GLOBAL SOLUTIONS to the NON-LOCAL NAVIER-STOKES EQUATIONS Joelma Azevedo Juan Carlos Pozo Arlúcio Viana (Communicated by Tomasz

DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2021146 DYNAMICAL SYSTEMS SERIES B

GLOBAL SOLUTIONS TO THE NON-LOCAL NAVIER-STOKES EQUATIONS

Joelma Azevedo Departamento de Matemática Universidade de Pernambuco Nazaréda Mata, Brazil Juan Carlos Pozo Departamento de Matemáticas,Facultad de Ciencias Universidad de Chile Santiago, Chile Arlucio´ Viana∗ Departamento de Matemática Universidade Federal de Sergipe SãoCristóvão,Brazil

(Communicated by Tomasz Dlotko)

Abstract. This paper is devoted to the study of the global well-posedness for a non-local-in-time Navier-Stokes equation. Our results recover in particular other existing well-posedness results for the Navier-Stokes equations and their time-fractional version. We show the appropriate manner to apply Kato’s strategy and this context, with initial conditions in the divergence-free σ d Lebesgue space Ld (R ). Temporal decay at 0 and ∞ are obtained for the solution and its gradient.

1. Introduction. Modeling fluid dynamics are of high scientific interest, and taking memory effects into account is a way to describe non-Newtonian fluids and viscoelastic materials. See e.g. [2,3, 14] and references therein. Such memory effects are mathematically represented through non-local-in-time operators. A more special case, which has drawn attention in the last years, is the fractional-in-time Navier-Stokes equation α d ∂t u − ∆u + (u · ∇)u + ∇p = f, t > 0, x ∈ Ω ⊂ R , d ∇ · u = 0, t > 0, x ∈ Ω ⊂ R , (1) d u(0, x) = u0(x), x ∈ Ω ⊂ R , α where ∂t u denotes the fractional derivative of u in the Caputo’s sense with order α ∈ (0, 1). Indeed, for numerical analysis of (1), see e.g. [9, 15, 17, 22, 25, 26]; for analysis of the existence of solutions of (1), including versions with delay and

2020 Mathematics Subject Classification. Primary: 35R09, 35R11, 35Q35, 35Q30; Secondary: 33E12, 76D03, 35B30, 35B40. Key words and phrases. Nonlocal Navier-Stokes, PDEs in connection with fluid mechanics, well-posedness, long-time behavior, uniqueness. ∗ Corresponding author: ArlúcioViana.

1 2 J. AZEVEDO, J. C. POZO AND A. VIANA stochastic noise, see e.g. [8, 18, 21, 24, 22, 23, 31, 27, 28, 29, 30,2, 32]. If the product (k ∗ v) denotes the convolution on the positive halfline R+ := [0, ∞) with R t respect to time variable, that is (k ∗ v)(t) = 0 k(t − s)v(s)ds, with t ≥ 0, then we α have ∂t u = g1−α ∗ ut, for an absolutely continuous function u , where gβ is the standard notation for the function tβ−1 g (t) = , t > 0, β > 0. β Γ(β) Toward the possibility of considering more general non-local-in-time effects, we will replace gα by k, and we assume as a general hypothesis that k is a kernel of type (PC), by which we mean that the following condition is satisfied.

(PC) k ∈ L1,loc(R+) is nonnegative and nonincreasing, and there exists a kernel ` ∈ L1,loc(R+) such that k ∗ ` = 1 on (0, ∞). We also write (k, `) ∈ PC. We point out that the kernels of type (PC) are divisors of the unit with respect to the temporal convolution. These kernels are also called Sonine kernels and they have been successfully used to study integral equations of first kind in the spaces of Höldercontinuous, Lebesgue and Sobolev functions, see [5]. Further, the condition (PC) covers several interesting integro-differential operators with respect to time that appear in the context of subdiffusion processes. For instance, a very important example of (k, `) ∈ (PC) is given by the pair (g1−α, gα) with α ∈ (0, 1). In this case the term ∂t(k ∗ v) becomes the Riemman-Liouville α fractional derivative Dt v of order α ∈ (0, 1). The Riemman-Liouville fractional derivative is closely related with a class of Montroll-Weiss continuous time random walk models and it has become one of the standard physical approaches to model anomalous diffusion processes. The details of the derivation of these equations from principles of physics and further applications of such models can be found in [16]. Another interesting and important example of kernels (k, `) ∈ (PC) is given by Z 1 k(t) = gα(t)ν(α)dα, t > 0, (2) 0 where ν is a continuous non-negative function that does not vanish in a set of positive measure. Under appropriate conditions on ν, the existence of a function 1 + ` ∈ Lloc(R ) such that k ∗ ` = 1, has been established in [13, Proposition 3.1]. In this case the operator ∂t(k ∗ ·) is a so-called operator of distributed order, and it have been applied in ultraslow diffusion equations. These equations have been successfully used in physical literature for modeling diffusion with a logarithmic growth of the mean square displacement, see [13, Theorem 4.3]. Therefore, in this paper we consider the following non-local-in-time Navier-Stoke- type equation d ∂t(k ∗ (u − u0)) − ∆u + (u · ∇)u + ∇p = f, t > 0, x ∈ R , (3) d ∇ · u = 0, t > 0, x ∈ R , (4) d u(0, x) = u0(x), x ∈ R , (5) where u(t, x) = (u1(t, x), u2(t, x), . . . , ud(t, x)) represents the velocity field and p = p(t, x) is the associated pressure of the fluid. The function u0(x) = u(0, x) is the initial velocity and f = (f1(t, x), f2(t, x), . . . , fd(t, x)) represents an external force. Clearly, if kˆ = 1, (3)–(5) reduces to the classical Navier-Stokes equation, while k = g1−α turns (3)–(5) into (1). NONLOCAL NAVIER-STOKES EQUATIONS 3

Throughout this paper, we assume that ` is a completely monotonic function according to the following deﬁnition.

Deﬁnition 1.1. A C∞ function g : (0, ∞) → R is called completely monotonic if n (n) ∞ (−1) g (λ) ≥ 0 for all n ∈ N0 and λ > 0. Further, a C function g : (0, ∞) → R is called a Bernstein function if g(λ) ≥ 0 for λ > 0 and g0 is a completely monotonic function. The class of completely monotonic functions will be denoted by CM. Returning to the problem (3)-(5), we see that it can be written in an abstract form as

∂t(k ∗ (u − u0)) + Apu = F (u, u) + P f, t > 0, (6) d σ d where Apu := P (−∆)u, P : Lp(R ) → Lp (R ) is well-known as Helmholtz-Leray’s projection, and the nonlinear term F (u, v) := −P (u · ∇)v. Equation (6) can be written as a Volterra equation of the form

u + (` ∗ Apu)(t) = u0 + (` ∗ [F (u, u) + P f])(t), t > 0, (7) by condition (k, `) ∈ PC. Next, we discuss the linear problem d ∂t(k ∗ (u − u0))(t) − ∆u(t) + ∇p(t) = f, t > 0, x ∈ R , d ∇ · u = 0, t > 0, x ∈ R , d u(0, x) = u0(x), x ∈ R , or equivalently the linear Volterra equation d u + ` ∗ (−∆u + ∇p) = u0 + ` ∗ f, t > 0, x ∈ R , d ∇ · u = 0, t > 0, x ∈ R , (8) by condition (PC). Applying the Helmholtz-Leray’s projection on (8), we have

u(t) + (` ∗ Apu)(t) = u0 + (` ∗ P f)(t) t > 0. (9) To obtain the fundamental solution of (9), we follow the same procedure by Kato [11]. Indeed, Duhamel’s principle yields its unique solution given by Z Z t Z u(t, x) := Z(t, x − y)u0(y)dy + Y (t − s, x − y)P f(s, y)dyds d d R 0 R where Z ∞ Z(t, x) := − w(t; dτ)H(τ, x) and Y (t, x) := ∂t(` ∗ Z(·, x))(t). (10) 0

