Weak global attractor for the 3D-Navier–Stokes equations via the globally modified Navier–Stokes equations

WEAK GLOBAL ATTRACTOR FOR THE 3D-NAVIER-STOKES EQUATIONS VIA THE GLOBALLY MODIFIED NAVIER-STOKES EQUATIONS M. C. BORTOLAN, A. N. CARVALHO, P. MAR´ IN-RUBIO, AND J. VALERO Abstract. In this paper we obtain the existence of a weak global attractor for the three-dimensional Navier- Stokes equations, that is, a weakly compact set with an invariance property, that uniformly attracts solutions, with respect to the weak topology, for initial data in bounded sets. To that end, we define this weak global attractor in terms of limits of solutions of the globally modified Navier-Stokes equations in the weak topology. We use the theory of semilinear parabolic equations and -regularity to obtain the local well posedness for the globally modified Navier-Stokes equations, the existence of the global attractor and its regularity. 1. Introduction The understanding of the asymptotic behavior of solutions for the three-dimensional Navier-Stokes equations is still a big challenge. In particular, the existence of the global attractor with respect to the strong topology of the phase space is up to now an open problem. The main difficulties in proving such results come from the facts that it is not known whether the weak solutions are continuous and also if uniqueness of the Cauchy problem is true. As proved in [4] (see also [22]) as a conditional result, if all the weak solution were continuous, then they would define a multivalued semiflow possessing a global attractor with respect to the strong topology (an extension of this to the stochastic framework is [20]). Later on, in [16] this result was extended by showing that if strong solutions were globally defined, then at least one weak solution would be continuous, ensuring then the existence of a multivalued semiflow having a global attractor, which is again a conditional theorem. Another approach that has been helpful in order to handle this problem is to use the theory of trajectory attractors (see [7,8,23,27,28] for instance, also [11] for the nonautonomous case, and [12] for the stochastic equation). The idea behind these types of attractors is to avoid the problem of the lack of continuity by using a weaker topology, namely, the topology of square integrable functions on finite intervals of time. The drawback of this kind of attractor is that the connection with the original physical phase space is lost. We observe that, as weak solutions are not known to be unique, it is possible to obtain several attractors by selecting different subsets of weak solutions. The key point is that, as weak solutions do not have enough regularity in order to get suitable estimates, it is necessary to select a subclass of weak solutions satisfying extra properties. In the literature, this is done by taking the weak solutions satisfying an energy inequality on some moments of time (the so-called Leray-Hopf solutions). The existence of the global attractor has been obtained when we consider the weak topology of the phase space. In view of the lack of uniqueness, the proper way to do that is to define a multivalued semiflow and prove at least the existence and the negative invariance of the attractor. This question is fraught with important difficulties. If we use the Leray-Hopf solutions satisfying the energy inequality almost everywhere for positive moments of time, but not for the initial one, then a multivalued semiflow is defined but we cannot obtain a