of them are so common that they can hardly be used to delimit the Ethiopian Linguistic Area, e.g. features (5-8), ( 17) and ( 22) in Table 1.

Other features have clearly a genetic origin, i.e. they are part of a common Afroasiatic ancestor language, e.g. features (3-4), (18-21) and probably ( 24) in Table 1. The distribution of feature (3) is very limited; it is usually found at the periphery, i.e. in North Ethiosemitic languages and in South Cushitic languages, but not in central areas of the Ethiopian Linguistic Area. Therefore, [START_REF] Crass | Ejectives and pharyngeal fricatives: two features of the Ethiopian language area[END_REF] argues that the reduction of the various Proto-Afroasiatic gutturals and pharyngeals to ʔ and h in the South Ethiosemitic and most Cushitic languages constitutes the areal feature. With regard to feature (4) -emphatic consonants (i.e. ejectives as in Ethiosemitic or pharyngealized consonants as in Arabic) -early research on Proto-Semitic had assumed that pharyngealized consonants represent the proto-form and the ejectives are innovations in Ethiosemitic due to language contact with Cushitic. However, the ejective pronunciation is also found in Semitic languages outside Ethiopia and is now considered to represent the original pronunciation in Proto-Semitic (see Kogan 2011: 59-61). Nevertheless, [START_REF] Crass | Ejectives and pharyngeal fricatives: two features of the Ethiopian language area[END_REF] argues that ejectives constitute an areal feature because they were lost diachronically in certain branches of Cushitic, particularly Agaw, but reintroduced through contact with Ethiosemitic (see e.g. Fallon 2015: 74-75 for Bilin).

Another group of features displays typological universal tendencies and is also not well suited to define a linguistic area, for instance the features (10-12) are typical for languages with SOV as common word order. Finally, some of the proposed areal features in Table 1, such as SOV word order, occurrence of postpositions, and use of quoting constructions, have a wider distribution in northern Africa and are probably part of a larger Chad-Ethiopia Linguistic Area (Güldemann 2010: 574-575;Heine 1975: 41-42). [START_REF] Tosco | Is there an "Ethiopian language area[END_REF] and recently Güldemann (2018: 464-470) express serious doubt regarding the existence of an Ethiopian Linguistic Area. [START_REF] Tosco | Is there an "Ethiopian language area[END_REF] main objection is that most languages of the area have the same genetic ancestor, Proto-Afroasiatic, and thus it is difficult to determine whether a feature is inherited or borrowed, while Güldemann (2018: 464-470) argues that [START_REF] Ferguson | The Ethiopian Language Area[END_REF] features actually do not delimit the Ethiopian Linguistic Area but have a wider distribution or represent universal tendencies in linguistic typology.

ADDITIONAL FEATURES OF THE ETHIOPIAN LINGUISTIC AREA

As the actual extent of a large Ethiopian Linguistic Area are hard to define, several works focus on research into smaller sub-areas or contact/convergence zones within it (see [START_REF] Zaborski | Ethiopian language subareas[END_REF]2010b), such as Northern Eritrea [START_REF] Tosco | A parsing view on inconsistent word order: articles in Tigre and its relatives[END_REF]2008), Highland East Cushitic-Gurage [START_REF] Crass | Ethiopia[END_REF]; and the contributions in Crass & Meyer 2007), Highland East Cushitic-North Omotic [START_REF] Treis | Switch-reference and Omotic-Cushitic language contact in Southwest Ethiopia[END_REF], Highland East Cushitic-Gurage-Ometo [START_REF] Tosco | The northern Highland East Cushitic verb in an areal perspective[END_REF], Surmic-Omotic-Cushitic [START_REF] Dimmendaal | A syntactic typology of Surmic from an areal and historical-comparative point of view[END_REF], and the Southwest Ethiopian Language Area [START_REF] Sasse | A southwest Ethiopian language area and its cultural background[END_REF][START_REF] Ongaye | The spread of punctual derivation in Dullay and Oromoid languages[END_REF]. The features suggested for these sub-areas also encompass complex constructions, which involve shared grammaticalization processes in different languages. For the Highland East Cushitic-Gurage sub-area, Crass & Meyer (2008: 244), for instance, describe the grammaticalization of an experiential perfect based on the verb 'know', as in (6), which [START_REF] Rapold | Exploring the periphery of the Central Ethiopian linguistic area: data from Yemsa and Benchnon[END_REF] also confirmed for Omotic.

(6) a. OROMO (CUSHITIC) ameerikaa deem-tan-i beek-tuu? America go-2PL.PFV-CNV know-2PL.IPFV.Q 'Have you ever been to America?'

b. AMHARIC (ETHIOSEMITIC) amerika hedä-h t-awk'-allä-h?

America go\CNV-2SM 2SM-know\IPFV-AUX.NPST-2SM

'Have you ever been to America?'