Here, −w(t; dτ) is a positive ﬁnite measure R+ such that Z ∞ − w(t; dτ) = 1, a.e. t > 0. 0 and 2 1 − |x| H(t, x) := e 4t I (4πt)d/2 is the heat kernel matrix that allows us to deﬁne the heat operator Z (e−tAp φ)(x) := H(t, x − y)φ(y)dy. d R 4 J. AZEVEDO, J. C. POZO AND A. VIANA

Another way to see (10) is to recall from [7, Th. 2.2] that condition (PC) is equiv- alent to ` be completely positive so that the subordination theory in [20, Ch. 4] applies to the convolution operators Z(t, ·)? and H(t, ·)?. We note that, if f is divergence-free, then P f = f. Recall from [11] we have −tAp Ap = −∆P is essentially the −∆ and e is essentially the heat semigroup. Having in mind these results, the solution u of (7) can be written by the formula of variation of constants for Volterra equations. In this direction, we have the following deﬁnition.

Definition 1.2. Consider 1 ≤ p < ∞. A continuous function u ∈ Cb([0, ∞); σ d Lp (R )) satisfying Z Z t Z u(t, x) = Z(t, x − y)u0(y)dy + Y (t − s, x − y)(F (u, u) + P f(s, y))dyds, d d R 0 R (11) σ d for t ∈ [0, ∞) is called a global mild solution to problem (7) in Lp (R ). σ d Here, the notation Lp (R ) describes the subspace of the divergence-free vector d fields in Lp(R ): σ d σ d d d d Lp (R ) = Lp (R ; R ) := {v ∈ Lp(R ): ∇ · v = 0 in R }. Besides, let J ⊂ R and X be a Banach space. We denote the space of the continuous and bounded functions from J to X by Cb(J; X), equipped with its usual norm. Moreover, k · kr is the norm of the vector-valued Lebesgue space d d d d Lr(R ) = Lr(R ; R ), and by Lr(R ), only when we need to stress that it is vector- valued. The symbol ’?’ will denote the convolution on Rd. We note that doing suitable changes on condition PC and on the definition of Z, the kernel k may be considered as a diagonal matrix with entries ki. Thus, the non-local Navier-Stokes equations (3)–(5) can describe mixed dynamics. That is new from the physical point of view. Although however, it seems not to be different or relevant from the mathematical point of view. Our existence result (see Theorem 3.3 below) generalizes Theorem 12 in [8] and complements it with stability results and nice convergences as t → 0+. The uniqueness result (see Theorem 3.4 below) seems to be completely new even for the fractional case. The strategy of its proof cannot follow the classical proof by Kato, but all the results are compatible with his results in [11] for the Navier-Stokes equations. The paper is organized as follows. We study the behavior of the convolution operators Z(t, ·)? and Y (t, ·)? in Sec.2, that is, we use ideas from Pozo and Vergara [19] for the non-local heat equation to obtain key estimates and time-continuity for σ the convolution operators in Lp . The global well-posedness of (3)–(5) is proved in Sec.3.

2. Key estimates for Z(t, ·) and Y (t, ·). In this section, we show key estimates that will be important in the proof of the existence theorem. In spite of this use, the estimates and continuity for the convolution operators Z(t, ·)? and Y (t, ·)? have self-interest. Next, we recall that there exist two positive finite measures −w(t, dτ) and η(t, dτ) on R+ such that we can represent the Fourier transforms of Z(t, ·) and Y (t, ·) by s and r, respectively, as follows: Z ∞ s(t; µ) = − e−µτ w(t; dτ), a.e. t > 0, and Re µ ≥ 0, (12) 0 NONLOCAL NAVIER-STOKES EQUATIONS 5 and Z ∞ − w(t; dτ) = 1, a.e. t > 0. 0 Further, Z ∞ r(t; µ) = e−µτ η(t; dτ), a.e. t > 0, and Re µ > 0, (13) 0 and Z ∞ η(t; dτ) = `(t), a.e. t > 0. 0 See [20, Chapter 4] and [6]. We have the following. Lemma 2.1. Let (k, `) ∈ PC and δ ∈ (0, 1). We assume that k is a completely monotone function on (0, ∞). Then there exist a Bernstein function e : (0, ∞) → ∞ R+ and a nonnegative function φδ ∈ C (0, ∞) such that 1 φˆ (λ) = `ˆ(λ)−δ, Re λ > 0. δ λ d Moreover, e(0) = 0 and φδ(t) = dt (g1−δ ∗ e)(t) = (g1−δ ∗ e˙)(t), t > 0. Proof. We consider the Bernstein functions a(t) = (1 ∗ k)(t), b(t) = 1, and c(t) = (1 ∗ g1−δ)(t). Then aˆ(λ) 1 eˆ(λ) =a ˆ(λ)dcˆ ( ) = λ−δ `ˆ(λ)−δ ˆb(λ) λ is the Laplace transform of a Bernstein function e, by [20, Lemma 4.3]. Hence the statement follows by the inversion of the Laplace transform. Lemma 2.2. Let (k, `) ∈ PC and δ ∈ (0, 1). We assume that k is a completely monotone function on (0, ∞). Let φδ as in Lemma 2.1. Then the solution ψδ of the Volterra equation of the first kind φδ(t) = (ψδ ∗ k)(t) and φδ can be represented as Z ∞ 1 −δ φδ(t) = − τ w(t; dτ), t > 0, (14) Γ(1 − δ) 0 and Z ∞ 1 −δ ψδ(t) = τ η(t; dτ), t > 0. (15) Γ(1 − δ) 0 Proof. We first note that the Laplace transform of the measures −w(t, dτ) and η(t, dτ) are given by ˆ ˆ − ωˆ(λ; dτ) = kˆ(λ)e−τλk(λ)dτ andη ˆ(λ; dτ) = e−τλk(λ)dτ, (16) by (12) and (13). On the other hand, we have ∞ ∞ Z Z ˆ 1 − τ −δωˆ(λ; dτ) = kˆ(λ) τ −δe−τλk(λ)dτ = ˆl(λ)−δΓ(1 − δ) < ∞, 0 0 λ by condition PC and provided that δ ∈ (0, 1). Hence, by Lemma 2.1 we have Z ∞ ˆ 1 ˆ −δ 1 −δ φδ(λ) = l(λ) = − τ ωˆ(λ; dτ). (17) λ Γ(1 − δ) 0 ˆ ˆ ˆ Now, since φδ(λ) = k(λ)ψδ(λ) we have that Z ∞ Z ∞ 1 1 1 −δ 1 −δ ψˆδ(λ) = φˆδ(λ) = τ (−ωˆ(λ; dτ)) = τ ηˆ(λ; dτ) kˆ(λ) kˆ(λ) Γ(1 − δ) 0 Γ(1 − δ) 0 6 J. AZEVEDO, J. C. POZO AND A. VIANA

by (17) and (16). The proof of the lemma follows then by the inversion of the Laplace transform.