bounded absorbing set (see [3, p.2610], [9, p.28]). Therefore, we are not able to prove the existence of a global attractor in this situation. On the other hand, if we consider only the Leray-Hopf solutions satisfying the energy inequality at the initial moment of time, then a bounded absorbing set exists, but the translation of solutions fails to be a solution, so we cannot define a multivalued semiflow. A detailed discussion of these difficulties is given in [3]. The first result in this direction was published in [14] (see [13] as well). The so-called universal attractor, consisting of the bounded Leray-Hopf solutions defined in the whole line (−∞,+∞) and satisfying the energy inequality almost everywhere for real moments of time, was defined. This set was proved to attract uniformly 2020 Mathematics Subject Classification. 35Q30, 35B41, 35K58, 76D05. Key words and phrases. 3D-Navier-Stokes equations, weak global attractors, globally modified Navier-Stokes equations, semilinear parabolic equations, -regularity. (M.C. Bortolan) This study was financed in part by the Coordena¸c˜ao de Aperfei¸coamento de Pessoal de N´ıvel Superior - Brasil (CAPES) - Finance Code 001, 88887.828519/2023-00 Program CAPES-PRInt, and by the Ministerio de Ciencia e Innovaci´on, Agencia Estatal de Investigaci´on (AEI) and FEDER under project PID2021-122991NB-C21. (A. N. Carvalho) Partially supported by FAPESP grants 2020/14075-6 and CNPq 308902/2023-8. (P. Mar´ın-Rubio) Partially supported by the Spanish Ministry of Science, Innovation and Universities, project PGC2018-096540- B-I00, by the Junta de Andaluc´ıa and FEDER, projects US-1254251 and P18-FR-4509. (J. Valero) Partially supported by the Spanish Ministry of Science, Innovation and Universities, project PGC2018-096540-B-I00, by the Spanish Ministry of Science and Innovation, project PID2019-108654GB-I00, and by the Junta de Andaluc´ıa and FEDER, project P18-FR-4509. 1 2 M. C. BORTOLAN, A. N. CARVALHO, P. MAR´ IN-RUBIO, AND J. VALERO in the weak topology of the phase space the Leray-Hopf solutions that start in a bounded set and which satisfy the energy inequality at the initial moment of time. In such a situation, as remarked before, a semiflow cannot be defined and the attractor was shown to be positively invariant but not negatively invariant. This problem was overcome in [16] (see also [9,10,11]) by restricting the phase space to a suitable ball, proving the existence of the weak global attractor in both the autonomous and nonautonomous cases. However, the question about the convergence of the solutions starting outside this ball to the global attractor remained open. In this paper, in order to define a weak global attractor in the whole phase space, we follow the approach of [6] by taking suitable approximations. In this way, we are able to define a suitable subclass of weak solutions possessing a global attractor which uniformly attracts, with respect to the weak topology, the solutions starting at any bounded set. Moreover, the solutions define a multivalued semiflow in the attractor (although not in the whole space) and this set is negatively and positively invariant with respect to it. Thus, by selecting a suitable subclass of weak solutions, we have been able to obtain a new global attractor with better properties than the ones known in the literature. We observe also that this new attractor is contained in the universal attractor given in [14], which coincides with the attractor from [16]. In order to be more specific about the