Other features contain additional modifications, not found in their original sense of the features as suggested by [START_REF] Ferguson | The Ethiopian Language Area[END_REF]. For instance, quotative constructions based on a direct quotation as complement to the verb 'say' (7a) is an often cited feature, although these constructions are not limited to the Ethiopian Linguistic Area [START_REF] Appleyard | The verb 'to say' as a "verb recycling" device in Ethiopian languages[END_REF][START_REF] Cohen | The grammaticalization of 'say' and 'do': an areal phenomenon in East Africa[END_REF][START_REF] Meyer | The quotative verb in Ethiosemitic languages and Oromo[END_REF]).

( and -m(a) (also augmented -m tannä > -nta) elsewhere (Meyer 2016: §4.4). Usually languages of the area have only a general converb, while Omotic, Highland East Cushitic, and certain Western Gurage languages also have specialized converbs [START_REF] Azeb | Converbs in an African perspective[END_REF][START_REF] Banti | Remarks on the typology of converbs and their functional equivalents in East Cushitic[END_REF]Hetzron 1975b;[START_REF] Völlmin | The so-called converbs in Gumer (Gurage)[END_REF]. According to Treis (2012: §5), converbs in Omotic and Highland East Cushitic distinguish between same-and different-subject, which is rare elsewhere in Africa. She therefore concludes that switch-reference, uniformly marked by *-n/*-m, is peculiar to certain Omotic languages, from where it spread to Highland East Cushitic.

Another probable feature is the use of the similative marker 'like' as a marker of purpose and complement clauses, which is found in Cushitic, Omotic and Ethiosemitic languages in central Ethiopia, but less so on the peripheral regions of the Ethiopian Linguistic Area [START_REF] Treis | Similative morphemes as purpose clause markers in Ethiopia and beyond[END_REF]. According to Treis (2017: 91), the similative-purpose multifunctionality is rare cross-linguistically.

Another type of features which might determine the Ethiopian Linguistic Area stem are lexical items which acquired multiple senses. [START_REF] Hayward | À propos patterns of lexicalization in the Ethiopian language area[END_REF]2000), for instance, identifies similar conceptualization patterns in the lexicon of Cushitic, Ethiosemitic and Omotic languages, e.g. 'go out'='go up', causative of 'enter'='marry' (see also Appleyard 2001: 7). Further research in such conceptualization patterns will certainly enhance the characterization of the Ethiopian Linguistic Area.

Other features, which seem to be specific to the Ethiopian Linguistic Area but still need further research, include (i) copula splits in main and subordinated clauses, i.e.

the occurrence of different copulas in according to the clause type, e.g. particle copulas with no or restricted agreement in main clauses vs. fully inflected verbal copulas in subordinate clauses, (ii) widespread use of associative plurals, i.e. affixes attached to proper names and pronouns expressing 'X and peoples associated to X', like Amharic ɨnnä-käbbädä 'Kebede and his associates' and ɨnnässu 'they' (from ɨnnä-ɨssu AP-3SM), (iii) fusion of case marking and definiteness.

Thus, despite the relatively extensive research over several decades, the Ethiopian Linguistic Area is still not sufficiently defined and needs further investigation.

  features of the Ethiopian Linguistic Area Existence of basic nouns with a plural/collective reading from which a singulative forms can be derived.

	EIGHT PHONOLOGICAL FEATURES
	1. The voiceless fricative f replaces the voiceless plosive p as counterpart to
	the voiced plosive b;
	2. Occurrence of ɗ as single implosive consonant;
	3. Occurrence of the pharyngeal fricatives ħ and ʕ as phonemes;
	4. A series of ejective consonants;
	5. Palatals occurs as phonemes and as result of palatalization, a phonological
	process most frequently occurring with alveolars;
	6. Gemination as a phonemic feature in the lexicon and the grammar;
	17. Occurrence of possessive suffixes;
	18. Masculine/feminine gender distinction on personal pronouns and verb
	indexes of the second and third person singular;
	19. Existence of a prefix conjugation in which the 2sm and 3sf subject is
	identically marked by t-;
	20. Use of non-linear (or non-concatinative) morphology for word formation;
	21. Occurrence of broken plurals;
	22. Reduplication marks a higher intensity;
	23. Occurrence of formal morphological distinctions between imperfective
	verbs in main and subordinate clauses;

7. Occurrence of one or two central vowels ɨ, ä; 8. Absence of consonant clusters with more than two consonants EIGHTEEN GRAMMATICAL FEATURES 9. SOV as unmarked word order; 10. Subordinate clauses precede main clauses; 11. High frequency of converbs; 12. High frequency of postposition; 13. Quoting clauses marked with the verb 'say'; 14. Compound verbs (i.e. complex predicates) with the verb 'say'; 15. Lexically/structurally distinct affirmative and negative copulas; 16. Use of singular (i.e. unmarked) nouns with numerals and quantifiers; 24. Use of singular feminine forms to mark plural; 25. Suppletive lexical items for expressing the imperative of the verb 'come'; 26.