Now, recall that e−tAp is essentially the heat semigroup as we told in Introduction so that Z(t, ·) in (10) is essentially the fundamental solution of the non-local heat equation. The convolution operator Z(t, ·)? has been estimated in [12, Theorem 5.1] and [19, Theorem 5.7]. Therefore, we can essentially recast [19, Theorems 5.7, 5.8, 5.9 and 5.10] to obtain the next result. Indeed, the L1 assumption in those theorems is only needed to guarantee the existence of the Fourier transform, but we can drop it oﬀ by working their proofs in the Schwartz space, and then, we can recover them by density. We will freely use the estimates of Lemma 2.3 in Sec.3.

d 1 1 Lemma 2.3. Let (k, `) ∈ PC, d ∈ N, 1 < r < q < ∞ and δ = 2 r − q . If σ d u0, g(t, ·) ∈ Lr (R ), for t > 0, then there is a constant C > 0 such that −δ kZ(t, ·) ? u k d ≤ C (1 ∗ `)(t) ku k d (18) 0 Lq (R ) 0 Lr (R ) and Z t Z t −δ Y (t − s, ·) ? g(s, ·)(x)ds ≤ C `(s)[(1 ∗ `)(s)] kg(t − s, ·)k d ds, Lr (R ) d 0 Lq (R ) 0 (19) for t > 0, provided that 0 < δ < 1. Moreover,

1 − 2 −δ k∇Z(t, ·) ? u k d ≤ C (1 ∗ `)(t) ku k d . (20) 0 Lq (R ) 0 Lr (R ) and

R t R t − 1 −δ ∇ Y (t − s, ·) ? g(s, ·)(x)ds ≤ C `(s)[(1 ∗ `)(s)] 2 kg(t − s, ·)kL ( d)ds, (21) 0 d 0 r R Lq (R ) 1 for t > 0, provided that 0 < δ < 2 . Proof. Recall (10). If we use the notation

Z ∞ 2 1 − |x| Z˜(t, x) := − w(t; dτ) e 4t , (22) d/2 0 (4πt) ˜ d d 1 d T then Z(t, x) = Z(t, x)I. Therefore, given u0 : R → R , say u0 = (u0, ··· , u0) , we have  ˜ 1  Z(t, ·) ? u0  .  Z(t, ·) ? u0 =  .  . (23) ˜ d Z(t, ·) ? u0 d d d In this proof, we denote Lp(R ) to be the vectorial Lebesgue space, so u0 : R → R d i d belongs to Lp(R ) if and only if u0 ∈ Lp(R ), for all i = 1, ··· , d, and we use the norm i kϕk = max kϕ kL . (24) Lp i=1,··· ,d p From Theorem 5.7 in [19], we have

i −δ i kZ˜(t, ·) ? u k d ≤ C (1 ∗ `)(t) ku k d , (25) 0 Lq (R ) i 0 Lr (R ) for t > 0, provided that 0 < δ < 1, with d ∈ N, d > 2, 1 < r < q < ∞, δ = d 1 1 i d d 2 r − q and u0 ∈ Lr(R ). Indeed, 0 < δ < 1 is the same as 1 < p < d−2 , d > 2 (or 1 < p < ∞, d = 1, 2) which is the assumption 1 < p < σ1(d, ρ) in [19, Th. 5.7], NONLOCAL NAVIER-STOKES EQUATIONS 7

d for ρ = 2. Now, we take u0 ∈ Lr(R ) and we keep in mind (23), (24) and (25) to obtain −δ kZ(t, ·) ? u k d ≤ C (1 ∗ `)(t) ku k d , (26) 0 Lq (R ) 0 Lr (R ) with C = maxi=1,··· ,d Ci. We have proved (18). A completely similar reasoning, using [19, Th. 5.9] instead of [19, Th. 5.7] leads to (20). Now, we look at the estimates for Y . The idea is inside the proofs of [19, Theo- rems 5.7 and 5.8] and the ﬁrst part of this proof. Let Y˜ = ∂t(` ∗ Z˜), and recall that r(t, |ξ|2) = FY (t, ξ). Then F(Y (t, ·) ? ϕ)(ξ) =[|ξ|2δr(t, |ξ|2)]|ξ|−2δϕˆ(ξ) =m ˜ (ξ)`(t)[(1 ∗ `)(t)]−δ|ξ|−2δϕˆ(ξ), (27) wherem ˜ (ξ) = [`(t)]−1[(1 ∗ `)(t)]δ|ξ|2δr(t, |ξ|2). The functionm ˜ (ξ) satisﬁes the Mikhlin condition by [19, Lemma 5.6], for any δ ∈ (0, 1). Thus, [10, Th. 5.2.7] p gives thatm ˜ ∈ Mq, for any 1 ≤ q ≤ ∞, where Mq denotes the space of the L Fourier multipliers. Then −1 ˆ km˜ kMq = sup kF (fm˜ )kLq ≤ C, (28) kfkLq ≤1 for some C depending on the dimension d and q. On the other hand, assume that d ∈ N, 1 < r < q < ∞ are such that d 1 1 δ := − < 1. 2 r q −δ Then, by [10, Th. 6.1.3], (−∆) ϕ ∈ Lq, for ϕ ∈ Lr and −δ k(−∆) ϕkLq ≤ CkϕkLr , 1 1 provided that 0 < 2δ < d (or r − q < 1) and 1 1 2δ − = . r q d Therefore, we can choose f = (−∆)−δϕ in (28) and use (27) to obtain

−δ −1 \−δ kY (t, ·) ? ϕkLq =`(t)[(1 ∗ `)(t)] kF ((−∆) ϕ m˜ )kLq −δ −δ ≤C `(t)[(1 ∗ `)(t)] k(−∆) ϕkLq −δ ≤C `(t)[(1 ∗ `)(t)] kϕkLr . The constant C > 0 above is not necessarily the same as in (28) but does not depend on t > 0. We have that −δ kY (t, ·) ? ϕkLq ≤ C `(t)[(1 ∗ `)(t)] kϕkLr , (29) for ϕ ∈ Lr, with a possibly diﬀerent C > 0. From this follows (19), by taking ϕ = g(t, ·). As before, a completely similar reasoning, using [19, Th. 5.10] instead of [19, Th. 5.8] leads to (20).