results proved here we consider the Navier-Stokes equations (NSE) (1.1)        ut−ν∆u+ (u· ∇)u=−∇p+f, t > 0, x ∈Ω, div u= 0, t > 0, x ∈Ω, u= 0, t > 0, x ∈∂Ω, u(0, x) = u0(x), x ∈Ω, where Ω ⊂R3is a bounded domain with smooth boundary ∂Ω, u= (u1, u2, u3) is the unknown velocity field of the fluid, ν > 0 is a given constant kinematic viscosity, pis the unknown pressure, u0is a initial velocity field and fis a constant external force taken in L2(Ω)3. We also consider the globally modified Navier-Stokes equations (GMNSE) (1.2)        ut−ν∆u+FN(u)(u· ∇)u=−∇p+f, t > 0, x ∈Ω, div u= 0, t > 0, x ∈Ω, u= 0, t > 0, x ∈∂Ω u(0, x) = u0(x), x ∈Ω, where Nis a fixed positive number and FN(u) = fN(kukL4(Ω)), where fN: [0,∞)→[0,∞) is defined by (1.3) fN(r) = min 1,N rfor r⩾0. The GMNSE were firstly introduced in [5] with the term fN(kukV) instead of fN(kukL4(Ω)). It follows in a similar way as in [5,21], for the case fN(kukV), that the problem (1.2) generates a family of continuous semigroups {SN(t): t⩾0}in Hgiven by SN(t)u0=uN(t, u0), where uN(·, u0) is the unique classical solution to (1.2). We will give a different approach to obtain the semigroup {SN(t): t⩾0}and the existence of its global attractors, which will provide additional regularity of solutions and of the global attractors. Following [19,25,26], if V={u∈C∞ 0(Ω)3: div u= 0}, where C∞ 0(Ω)3is the set of smooth functions u: Ω →R3with compact support in Ω we define H= the closure of Vin L2(Ω)3, with inner product (·,·) and norm k·kHwhere, for u, v ∈L2(Ω)3, (u, v) = 3 X j=1 ZΩ ujvjdx. and V= the closure of Vin H1 0(Ω)3, with scalar product ((·,·)) and norm k·kVwhere, for u, v ∈H1 0(Ω)3, ((u, v)) = 3 X i,j=1 ZΩ ∂uj ∂xi ∂vj ∂xi dx. Consider the Leray projection P:L2→Hand A=P∆: D(A)⊂H→Hthe Stokes operator (we have D(A)⊂W2,2(Ω)3∩H, see [18] for instance). Since Ais closed and densely defined (Ais, in fact, sectorial), we can consider the interpolation-extrapolation scale generated by (H, A), given by {(Hα, Aα): α⩾−1}. Since His reflexive, thanks to [1, Theorem V.1.5.12], we have (1.4) H−α= (Hα)∗for α∈[0,1]. WEAK GLOBAL ATTRACTORS 3 In this way, we have found A−1, with D(A−1) = H, (1.5) A−1:H⊂H−1→H−1, satisfying A−1u(φ) = ZΩ uAφ for each u∈Hand φ∈D(A), and this operator is sectorial (see [29], for instance). With this setting, we prove the following result concerning the local well posedness of (1.2), which will be proved at the end of Section 3. Theorem 1.1. Assume that Pf ∈H−1 2. For each N > 0, equation (1.2)generates a semigroup SN= {SN(t): t⩾0}in Hwith a global attractor AN. For each u0∈H, if u(t) = SN(t)u0for t⩾0, we have u∈C([0,∞), H)∩C((0,∞), V )∩C1((0,∞), H−1 2), and uis a classical solution of (1.2). Moreover, for any T > 0,0< η < 1 8and p(1 2+η)<1, du dt ∈Lp(0, T;Hη−1 2). Now we introduce some terminology to be able to state our main result concerning the existence of a weak global attractor for (1.1). First we define the set (the candidate for the weak global attractor) A ⊂ Has follows A=   y∈H: there are sequences tj j→∞ −→ ∞,{uj 0}j∈N⊂B0 and Nj j→∞ −→ ∞ such that SNj(tj)uj 0→yweakly in H.  , where B0is a suitable closed ball in H(see (3.13)). Let Kbe the set of all weak solutions of (1.1) (we recall its meaning in Definition 4.1). We now define the subset of weak solutions that will be of our interest. Definition 1.2. We say that u∈ KN if u∈ K and one of the following conditions holds: (1) u(0) ∈ A and there exist tj j→∞ −→ ∞,{uj 0}j∈N⊂B0and Nj j→∞ −→ ∞ such that for any t0⩾0 and [0,∞)3sj j→∞ −→ t0we have SNj(sj+tj)uj 0 j→∞ −→ u(t0) weakly in H. (2) u(0) ∈H\ A and there exists Nj j→∞ −→ ∞ such that for any t0⩾0 and [0,∞)3sj j→∞ −→ t0we have SNj(sj)u(0) j→∞ −→ u(t0) weakly in H. With this we prove the following result. Theorem 1.3. Assume that P f ∈H−1 2. The set Ais a weak global attractor for the solutions in KN , which means that: (a) Ais weakly compact in H. (b) Ais negatively invariant, that is, for any y∈ A and t⩾0there exists u∈ KN such that u(0) ∈ A and u(t) = y. (c) Ais positively invariant, that is, given t⩾0we have {u(t): u∈ KN , u(0) ∈ A} ⊂ A. (d) Aweakly attracts KN, that is, for any bounded subset Bof Hwe have sup u∈KN, u(0)∈B distw(u(t),A)t→∞ −→ 0, where distw(·,·)is the Hausdorff semidistance defined using the metric induced by the weak topology of Hin the ball B0(see (4.4)). Finally, in Proposition 4.12, we obtain that the global attractors for the globally modified Navier-Stokes equations behave upper semicontinuously with respect the weak global attractor A. The paper is organized as follows: in Section 2we introduce the notion of -regular maps and a local existence result for semilinear parabolic problems with nonlinearities being −regular maps. In addition, we present a new result (see Theorem 2.7) regarding the regularity of solutions. In Section 3we prove that the nonlinearities associated to the GMNSE are 3 8-regular maps, and obtain the local existence and regularity results for the very weak formulation of (1.2), i.e. Theorem 1.1 will be proved. Finally, in Section 4, we obtain the result on existence of the weak global attractor, prove Theorem 1.3 and obtain the upper semicontinuity of the attractors (see Proposition 4.12). 4 M. C. BORTOLAN, A. N. CARVALHO, P. MAR´ IN-RUBIO, AND J. VALERO 2. Local existence and regularity results for abstract semilinear parabolic problems In what follows we will introduce some notation and terminology in order to state the main results of [2, Section 2] that will be used to obtain the global well posedness, existence of the global attractor and its regularity for (1.2), and also present new regularity results (Lemma 2.5 and Theorem 2.7). Let (X, k·kX) be a Banach space. Consider a positive sectorial operator −A:D(A)⊂X→X, its associated scale of fractional power spaces Xα= (D(Aα),k·kXα) for α⩾0 (where kxkXα=kAαxkXfor x∈Xα), and the analytic semigroup {eAt :t⩾0}generated by Ain X. Without loss of generality, we can assume that there exists a constant M⩾1 such that (2.1) keAtxkXα⩽MkxkXαfor t⩾0, x ∈Xαand α⩾0, and (2.2) keAtxkXα⩽Mt−αkxkXfor t > 0, x ∈Xand α⩾0. With this, we can see that for 0 ⩽α⩽βwe have (2.3) keAtxkXβ⩽Mtα−βkxkXαfor all t > 0 and x∈Xα. Indeed, keAtxkXβ=kAαeAtxkXβ−α⩽Mt−(β−α)kAαxkX=Mtα−βkxkXα. Given ,ρand γ() positive constants, with ρ > 1 and ρ ⩽γ()<1, we define F(, ρ, γ()) as the family of functions f:X1+→Xγ(), such that, for all x, y ∈X1+, satisfy (2.4) kf(x)−f(y)kXγ()⩽ckx−ykX1+(kxkρ−1 X1++kykρ−1 X1++ 1) and (2.5) kf(x)kXγ()⩽c(kxkρ X1++ 1), for some constant c > 0. A function f∈ F(, ρ, γ()) is called an -regular map relative to the pair (X1, X0). We consider the local well posedness for the abstract parabolic problem (2.6) (˙x=Ax +f(x), t > 0 x(0) = x0, when f∈ F(, ρ, γ()). We say that a function x: [0, τ]→X1is an -regular mild solution of (2.6) in [0, τ] if x∈C([0, τ], X1)∩C((0, τ], X1+) and (2.7) x(t) = eAtx0+Zt 0 eA(t−s)f(x(s))ds for each t∈[0, τ]. With these definitions, we have the following: Theorem 2.1 (See [2, Section 2]).Let f∈ F(, ρ, γ()). Given y0∈X1, there exist r > 0and τ0>0such that for any x0∈BX1(y0, r)there exists