  Table 1 -the morphological distinction between imperfective verbs in main and subordinate clauses -is a common phenomenon in Cushitic, Ethiosemitic and Omotic. That means many languages grammaticalized clause-type markers distinguishing between subordinate and main clause verbs, e.g. the contrast between main-and relative clause verbs in (3). The lack of any additional marker on the verb in the Libido examples (3b) indicates that

	(5) ZAY (ETHIOSEMITIC)
	a. ʃäfät näk'äl-ä-n-u		b. ʃäfät näk'äl-ä-n
	canoe take\PFV-3SM-FOC-DCL	canoe take\PFV-3SM-FOC
	'He took the canoe.'		'Did he take the canoe?'
	c. ʃäfät ʔal-nɨqäl-o.
	canoe NEG-take\JUSS-3SM:DCL
	(3) LIBIDO (CUSHITIC)
	a. ʔit-t-aa-tt-oo	vs. b. hin-t-aa
	eat-2SG-IPFV-2SG-IPFV.DCL		dig-2SG-IPFV[REL]
	'you eat'		'you who digs' (Crass 2014: 184, 186)
	AMHARIC (ETHIOSEMITIC)
	c. tɨbälalläh	vs. d. jämmɨ-ttɨ-bäla
	tɨ-bäla-allä-h
	2SM-eat\IPFV-AUX.NPST-2SM	REL-2SM-eat\IPFV
	'you eat'		'you who eats'
	Feature (23) is probably not limited to imperfective verbs and the distinction
	between main and subordinated verbs, but also applies the marking of illocutionary
	force, i.e. the distinction between declarative and interrogative clauses (4) and (5),
	and the distinction between affirmative and negative clauses (5): 3
	(4) HAMAR (OMOTIC)
	a. ʊʧ-ʌ=tɛ=kɪ=d-ʌ=d-ɛ		b. taki ʊʧ-ʌ=tɛ=d-ʌ	ʌynʌ
	drink-PFV=LOC=3=be-PFV=be-IPFV	now drink-PFV=LOC=be-PFV who
	'He is drinking.'			'Who is drinking now?'
				(Cupi et al. 2012: 188)

it is a dependent relative verb, while the main clause imperfective verb in (3a) contains an additional suffix. In other languages, like Amharic (3c-d), main and subordinate imperfective verbs contain additional markers, but their form differs.

  Converbs -i.e. dependent verbs with restricted person, TAM, or polarity marking used for clause-chaining or adverbial modification -are another often-cited feature of the Ethiopian Linguistic Area. Within Cushitic, Beja, Agaw and Saho-Afar have a separate converb paradigm, while Oromo changes a regular perfective verb to a pseudo-converb by lengthening its final vowel or adding epenthetic i to a final consonant, as in (6a). Ethiosemitic innovated converbs due to contact with Cushitic.

	b. bär-u	«alɨ-kkäffät»	ʔal-ä-ɲɲ.
	door-DEF.M	NEG.1SG-be_open\IPFV	say\PFV-SBJ.3SM-OBJ.1SG
	'I could not open the door (lit. The door said to me, "I won't be opened.")'
	c. lä-ne blo	näw	jä-mätt'a-w
	DAT-1SG SAY\CNV.3SM COP.3SM REL-COME\PFV.3SM-DEF
	'He came for the sake of me.'
	Quoting clauses further grammaticalized into a benefactive focus construction (7c)
	(Crass & Meyer 2008: 242), and other complex predicates, like the negative
	circumstantial in (7b), in which 'say' functions as light verb, while the coverb is
	typically an ideophone.
	7) AMHARIC
	a. ʔasfa «gɨbɨr al-käfl-ɨmm»	blo	näggär-at.
	Asfa tax	NEG.1SG-pay\IPFV-NEG say\CNV.3SM tell\PFV-SBJ.3SM:OBJ.3SF
	'Asfa told her that he won't pay taxes.'

Geez, Tigrinya, Amharic and Argobba grammaticalized a verbal noun plus possessive suffix as converb. The remaining South Ethiosemitic languages have pseudo-converbs, marked by a suffix on inflected verbs: -ä/-ani in Wolane/Silt'e,

An online database and description of typological linguistic features in a vast selection of the world's languages isDryer & Haspelmath (2013).

Meyer & Wolff (2019: 297-300) -in reference to[START_REF] Bender | Northeast Africa: a case study in genetic and areal linguistics[END_REF] -call this area Northeast African Linguistic Area, as it currently encompasses Ethiopia and the surrounding countries.

For further information on focus and illocutionary force in Zay, see[START_REF] Meyer | To be or not to be' -Is there a copula in Zay?[END_REF] 2006; 2014a).

Note that cleft sentences are generally a very productive means for marking focus in the Ethiopian Linguistic Area[START_REF] Appleyard | The relative verb in focus constructions: an Ethiopian areal feature[END_REF]).