Corollary 1. Let (k, `) ∈ PC, d ∈ N, 2 ≤ d < q < ∞. Then 1 − d lim sup[(1 ∗ `)(t)] 2 2q kZ(t, ·) ? vk d = 0, (30) Lq (R ) t→0+ σ d uniformly for v in bounded sets of Ld (R ). 8 J. AZEVEDO, J. C. POZO AND A. VIANA

Proof. Assume ﬁrst that v is a divergence-free test function. In particular, v ∈ σ d Lq (R ) and (18) gives

1 − d 1 − d [(1 ∗ `)(t)] 2 2q kZ(t, ·) ? vk d ≤ [(1 ∗ `)(t)] 2 2q kvk d → 0, (31) Lq (R ) Lq (R ) + σ d as t → 0 , since d < q. Now, for v ∈ Ld (R ), there is a sequence (vn) of divergence- free test functions converging to v. We have then

1 − d 1 − d [(1 ∗ `)(t)] 2 2q kZ(t, ·) ? vk d ≤ [(1 ∗ `)(t)] 2 2q kZ(t, ·) ? v − Z(t, ·) ? v k d Lq (R ) n Lq (R ) 1 − d + [(1 ∗ `)(t)] 2 2q kZ(t, ·) ? v k d . (32) n Lq (R ) The second portion of the right-hand side of (32) goes to 0 as t → 0+, by (31). On the other hand, we can use (18) to obtain

1 − d [(1 ∗ `)(t)] 2 2q kZ(t, ·) ? v − Z(t, ·) ? v k d ≤ Ckv − v k d → 0, n Lq (R ) n Ld(R ) as n → ∞. These observations lead to (30).

Lemma 2.4. Let a, b ∈ (0, 1), and ` be completely monotonic function. Then, there exists C > 0 such that Z t `(t − s)[(1 ∗ `)(t − s)]a−1[(1 ∗ `)(s)]b−1ds < C[(1 ∗ `)(t)]a+b−1. (33) 0 Proof. Let Z t/2 I = `(t − s)[(1 ∗ `)(t − s)]a−1[(1 ∗ `)(s)]b−1ds 0 Z t + `(t − s)[(1 ∗ `)(t − s)]a−1[(1 ∗ `)(s)]b−1ds t/2

:=I1 + I2.

a−1 a−1 Recall that ` is decreasing and maxs∈[0,t/2][(1 ∗ `)(t − s)] = [(1 ∗ `)(t/2)] . Then, Z t/2 a−1 b−1 I1 ≤[(1 ∗ `)(t/2)] `(t − s)[(1 ∗ `)(s)] ds 0 (1 ∗ `)(t) ≤ [(1 ∗ `)(t)]a+b−1. b(1 ∗ `)(t/2) Similarly, Z t b−1 a−1 I2 ≤[(1 ∗ `)(t/2)] `(t − s)[(1 ∗ `)(t − s)] ds t/2 (1 ∗ `)(t) ≤ [(1 ∗ `)(t)]a+b−1. a(1 ∗ `)(t/2) Then, 1 1 (1 ∗ `)(t) I ≤ 2 max , [(1 ∗ `)(t)]a+b−1. (34) a b (1 ∗ `)(t/2) (1∗`)(t) Now, observe that h(t) = (1∗`)(t/2) is a continuous function on (0, ∞). Plus, h(t) ≥ 1, and lim supt→∞ h(t) ≤ 2. The ﬁrst is a consequence of ` being nonnegative so that (1∗`) is non-decreasing; and the second is justiﬁed by the following NONLOCAL NAVIER-STOKES EQUATIONS 9 computation: `(t) lim sup h(t) ≤ sup ≤ 2, t→∞ t>a>0 `(t/2) · 1/2 since ` is non increasing. This gives that h is bounded on (0, ∞), say |h(t)| ≤ ϑ, for 1 1 some ϑ > 0. Put C = 2ϑ max a , b into (34) and (33) is proved.

R ∞ e−st Remark 1. It is easy to see that ` = gβ and ` = 0 1+s ds both are completely monotonic. The first ` is associated with the fractional subdiffusion, and the latter one, with the ultraslow diffusion (with k given by (2)), respectively. We close this section with a result concerning the temporal continuity of the convolution operators Z(t, ·)? and Y (t, ·)?.

σ d σ d Lemma 2.5. Let d ∈ N, 2 ≤ d < q < ∞, v ∈ Ld (R ) and g ∈ C((0, ∞); L qd (R )), q+d such that −1+ d kg(t, ·)k d ≤ κ[(1 ∗ `)(t)] 2q , (35) L qd (R ) q+d for some κ > 0 and all t > 0. Then, it holds that σ d σ d t 7−→ Z(t, ·) ? v ∈ C([0, ∞); Ld (R )) ∩ C((0, ∞); Lq (R )), (36) σ d t 7−→ ∇Z(t, ·) ? v ∈ C((0, ∞); Ld (R )), (37) Z t σ d σ d t 7−→ Y (t − s, ·) ? g(s, ·)ds ∈ C((0, ∞); Ld (R ) ∩ C((0, ∞); Lq (R )), (38) 0 and Z t σ d t 7−→ ∇ Y (t − s, ·) ? g(s, ·)ds ∈ C((0, ∞); Ld (R )). (39) 0 σ d The continuities (36) and (37) are uniform for v in bounded sets of Ld (R ). σ d Proof. Let ε > 0 and v be in a bounded set L ( ), say kvk σ d ≤ µ. We ﬁrst d R Ld (R ) d claim that the kernel t 7→ Z(t, ·) is continuous in L1(R ). Indeed, we can see that the −τλkˆ(λ) measure −w(t, dτ) is induced by the function (k ∗ϕτ )(t)dt, whereϕ ˆτ (λ) = e , is a.e. continuous, for t > 0, provided that k is completely monotonic. Indeed, in ˆ e−τλk(λ) our case, the propagation function w satisﬁesw ˆ(λ; τ) = λ , see [20, Ch. 4]. Hence, Z kZ(t, ·) − Z(t , ·)k d = |Z(t, x) − Z(t , x)|dx 0 L1(R ) 0 d R Z Z ∞

= −w(t, dτ)H(τ, x) + w(t0, dτ)H(τ, x) dx d R 0 Z ∞ Z ≤ | − w(t, dτ) + w(t0, dτ)| H(τ, x)dx d 0 R Z ∞ = |w(t, dτ) − w(t0, dτ)| 0 →0, as t → t0, by the Lebesgue dominated convergence theorem. Therefore, there exists η > 0 such that |t − t0| < η implies ε kZ(t, ·) − Z(t , ·)k d < , 0 L1(R ) µ 10 J. AZEVEDO, J. C. POZO AND A. VIANA which yields

kZ(t, ·) ? v − Z(t , ·) ? vk d ≤ kZ(t, ·) − Z(t , ·)k d kvk d < ε. 0 Ld(R ) 0 L1(R ) Ld(R ) d d We can see that the kernel t 7→ Z(t, ·) is continuous in Lp(R ), 1 < p < d−2 , d > 2 or 1 < p < ∞, d = 1, 2. In fact, Z ∞ p p |Z(t, x) − Z(t0, x)| ≤ H(τ, x)|w(t, dτ) − w(t0, dτ)| . 0 Then, by Minkowski’s inequality for integrals, we have 1 Z Z ∞ p p kZ(t, ·) − Z(t , ·)k d ≤ H(τ, x)|w(t, dτ) − w(t , dτ)| 0 Lp(R ) 0 d R 0 1 Z ∞ Z p p ≤ H(τ, x) dx |w(t, dτ) − w(t0, dτ)| d 0 R 1 ∞ d ! p Z (4πτ/p) 2 = dp |w(t, dτ) − w(t0, dτ)| 0 (4πτ) 2 Z ∞ d ( 1 −1) =C(d, p) τ 2 p |w(t, dτ) − w(t0, dτ)|. 0 d 1 Putting δ = 2 1 − p , the assumptions on p gives δ ∈ (0, 1). Then, Z ∞ d ( 1 −1) τ 2 p |w(t, dτ) − w(t0, dτ)| → 0, (40) 0 as t → t0 > 0, by the Lebesgue dominated convergence theorem. Indeed, ξ(t) := d ( 1 −1) −δ −δ τ 2 p |w(t, dτ) − w(t0, dτ)| is dominated by ζ(t) := −τ w(t, dτ) − τ w(t0, dτ), whose integral is Γ(1 − δ)(φδ(t) + φδ(t0)), by (14). We see that ξ(t) → 0 and −δ ζ(t) → −2τ w(t0, dτ) a.e., as t → t0 > 0. From Lemma 2.1, the function φδ given is smooth. Hence, Z ∞ Z ∞ −δ ζ(t, dτ) = Γ(1 − δ)(φδ(t) + φδ(t0)) → 2Γ(1 − δ)φδ(t0) = −2τ w(t0, dτ), 0 0 as t → t0 > 0. Now, the Lebesgue dominated convergence theorem gives that (40). Therefore,