a continuous function x(·, x0): [0, τ0]→X1, which is the unique -regular mild solution of (2.6). This solution satisfies x∈C((0, τ0], X1+θ)for 0⩽θ < γ(), and tθkx(t, x0)kX1+θ t→0+ −−−−→ 0for 0< θ < γ(). Moreover, given 0⩽θ0< γ(), there exists C > 0such that if x0, z0∈BX1(y0, r)then tθkx(t, x0)−x(t, z0)kX1+θ⩽Ckx0−z0kX1, for all t∈[0, τ0]and 0⩽θ⩽θ0. Also, if γ()> ρ, the time of existence is uniform on bounded subsets of X1and, furthermore, x∈C((0, τ0], X1+γ())∩C1((0, τ0], Xγ()), xis a classical solution of (2.6)and, if τmis the maximal time of existence for x(t, x0), either τm=∞or lim t→τ− m kx(t, x0)kX1=∞. Remark 2.2. We stress that it is not assumed that fis defined on X1. The only requirement on fis that it is an -regular map relative to the pair (X1, X0) for some  > 0. Hence, local well posedness in X1is obtained without requiring that the nonlinearity fis defined on X1. WEAK GLOBAL ATTRACTORS 5 2.1. Regularity results. Continuing the work done in [2], we present a new result regarding the regularity of -regular mild solutions of (2.6). We begin with a set of technical lemmas. Lemma 2.3. If x: [0, τ0]→X1is an -regular mild solution of (2.6)then for each 0⩽θ < γ()there exists a constant C⩾0such that kx(t)kX1+θ⩽Ct−θfor all t∈(0, τ0]. Proof. Since xis an -regular mild solution of (2.6), from Theorem 2.1 we obtain sup s∈(0,τ0] skx(s)kX1+<∞. Hence, using [2, Lemma 2], we have sup t∈(0,τ0] tθ   Zt 0 eA(t−s)f(x(s))ds   X1+θ ⩽C1, for some constant C1⩾0. Thus, using (2.7) and (2.3), we have kx(t)kX1+θ⩽keAtx0kX1+θ+   Zt 0 eA(t−s)f(x(s))ds   X1+θ ⩽Mt−θkx0kX1+C1t−θ, and the estimate follows, taking C=Mkx0kX1+C1. Lemma 2.4. Fix 0< T < ∞and let g: (0, T ]→Xγ()be locally H¨older continuous with Rr 0kg(s)kXγ()ds < ∞ for some r > 0. For 0⩽t⩽T, define (2.8) G(t) = Zt 0 eA(t−s)g(s)ds. Then Gis continuous on [0, T], continuously differentiable on (0, T), with G(t)∈D(A)for t∈(0, T)and dG dt (t) = AG(t) + g(t)for t∈(0, T), with G(t)t→0+ −→ 0in Xγ(). Proof. Apply [15, Lemma 3.2.1] with gand Xγ()in place of fand X. Lemma 2.5. Fix 0< T < ∞, and assume that g: (0, T]→Xγ()satisfies kg(θ)−g(r)kXγ()⩽min{K(r), K(θ)}|θ−r|δfor all θ, r ∈(0, T], where δ > 0is a fixed constant and K: (0, T ]→[0,∞)is continuous, and integrable in (0, T ). Then the function Gdefined in (2.8)is continuously differentiable from (0, T )into Xγ()+µfor any 0⩽µ<δ, and (2.9)     dG dt (t)   Xγ()+µ ⩽Mt−µkg(t)kXγ()+MZt 0 K(s)(t−s)−µ+δ−1ds for t∈(0, T), with Mindependent from µand g. Furthermore (0, T)3t7→ dG dt (t)∈Xγ()+µ is locally H¨older continuous, provided Rr 0K(s)ds =O(rϕ)as r→0+, for some ϕ > 0. Proof. From Lemma 2.4 we know that dG dt (t) = AG(t) + g(t) = eAtg(t) + H(t) for all t∈(0, T), where H(t) = Zt 0 AeA(t−s)(g(s)−g(t))ds. Hence, for 0 ⩽µ < δ, we have     dG dt (t)   Xγ()+µ ⩽keAtg(t)kXγ()+µ+Zt 0 kAeA(t−s)(g(s)−g(t))kXγ()+µds ⩽Mt−µkg(t)kXγ()+MZt 0 K(s)(t−s)−µ+δ−1ds, and (2.9) follows. For the second part fix r0>0 such that Rr 0K(s)ds ⩽Crϕ, for 0 ⩽r⩽r0. Hence, for a fixed t > 0 and 0 < h ⩽r0, we have H(t+h)−H(t) = Zt+h 0 AeA(t+h−s)(g(s)−g(t+h))ds −Zt 0 AeA(t−s)(g(s)−g(t))ds =Zh 0 AeA(t+h−s)(g(s)−g(t+h))ds +Zt 0 AeA(t−s)(g(s+h)−g(s)−g(t+h) + g(t)) | {z } (∗) ds. 6 M. C. BORTOLAN, A. N. CARVALHO, P. MAR´ IN-RUBIO, AND J. VALERO For the term (∗) (2.10) kg(s+h)−g(s)−g(t+h) + g(t)kXγ() ⩽kg(s+h)−g(s)kXγ()+kg(t+h)−g(t)kXγ()⩽K(s)hδ+K(t)hδ, and (2.11) kg(s+h)−g(s)−g(t+h) + g(t)kXγ() ⩽kg(s+h)−g(t+h)kXγ()+kg(t)−g(s)kXγ() ⩽K(s+h)(t−s)δ+K(s)(t−s)δ. Choosing ω∈(0,1) such that µ < δω, interpolating (2.10) and (2.11) we obtain kg(s+h)−g(s)−g(t+h) + g(t)kXγ() ⩽(K(s) + K(t))1−ω(K(s+h) + K(s))ωh(1−ω)δ(t−s)δω. Thus kH(t+h)−H(t)kXγ()+µ ⩽MZh 0 K(s)(t+h−s)−µ+δ−1ds +Mhδ(1−ω)Zt 0 (K(s) + K(t))1−ω(K(s+h) + K(s))ω(t−s)−µ+δω−1ds ⩽MCt−µ+δ−1hϕ +Mhδ(1−ω)Zt 0 (K(s) + K(t))1−ω(K(s+h) + K(s))ω(t−s)−µ+δω−1ds. Using −µ+δω −1<−1, K∈L1(0, T ) and H¨older’s inequality with exponents p= 1/ω, p0= 1/(1 −ω) we easily obtain that Zt 0 (K(s) + K(t))1−ω(K(s+h) + K(s))ω(t−s)−µ+δω−1ds < ∞ for t∈(0, T) fixed. Also, the map (0, T)3t7→ eAtg(t)∈Xγ()+µis locally H¨older continuous. Indeed,  eAtg(t)−eAsg(s) Xγ()+µ⩽ eAt −eAsg(t) Xγ()+µ+ eAs (g(t)−g(s)) Xγ()+µ ⩽Cµ+γ()|t−s|s−1−µ−γ()kg(t)kX+Ms−µmin{K(s), K(t)} |t−s|δ, where we have used (2.3), the locally H¨older property of g(·) and the Lipschitz continuity of eAtg(t) (see [24, p.97, (37.13)]). Hence, the result follows.  Lastly, we will need the following singular version of the Gr¨onwall inequality, which proof is analogous to [15, Lemma 7.1.1]. Lemma 2.6 (Singular Gr¨onwall Inequality).Let a, b, c ⩾0,0⩽α, β, γ < 1and u: (0, T )→Rbe an integrable function with (2.12) 0 ⩽u(t)⩽at−α+bt−β+cZt 0 (t−s)−γu(s)ds for a.e. t∈(0, T). Then, there exists a constant K⩾0that depends only on c, γ, T such that u(t)⩽K 1−αat−α+K 1−βbt−βfor a.e. t∈(0, T). With these lemmas, we can present the main result of this section. Theorem 2.7. Let x0∈X1,fbe an -regular map relative to the pair (X1, X0)and x: [0, τ0]→X1be the unique -regular mild solution of (2.6). Then for each 0< η < γ()−the map (0, τ0)3t7→ dx dt (t)∈Xγ()+η is locally H¨older continuous and there exists a constant C⩾0such that   dx dt (t)  Xγ()+η ⩽Ct−γ()−ηfor t∈(0, τ0). In particular, (0, τ0)3t7→ dx dt (t)∈Xγ()+ηis in Lp(0, τ0;Xγ()+η), provided that p(γ() + η)<1. WEAK GLOBAL ATTRACTORS 7 Proof. Fix τ∈(0, τ0) and, for t∈(0, τ0−τ], we define g(t) = f(x(t+τ)). Since x∈C([τ, τ0], X1+) and fis an -regular map relative to the pair (X1, X0), for t, s ∈(0, τ0−τ] we have kg(t)−g(s)kXγ()=kf(x(t+τ)) −f(x(s+τ))kXγ() ⩽ckx(t+τ)−x(s+τ)kX1+(1 + kx(t+τ)kρ−1 X1++kx(s+τ)kρ−1 X1+) ⩽C1kx(t+τ)−x(s+τ)kX1+, and kg(t)kXγ()=kf(x(t+τ))kXγ()⩽c(kx(t+τ)kρ X1++ 1) ⩽C2, for some constants C1, C2⩾0. Now, for 0 < t < t +h⩽τ0−τwe have x(t+τ+h)−x(t+τ) = (eAh −I)eAtx(τ) + Zh 0 eA(t+h−s)g(s)ds +Zt 0 eA(t−s)(g(s+h)−g(s))ds. Since (eAh −I)eAtx(τ) = Zh 0 d dreA(t+r)x(τ)dr =Zh 0 AeA(t+r)x(τ)dr, we have k(eAh −I)eAtx(τ)kX1+⩽Mhtη−1kx(τ)kX1++η, and for the second term we obtain Zh 0 keA(t+h−s)g(s)kX1+ds ⩽MC2htγ()−−1, where we have used (2.1) and (2.3). For the last term, we have Zt 0 keA(t−s)(g(s+h)−g(s))kX1+ds ⩽MZt 0 (t−s)γ()−−1kg(s+h)−g(s)kXγ()ds, therefore kg(t+h)−g(t)kXγ()⩽C1kx(t+τ+h)−x(t+τ)kX1+ ⩽MC1htη−1kx(τ)kX1++η+MC1C2htγ()−−1 +MC1Zt 0 (t−s)γ()−−1kg(s+h)−g(s)kXγ()ds. From Lemma 2.6 there exists a constant K⩾0 such that kg(t+h)−g(t)kXγ()⩽Khtη−1kx(τ)kX1++η+tγ()−−1. Setting K(s) = Ksη−1kx(τ)kX1++η+sγ()−−1for s∈(0, τ0−τ], we see that kg(t)−g(s)kXγ()⩽min{K(t), K(s)} |t−s| and, moreover, Zr 0 K(s)ds =Krη ηkx(τ)kX1++η+rγ()− γ()−. Hence, it follows from Lemma 2.5 with δ= 1 that (2.13)     dG dt (t)   Xγ()+η ⩽Mt−ηkg(t)kXγ()+MZt 0 K(s)(t−s)−ηds ⩽MC2t−η+MKB(η, 1−η)kx(τ)kX1++η +MKB(γ()−, 1−η)tγ()−−η, since 0 < η < γ()− < 1 (here Bdenotes the beta function), and that (0, T)3t7→ dG dt (t)∈Xγ()+ηis locally H¨older continuous. Since kAeAtx(τ)kXγ()+η⩽Mt−γ()+kx(τ)kX1++η, together with (2.13) and the fact that dx dt (t+τ) = AeAtx(τ) + dG dt (t), we obtain a constant C3⩾0 such that     dx dt (t+τ)   Xγ()+η ⩽C3ht−η+t−γ()+kx(τ)kX1++η+tγ()−−ηi, and that (0, T)3t7→ dx dt (t)∈Xγ()+ηis locally H¨older continuous. From Lemma 2.3, since 0 < η < γ()−, there exists a constant C⩾0 such that kx(τ)kX1++η⩽Cτ−−η, 8 M. C. BORTOLAN, A. N. CARVALHO, P. MAR´ IN-RUBIO, AND J. VALERO and, therefore, there exists a constant C4⩾0 such that     dx dt (t+τ)   Xγ()+η ⩽C4[t−η+t−γ()+τ−−η+tγ()−−η] for all τ∈(0, τ0) and t∈(0, τ0−τ). From this, for each t∈(0, τ0), replacing both tand τby t 2in the previous estimate it follows immediately that     dx dt (t)   Xγ()+η ⩽4C4[t−η+t−γ()−η+tγ()−−η]⩽C5t−γ()−η, for all t∈(0, τ0), for some constant C5⩾0.  