kZ(t, ·) − Z(t , ·)k d → 0, 0 Lp(R ) as t → t0 > 0. Therefore, there exists η > 0 such that |t − t0| < η implies ε kZ(t, ·) − Z(t , ·)k d < , 0 Lp(R ) µ which yields

kZ(t, ·) ? v − Z(t , ·) ? vk d ≤ kZ(t, ·) − Z(t , ·)k d kvk d < ε. 0 Lq (R ) 0 Lp(R ) Ld(R ) To conclude the proof of (36), it only remains to prove the continuity at t = 0. To do this, we use that w(t) = Z(t, ·) ? v solves (9) with u0 ≡ v, that is,

u + (` ∗ Apu)(t) = v. (41) ∞ d Indeed, assume ﬁrst that v ∈ C ( ) is divergence-free. Then, kA vk d < ∞, 0 R p Ld(R ) and we use (41) and the uniform boundedness, given by (18), to obtain Z t kZ(t, ·) ? v − vk d ≤ C `(s)dskA vk d → 0, (42) Ld(R ) p Ld(R ) 0 NONLOCAL NAVIER-STOKES EQUATIONS 11

+ as t → 0 . Observe that we are not saying that Ap is bounded but using that ∆v has compact support. A density argument, as what was done to show (30), shows σ d that (42) holds for v ∈ Ld (R ). This ends the proof of (36). In order to prove (37), we recall that

Z − 1 |∇H(τ, x)|dx = Cτ 2 . d R

Hence,

as t → t0 > 0, by the Lebesgue dominated convergence theorem and Lemma 2.1. Since

k∇Z(t, ·) ? v − ∇Z(t , ·) ? vk d ≤ k∇Z(t, ·) − ∇Z(t , ·)k d kvk d , 0 Ld(R ) 0 L1(R ) Ld(R )

(37) follows in the same way that (36). Now, we ﬁx t0 > 0. For t > t0, we estimate

Z t Z t0

Y (t − s, ·) ? g(s, ·)ds − Y (t0 − s, ·) ? g(s, ·)ds d 0 0 Ld(R ) Z t Z t0 ≤ kY (s, ·) ? g(t − s, ·)k d ds + kY (s, ·) ? [g(t − s, ·) − g(t0 − s, ·)] k d ds Ld(R ) Ld(R ) t0 0 Z t − d ≤ `(s)[(1 ∗ `)(s)] 2q kg(t − s, ·)k d ds L qd (R ) t0 q+d Z t0 − d + `(s)[(1 ∗ `)(s)] 2q kg(t − s, ·) − g(t0 − s, ·)k d ds L qd (R ) 0 q+d Z t − d −1+ d ≤κ `(s)[(1 ∗ `)(s)] 2q [(1 ∗ `)(t − s)] 2q ds t0 Z t0 − d + `(s)[(1 ∗ `)(s)] 2q kg(t − s, ·) − g(t0 − s, ·)k d ds. L qd (R ) 0 q+d

The second term of the right-hand side above goes to zero, as t → t0, by Lebesgue t dominated convergence theorem. On the other hand, we may assume that 2 < t0 < s < t so that t − s < s and `(s) ≤ `(t − s). Then, the change of variables 12 J. AZEVEDO, J. C. POZO AND A. VIANA

(1 ∗ `)(s) = η gives Z t − d −1+ d `(s)[(1 ∗ `)(s)] 2q (1 ∗ `)(t − s) 2q ds t0 Z t − d −1+ d ≤[(1 ∗ `)(t0)] 2q [(1 ∗ `)(t − s)] 2q `(t − s)ds t0 Z t−t0 − d −1+ d =[(1 ∗ `)(t0)] 2q [(1 ∗ `)(s)] 2q `(s)ds 0 Z (1∗`)(t−t0) − d −1+ d =[(1 ∗ `)(t0)] 2q η 2q dη 0 − d 2q[(1 ∗ `)(t0)] 2q d = [(1 ∗ `)(t − t )] 2q d 0 →0, Z t + σ d as t → t0 . Therefore, t 7→ Y (t − s, ·) ? g(s, ·)ds is right-continuous in Ld (R ), 0 for all t > 0. It is similar to prove the it is left-continuous. Also, we have

Z t Z t0

Y (t − s, ·) ? g(s, ·)ds − Y (t0 − s, ·) ? g(s, ·)ds d 0 0 Lq (R ) Z t − 1 −1+ d ≤κ `(s)[(1 ∗ `)(s)] 2 (1 ∗ `)(t − s) 2q ds t0 Z t0 − 1 + `(s)[(1 ∗ `)(s)] 2 kg(t − s, ·) − g(t − s, ·)k d ds 0 L qd (R ) 0 q+d and Z t − 1 − 1 −1+ d 2q[(1 ∗ `)(t0)] 2 d `(s)[(1 ∗ `)(s)] 2 (1 ∗ `)(t − s) 2q ds ≤ [(1 ∗ `)(t − t0)] 2q → 0, t0 d as t → t0 > 0. These facts are suﬃcient to reproduce the above argument and Z t σ d prove that t 7→ Y (t − s, ·) ? g(s, ·)ds is continuous in Lq (R ), for all t > 0. This 0 ﬁnishes the proof of (38). Finally,

Z t Z t0

∇ Y (t − s, ·) ? g(s, ·)ds − ∇ Y (t0 − s, ·) ? g(s, ·)ds d 0 0 Ld(R ) Z t − d − 1 −1+ d ≤κ `(s)[(1 ∗ `)(s)] 2q 2 (1 ∗ `)(t − s) 2q ds t0 Z t0 − d − 1 + `(s)[(1 ∗ `)(s)] 2q 2 kg(t − s, ·) − g(t − s, ·)k d ds. 0 L qd (R ) 0 q+d

As previously, the second term of the right-hand side above goes to zero, as t → t0, by Lebesgue dominated convergence theorem and the ﬁrst one

Z t − d − 1 − d − 1 −1+ d 2q[(1 ∗ `)(t0)] 2q 2 d `(s)[(1 ∗ `)(s)] 2q 2 (1 ∗ `)(t − s) 2q ds ≤ [(1 ∗ `)(t − t )] 2q → 0, d 0 t0

as t → t0 > 0. Therefore, (39) holds and we complete the proof of Lemma 2.5. NONLOCAL NAVIER-STOKES EQUATIONS 13

3. The global well-posedness. In this section we investigate the existence and uniqueness of global mild solutions for equation (7). Before we state the main result, we introduce space where the mild solution will dwell. Let d ∈ N. For any 2 ≤ d < q < ∞, consider the space Xq of the functions v satisfying

σ d v ∈ Cb((0, ∞); Ld (R )), 1 d 2 − 2q σ d (1 ∗ `) v ∈ Cb((0, ∞); Lq (R )), 1 2 σ d (1 ∗ `) ∇v ∈ Cb((0, ∞); Ld (R )),

which is a Banach space with norm kvkXq given by

n 1 − d 1 o max sup kv(t)k σ d , sup [(1 ∗ `)(t)] 2 2q kv(t)k σ d , sup [(1 ∗ `)(t)] 2 k∇v(t)k σ d . (43) t>0 Ld (R ) t>0 Lq (R ) t>0 Ld (R )

We use the following lemma to obtain estimates for the nonlinear part of the integral formulation (11). See [11].