3. Very weak formulation of the globally modified Navier-Stokes equation In this section we show that the nonlinearities of the GMNSE are 3 8-regular maps, obtain the very weak formulation of the GMNSE, prove Theorem 1.1, and guarantee the existence of the global attractor in Hfor the GMNSE. 3.1. Nonlinearity of the GMNSE. We begin with some results involving the map fNdefined in (1.3), to study the nonlinear term FN(u)(u· ∇u) of the GMNSE. We note that, from [5, Lemma 4], for every N > 0 and s, t ⩾0 with s+t > 0 we have (3.1) |fN(s)−fN(t)|⩽1 max{s, t}|s−t|. Writing, for r⩾1, Lr=Lr(Ω)3, to simplify the notation, we point out that for each N > 0 and u∈L4we have (3.2) FN(u)kukL4⩽N, and, for any N > 0 and u, v ∈L4(Ω) with kukL4(Ω) +kvkL4(Ω) >0, |FN(u)−FN(v)|⩽1 max{kukL4(Ω),kvkL4(Ω)}ku−vkL4(Ω), just using (3.1) and noting that |kukL4(Ω) − kvkL4(Ω)|⩽ku−vkL4(Ω). Before continuing, we note that the tensor product in R3will be of great importance and we recall it here. Here, M3(R) denotes the space of 3 ×3 real matrices and | · |M3(R)stands for its norm, given as the sum of the absolute value of all its entries. The tensor product between u= (u1, u2, u3), v = (v1, v2, v3)∈R3is the 3×3 real matrix given by u⊗v=  u1v1u1v2u1v3 u2v1u2v2u2v3 u3v1u3v2u3v3 = (uivj)3 i,j=1. Clearly, the following properties hold: (i) (u+v)⊗w=u⊗w+v⊗w; (ii)|u⊗v|M3(R)⩽|u|R3|v|R3; (iii)u⊗v= (v⊗u)t, where Atdenotes the transposed matrix of A∈M3(R). Lemma 3.1. For all u, v ∈L4, we have kFN(u)u⊗u−FN(v)v⊗vkL2⩽3Nku−vkL4, and kFN(u)u⊗ukL2⩽NkukL4. Proof. Note that kFN(u)u⊗ukL2⩽FN(u)kukL4kukL4⩽NkukL4, where in the last inequality we used (3.2). Now we deal with the first estimate. Firstly, we treat the case where kukL4,kvkL4⩽N. In this case, we have FN(u) = FN(v) = 1 and kFN(u)u⊗u−FN(v)v⊗vkL2=ku⊗u−v⊗vkL2⩽ku⊗(u−v)kL2+k(u−v)⊗vkL2 ⩽kukL4ku−vkL4+ku−vkL4kvkL4⩽(kukL4+kvkL4)ku−vkL4⩽2Nku−vkL4. WEAK GLOBAL ATTRACTORS 9 Now we assume that kvkL4,kukL4⩾N. We can assume, without loss of generality, that kvkL4⩾kukL4. If this case, we have kFN(u)u⊗u−FN(v)v⊗vkL2=N  u⊗u kukL4 −v⊗v kvkL4  L2 ⩽Nku⊗(u−v)kL2 kukL4 +Nku⊗vkL21 kukL4 −1 kvkL4+Nk(u−v)⊗vkL2 kvkL4 ⩽NkukL4ku−vkL4 kukL4 +NkukL4kvkL4 |kukL4− kvkL4| kukL4kvkL4 +NkvkL4ku−vkL4 kvkL4 ⩽3Nku−vkL4. For the last case, we can assume, without loss of generality, that kukL4⩽N⩽kvkL4. Hence FN(u)u⊗u−FN(v)v⊗v=1 kvkL4kvkL4u⊗u−Nv ⊗v =1 kvkL4kvkL4u⊗(u−v)+(kvkL4−N)u⊗v+N(u−v)⊗v. Estimating the L2norm and using the fact that kukL4⩽Nand that kvkL4−N⩽kvkL4− kukL4⩽ku−vkL4, we obtain kFN(u)u⊗u−FN(v)v⊗vkL2⩽3Nku−vkL4, and the proof is complete.  We recall the following Sobolev type embeddings for a bounded domain Ω ⊂R3with smooth boundary ∂Ω. Given s > 0 we consider mthe smallest integer greater than or equal to s. Then [Lp(Ω)3, Wm,p(Ω)3]s m=Hs,p(Ω)3. Additionally, for 1 ⩽p⩽q= 3p/(3 −sp), we have (3.3) Hs,p(Ω)3⊂Lq(Ω)3. Hence, we arrive at the following result without having to know explicitly Hα. Proposition 3.2. If α∈(0,1) and q=6 3−4α, we have Hα⊂H2α,2(Ω)3and Hα⊂Lq(Ω)3. Proof. By construction D(A)⊂W2,2(Ω)3and since Ais sectorial, the inclusion holds with equivalent norms. Interpolating these spaces we arrive at Hα⊂H2α,2(Ω)3and apply (3.3).  