Proposition 1. Let p1, p2 and p3 be real numbers such that

1 1 1 1 < p1, p2, p3 < ∞ and = + . p3 p1 p2

There exists a constant C = C(p1, p2, p3, d) > 0 such that

kF (v, w)k d ≤ Ckvk d k∇wk d , Lp3 (R ) Lp1 (R ) Lp2 (R ) for any v ∈ Lσ ( d) and ∇w ∈ Lσ ( d). p1 R p2 R

We rewrite (11) as

Z t u(t) = Z(t, ·) ? u0 + Y (t − s, ·) ? P f(s, ·) ds + L(u(t), u(t)), 0 where the bilinear operator L is given by

Z t L(v(t), w(t)) := Y (t − s, ·) ?F (v, w)ds. (44) 0

To do this, we have to ensure that L : Xq × Xq → Xq is well-deﬁned and this is the purpose of the following lemma.

Lemma 3.1. Let d ∈ N, 2 ≤ d < q < ∞ and assume that ` is a completely monotonic function. Then, L : Xq × Xq → Xq given by (44) is well-deﬁned and satisﬁes (50).

Proof. We must to verify that (50) holds, where η will be explicit later. Indeed, qd given v, w ∈ Xq, we use Proposition1 with ( p1, p2, p3) = (q, d, q+d ) after (19) with 14 J. AZEVEDO, J. C. POZO AND A. VIANA

qd (q, r) = (q, q+d ) to estimate Z t

kL(v, w)k d = Y (t − s, ·) ?F (v, w) ds Lq (R ) d 0 Lq (R ) Z t − d · 1 ≤ C `(s)[(1 ∗ `)(s)] 2 d kF (v, w)k d ds L qd (R ) 0 q+d Z t − 1 ≤ C `(s)[(1 ∗ `)(s)] 2 kv(t − s)k d k∇w(t − s)k d ds Lq (R ) Ld(R ) 0 t Z 1 d − 2 2q −1 ≤ C `(s)[(1 ∗ `)(s)] [(1 ∗ `)(t − s)] ds kvkXq kwkXq 0 1 d − 2 + 2q ≤ M1[(1 ∗ `)(t)] kvkXq kwkXq , (45) where M1 combines constants from Lemmas 2.3 and 2.4, and Proposition1. Sim- qd ilarly, we use Proposition1 with ( p1, p2, p3) = (q, d, q+d ) after (21) with (q, r) = qd (d, q+d ) to estimate

k∇L(v, w)k d (46) Ld(R ) Z t

= ∇Y (t − s, ·) ?F (v, w) ds d 0 Ld(R ) Z t − 1 − d · 1 ≤ C `(s)[(1 ∗ `)(s)] 2 2 q kF (v, w)k d ds L qd (R ) 0 q+d Z t − 1 − d ≤ C `(s)[(1 ∗ `)(s)] 2 2q kv(t − s)k d k∇w(t − s)k d ds Lq (R ) Ld(R ) 0 t Z 1 d d − 2 − 2q 2q −1 ≤ C `(s)[(1 ∗ `)(s)] [(1 ∗ `)(t − s)] ds kvkXq kwkXq 0 1 − 2 ≤ M2[(1 ∗ `)(t)] kvkXq kwkXq , (47) and Z t − d · 1 kL(v, w)k d ≤ C `(s)[(1 ∗ `)(s)] 2 q kF (v, w)k d ds Ld(R ) L qd (R ) 0 q+d

≤ M3kvkXq kwkXq . (48)

Estimates (45), (47) and (48) imply that kL(v, w)kXq ≤ ηkvkXq kwkXq , where η := max{M1,M2,M3}. It only remains to prove that the map t 7→ L(v(t), w(t)) belongs to C((0, ∞); σ d σ d σ d Ld (R )) ∩ C((0, ∞); Lq (R )) and t 7→ ∇L(v(t), w(t)) belongs to C((0, ∞); Ld (R )). σ d Indeed, given v, w ∈ Xq, we have that the map g(t) = F (v(t), w(t)) ∈ L qd (R ) q+d satisﬁes

kg(t, ·) − g(t0, ·)k σ d ≤ kv(t) − v(t0)kL k∇w(t)kL + kv(t0)kL k∇w(t) − ∇w(t0)kL L qd (R ) q d q d q+d (49) and −1+ d kg(t, ·)k d ≤ [(1 ∗ `)(t)] 2q kvk kwk , L qd (R ) Xq Xq q+d σ d by Proposition1. From v, w ∈ Xq and (49), we obtain that g ∈ C((0, ∞); L qd (R )). q+d Now, one can apply Lemma 2.5 to conclude that L(v, w) ∈ Xq. NONLOCAL NAVIER-STOKES EQUATIONS 15

Remark 2. We observe that Mi = ϑiCiC˜i, where C˜i is the constant in Proposition 1 d 1 d d 1, Ci is in Lemma 2.3, and ϑi is in Lemma 2.4, with (a, b) being ( 2 , 2q ), ( 2 − 2q , 2q ) d d and (1 − 2q , 2q ), for i = 1, 2, 3, respectively. Then, the constant η depends on d, q. We denote ϑ = maxi=1,2,3 ϑi, C = maxi=1,2,3 Ci, and C˜ = maxi=1,2,3 C˜i. The existence of the mild solutions for (7) will be a consequence of the following ﬁxed point lemma (see [4, Lemma 1.5]). Lemma 3.2. Let X be an abstract Banach space with norm k·k and L : X×X → X be a bilinear operator. Assume that there exists η > 0 such that, given x1, x2 ∈ X, we have

kL(x1, x2)k ≤ ηkx1kkx2k. (50) Then, for any y ∈ X, such that 4ηkyk < 1, (51) the equation x = y + L(x, x) (52) has a solution x in X. Moreover, this solution x is the only one such that 1 − p1 − 4ηkyk kxk ≤ . (53) 2η Theorem 3.3. Let d ∈ N, 2 ≤ d < q < ∞, η the constant given by Lemma 3.1 and d f ∈ C((0, ∞); L qd ( ) be such that q+d R

1− d α := sup[(1 ∗ `)(t)] 2q kf(t)k d < ∞. (54) L qd (R ) t>0 q+d

σ d 1−4αϑCη For u0 ∈ Ld (R ) and α > 0 suﬃciently small, there exists 0 < λ < 4η such −1 that if ku k d ≤ min{1,C }λ, then the problem (7) has a global mild solution 0 Ld(R ) u ∈ Xq that is the unique one satisfying (53). In particular, p 1 − 1 − 4η(λ + αϑC) − 1 + d ku(t, ·)k d ≤ [(1 ∗ `)(t)] 2 2q (55) Lq (R ) 2η and p 1 − 1 − 4η(λ + αϑC) − 1 k∇u(t, ·)k d ≤ [(1 ∗ `)(t)] 2 . (56) Ld(R ) 2η If, in addition, f ≡ 0, we have

1 − d [(1 ∗ `)(t)] 2 2q ku(t, ·)k d → 0 (57) Lq (R ) and 1 [(1 ∗ `)(t)] 2 k∇u(t, ·)k d → 0 (58) Ld(R ) + as t → 0 . Furthermore, let u, v ∈ Xq be two solutions given by the existence part corresponding to the initial data u0 and v0, respectively. Then, C ku − vkX ≤ ku0 − v0kL ( d), (59) q p1 − 4η(λ + αϑC) d R where C > 0 is the constant of Lemma 2.3. 16 J. AZEVEDO, J. C. POZO AND A. VIANA

Proof. Recall that we are going to use Lemma 3.2. In view of Lemmas 2.5 and 3.1, we only need to verify (51) to get the existence of the solution. Indeed, if we use (54) and repeat the reasonings in Lemma 3.1, we obtain Z t

Y (t − s, ·) ? P f(s, ·) ds ≤ αϑC, 0 Xq with the notation of Remark2. Plus, kZ(t, ·) ? u0kXq ≤ λ is a direct consequence of (18) and (20) in Lemma 2.3. If Z t y = Z(t, ·) ? u0 + Y (t − s, ·) ? P f(s, ·) ds 0 then

4ηkykXq ≤ 4η(λ + αϑC) < 1, 1−4αϑCη for 0 < λ < 4η . Then, (51) holds and we can use Lemma 3.2 to conclude that there exists a solution u ∈ Xq which is the unique one satisfying (53). The decays (55) and (56) are now consequence of u ∈ Xq and (43). Now, let u, v ∈ Xq be two solutions corresponding to the initial data u0 and v0, respectively. Then Z t u(t, x) − v(t, x) = Z(t, ·) ? (u0 − v0)(x) + Y (t − s, ·) ? F (u, u) − F (v, v) ds. 0 From Proposition1, it is easy to see that

d d d d kF (u, u)−F (v, v)kL qd ≤ ku−vkL ( )k∇ukL ( ) +kvkL ( )k∇(u−v)kL ( ). ( d) q R d R q R d R q+d R Thus, it is possible to use Lemma 2.3 as done in Lemma 3.1 to estimate

ku(t, ·) − v(t, ·)k d Ld(R ) Z t − d ≤ Cku − v k d + C `(s)[(1 ∗ `)(s)] 2q kF (u, u) − F (v, v)k d ds 0 0 Ld(R ) L qd (R ) 0 q+d

≤ Cku − v k d 0 0 Ld(R ) Z t − d +C `(s)[(1 ∗ `)(s)] 2q ku(t − s) − v(t − s)k d k∇v(t − s)k d ds Lq (R ) Ld(R ) 0 Z t − d +C `(s)[(1 ∗ `)(s)] 2q ku(t − s)k d k∇(u − v)(t − s)k d ds Lq (R ) Ld(R ) 0

≤ Cku − v k d 0 0 Ld(R ) t Z d 1 d 1 − 2q − 2 + 2q − 2 +C[kukXq + kvkXq ]ku − vkXq `(s)[(1 ∗ `)(s)] [(1 ∗ `)(t − s)] ds 0 ≤ Cku − v k d + η kuk + kvk ku − vk , (60) 0 0 Ld(R ) Xq Xq Xq

ku(t, ·) − v(t, ·)k d Lq (R ) − 1 + d ≤ C[(1 ∗ `)(t)] 2 2q ku − v k d (61) 0 0 Ld(R ) t Z 1 − 2 +C `(s)[(1 ∗ `)(s)] kF (u, u) − F (v, v)kL qd ds ( d) 0 q+d R − 1 + d ≤ Cku − v k d [(1 ∗ `)(t)] 2 2q (62) 0 0 Ld(R ) 1 d − 2 + 2q +η[(1 ∗ `)(t)] kukXq + kvkXq ku − vkXq , (63) NONLOCAL NAVIER-STOKES EQUATIONS 17 and

k∇(u(t, ·) − v(t, ·))k d Ld(R ) − 1 ≤ C[(1 ∗ `)(t)] 2 ku − v k d (64) 0 0 Ld(R ) Z t − 1 − d +C `(s)[(1 ∗ `)(s)] 2 2q kF (u, u) − F (v, v)k d ds L qd (R ) 0 q+d − 1 ≤ Cku − v k d [(1 ∗ `)(t)] 2 0 0 Ld(R ) 1 − 2 +η[(1 ∗ `)(t)] kukXq + kvkXq ku − vkXq . (65) Therefore, the estimates (60), (63) and (65) give ku − vk ≤ Cku − v k d + η kuk + kvk ku − vk . Xq 0 0 Ld(R ) Xq Xq Xq

We now use that 1 − 4ηkykXq ≥ 1 − 4η(λ + αϑC) and (53) to yield (59). Proceeding as in (63), one can see that 1 − d [(1 ∗ `)(t)] 2 2q ku(t, ·)k d (66) Lq (R ) 1 − d ≤ [(1 ∗ `)(t)] 2 2q kZ(t, ·) ? u k d 0 Lq (R ) 1 − d +ηkuk sup [(1 ∗ `)(s)] 2 2q ku(s, ·)k d Xq Lq (R ) s∈(0,t) 1 − d ≤ [(1 ∗ `)(t)] 2 2q kZ(t, ·) ? u k d 0 Lq (R ) p 1 − 1 − 4η(λ + αϑC) 1 − d + sup [(1 ∗ `)(s)] 2 2q ku(s, ·)k d . Lq (R ) 2 s∈(0,t) It follows that 1 − d sup [(1 ∗ `)(s)] 2 2q ku(s, ·)k d Lq (R ) s∈(0,t)

2 1 − d ≤ sup [(1 ∗ `)(s)] 2 2q kZ(s, ·) ? u k d . (67) p 0 Lq (R ) 1 + 1 − 4η(λ + αϑC) s∈(0,t) Then, (30) yields (57). It is similar to prove (58). This ﬁnishes the proof of Theorem 3.3.

T Let d ∈ N. For any 2 ≤ d < q < ∞, we consider, for any T > 0, the space Xq of the functions v satisfying σ d v ∈ Cb((0,T ]; Ld (R )), 1 d 2 − 2q σ d (1 ∗ `) v ∈ Cb((0,T ]; Lq (R )), 1 2 σ d (1 ∗ `) ∇v ∈ Cb((0,T ]; Ld (R )), which is a Banach space with norm kvk T given by Xq

n 1 − d 1 o max sup kv(t)k σ d , sup [(1 ∗ `)(t)] 2 2q kv(t)k σ d , sup [(1 ∗ `)(t)] 2 k∇v(t)k σ d t∈(0,T ) Ld (R ) t∈(0,T ) Lq (R ) t∈(0,T ) Ld (R ) (68) T Denote by Y the class of all functions in Xq (or Xq ) satisfying (57) and (58). Theorem 3.4. Under the assumptions of Theorem 3.3, the mild solution of (7) is unique in Y. Before we give the proof of Theorem 3.4, we recall the following generalized Gronwall inequality due to Amann [1, Lemma 3.3]. 18 J. AZEVEDO, J. C. POZO AND A. VIANA

1 Lemma 3.5. Suppose that 0 ≤ τ < τ¯, that b ∈ L ([0, τ¯], R+) is decreasing, and that a, ϕ ∈ C([τ, τ¯], R+). Moreover, suppose that

Z t ϕ(t) ≤ a(t) + b(t − s)ϕ(s)ds, ∀ t ∈ [τ, τ¯]. τ

Then there exists a constant β > 0, depending only on b, such that

ϕ(t) ≤ 2 max a(t) exp[β(t − τ)], for τ ≤ t ≤ τ.¯ t∈[τ,τ¯]

Proof of Theorem 3.4. The proof of the global existence in Theorem 3.3 can be T repeated for a ball in Xq , for any T > 0, so that, there exists a unique mild T solution w ∈ Xq for (7) that satisﬁes (53). On the other hand, the mild solution u T found in Theorem 3.3 is in Xq , in particular. Hence, w ≡ u on [0,T ]. Let v be any global mild solution of (7) belonging to Y. From (57) and (58), there exists T˜ > 0 T˜ such that v ∈ Xq satisﬁes (53). Now, the T > 0 at the beginning of this proof is ˜ ˜ taken to be T > T . Thus, v ≡ u on [0, T ]. Letη ˜ = max{kukXq , kvkXq }. Then, for t > T˜, we have

ku(t, ·) − v(t, ·)k d + k∇(u(t, ·) − v(t, ·))k d Lq (R ) Ld(R ) Z t − 1 ≤ C `(t − s)[(1 ∗ `)(t − s)] 2 ku(s) − v(s)k d k∇u(s)k d ds Lq (R ) Ld(R ) 0 Z t − 1 +C `(t − s)[(1 ∗ `)(t − s)] 2 kv(s)k d k∇(u(s) − v(s))k d ds Lq (R ) Ld(R ) 0 Z t − 1 − d +C `(t − s)[(1 ∗ `)(t − s)] 2 2q ku(s) − v(s)k d k∇u(s)k d ds Lq (R ) Ld(R ) 0 Z t − 1 − d +C `(t − s)[(1 ∗ `)(t − s)] 2 2q kv(s)k d k∇(u(s) − v(s))k d ds Lq (R ) Ld(R ) 0 Z t − 1 − 1 ≤ Cη˜ `(t − s)[(1 ∗ `)(t − s)] 2 [(1 ∗ `)(s)] 2 sup ku(τ) − v(τ)k d ds Lq (R ) T˜ τ∈(T˜ ,s) Z t − 1 d − 1 +Cη˜ `(t − s)[(1 ∗ `)(t − s)] 2 [(1 ∗ `)(s)] 2q 2 (69) T˜

× sup k∇u(τ) − ∇v(τ)k d ds Ld(R ) τ∈(T˜ ,s) Z t − 1 − d − 1 +Cη˜ `(t − s)[(1 ∗ `)(t − s)] 2 2q [(1 ∗ `)(s)] 2 (70) T˜

× sup ku(τ) − v(τ)k d ds Lq (R ) τ∈(T˜ ,s) Z t − 1 − d d − 1 +Cη˜ `(t − s)[(1 ∗ `)(t − s)] 2 2q [(1 ∗ `)(s)] 2q 2 (71) T˜

× sup k∇u(τ) − ∇v(τ)k d ds Ld(R ) τ∈(T˜ ,s)

n d − 1 − 1 o ≤ Cη˜max [(1 ∗ `)(T˜)] 2q 2 , [(1 ∗ `)(T˜)] 2 (72) NONLOCAL NAVIER-STOKES EQUATIONS 19

Z t − 1 − 1 − d × `(t − s) [(1 ∗ `)(t − s)] 2 + [(1 ∗ `)(t − s)] 2 2q T˜ #

× sup ku(τ) − v(τ)k d + sup k∇u(τ) − ∇v(τ)k d ds. Lq (R ) Ld(R ) τ∈(T˜ ,s) τ∈(T˜ ,s) In Lemma 3.5, let ϕ :[T,˜ ∞) → [0, ∞) given by

ϕ(t) = ku(t, ·) − v(t, ·)k d + k∇(u(t, ·) − v(t, ·))k d , Lq (R ) Ld(R ) a ≡ 0, τ = T˜ and b(t) is given by

n d − 1 − 1 o − 1 − 1 − d Cη˜max [(1 ∗ `)(T˜)] 2q 2 , [(1 ∗ `)(T˜)] 2 `(t) [(1 ∗ `)(t)] 2 + [(1 ∗ `)(t)] 2 2q .

Since u, v ∈ Xq, we have that ϕ is continuous and one can easily check that b is decreasing. Therefore, we can apply Lemma 3.5 to conclude that ϕ ≡ 0 on [T,T˜ ], for any T > T˜. Accordingly, u ≡ v, and we have proved the uniqueness.

Remark 3. 1. If k = g1−α then ` = gα. Therefore, Theorem 3.3 gives in particular the existence result in [8, Th. 12] and adds to it the following stability inequalities − α + αd Ct 2 2q ku(t, ·) − v(t, ·)kLσ ( d) ≤ ku0 − v0kLσ ( d) q R p1 − 4η(λ + αϑC) d R and − α Ct 2 k∇u(t, ·) − ∇v(t, ·)kLσ ( d) ≤ ku0 − v0kLσ ( d), q R p1 − 4η(λ + αϑC) d R

for t > 0, where u, v ∈ Xq are two solutions starting at the initial data u0 and v0, respectively. The uniqueness in Theorem 3.4 also seems to be completely new for the fractional-in-time Navier-Stokes equations (1). 2. The uniqueness result in Theorem 3.4 is compatible with the uniqueness result in [11], although Kato’s essential arguments do not work here because of the non-local character of problem (8). 3. Theorem 3.3 can be generalized to reach the global existence for the more general problem β d ∂t(k ∗ (u − u0)) + (−∆) u + (u · ∇)u + ∇p = f, t > 0, x ∈ R , d ∇ · u = 0, t > 0, x ∈ R , d u(0, x) = u0(x), x ∈ R , where β ∈ (0, 2). To do this, one must adapt the proofs of Lemmas 2.3 and 2.5. This can be done by using ideas of the theorems in [19, Sec. 5], which is valid for β 6= 1. 4. If the smallness of u0 is replaced by the smallness of the time of existence T , suitable changes on the proof of Theorem 3.3 leads to the existence of local mild solutions.

Acknowledgments. We warmly thank the referee for his/her time spent carefully reading our manuscript, and for his/her suggestions, which drove us to a better version of the work. We are indebted to Professor Vicente Vergara, who proposed this problem. We thank him for the fruitful discussions. This paper is partially 20 J. AZEVEDO, J. C. POZO AND A. VIANA supported by CNPq under grant number 408194/2018-9. J. C. Pozo is partially supported by by FONDECYT grant 1181084.

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