Now we define the spaces X1and X0that we use for the rest of the paper. Set (3.4) X1=Hand X0=H−1. Hence, choosing (3.5) =3 8and γ() = 1 2, we have Xγ()=X1 2=H−1 2and X1+=X11 8=H3 8. Note that from Proposition 3.2 we obtain X1+=H3 8⊂L4. Also, we point out that, from (1.4), Xγ()=H−1 2= (H1 2)∗=V∗. Theorem 3.3. For each N > 0, the map GN:X1+→Xγ()given by (3.6) GN(u)φ=ZΩ FN(u)u⊗u· ∇φfor u∈X1+and φ∈V, is a well-defined 3 8-regular map relative to the pair (X1, X0), and we can choose any ρ∈(1,4 3]. Proof. Since X1+⊂L4, for u∈X1+and φ∈V, using Lemma 3.1 we have |GN(u)φ|=ZΩ FN(u)u⊗u· ∇φ⩽kFN(u)u⊗ukL2kφkV ⩽NkukL4kφkV⩽NkkukH3 8 kφkV, where kis the embedding constant of H3 8into L4. Then, we obtain |GN(u)φ|⩽NkkukX1+kφkV, 16 M. C. BORTOLAN, A. N. CARVALHO, P. MAR´ IN-RUBIO, AND J. VALERO Proof. Denote B=φ(R) and fix t0∈R. Denote uτ(·) = φ(·+t0−τ) for τ⩾0. Then, as φ(t0) = uτ(τ), we deduce by the attracting property of Athat distw(φ(t0),A) = distw(uτ(τ),A) ⩽sup u∈KN, u(0)∈B distw(u(τ),A)τ→∞ −→ 0. Therefore, φ(t0)∈ A. Since t0∈Ris arbitrary, the result follows.  Theorem 4.10. The weak global attractor Acan be characterized by the union of all bounded complete trajectories of (1.1)in KN , that is, A={φ(0): φis a bounded complete trajectory of (1.1)in KN}. Proof. Let z∈ A be arbitrary. We take a sequence tn n→∞ −→ ∞. Since Ais negatively invariant, there exist un∈ KN such that un(tn) = zand un(0) ∈ A. We define the functions v0 n(·) = un(·+tn), which belongs to KN by the translation property. As the attractor is positively invariant, v0 n(t)∈ A for all t⩾0. In view of Lemma 4.8, up to a subsequence, v0 n(sn)n→∞ −→ v0(t)∈ A weakly in Hfor all t⩾0 and [0,∞)3sn n→∞ −→ t, where v0(·)∈ KN . In addition, v0(0) = z. Consider now the sequence v1 n(·) = un(·+tn−1), which again belongs to KN and satisfies v1 n(t)∈ A for all t⩾0. Using again Lemma 4.8, up to a subsequence we have v1 n(sn)n→∞ −→ v1(t)∈ A weakly in Hfor all t⩾0 and [0,∞)3sn n→∞ −→ t, where v1(·)∈ KN . It follows that v1(t+ 1) = v0(t) for all t⩾0 as well. In this way, we construct inductively a sequence of solutions from KN, denoted by {vj(·)}j∈N, such that vj(t)∈ A and vj(t+ 1) = vj−1(t) for all t⩾0 and j. Define φ:R→Hto be at any tthe common value of the functions vj, that is, φ(t) = vj(t+j) if t⩾−j. Then φis a bounded complete trajectory of (1.1) in KN. Thus, A⊂{φ(0): φis a bounded complete trajectory of (1.1) in KN }. The converse inclusion follows from Lemma 4.9, and the result is proved.  We observe that Lemma 4.4 implies any complete trajectory is a Leray-Hopf solution defined in the whole line (−∞,+∞) and satisfying the energy inequality almost everywhere for real moments of time. Thus, the following result follows immediately. Corollary 4.11. The global attractor Abelongs to the universal attractor Awfrom [14]. Also, it is not difficult to show that the global attractor AR0defined in [16] in the ball of radius R0=kfk2 ν2λ2 1 coincides with the universal attractor Aw, so A⊂AR0as well. Finally, we prove that the global attractors AN of (3.9) behave upper semicontinuously with respect to the weak global attractor Aof (1.1). Proposition 4.12. If ANis the global attractor of SNthen distw(AN,A)N→∞ −→ 0. Proof. If Nj j→∞ −→ ∞ and uj∈ ANjfor each j∈N, since ∪N>0AN⊂B0, there exists u0∈Hsuch that, up to a subsequence, uj→u0weakly in H. For each ujthere exists a global solution ξjof the semigroup SNjwith ξ(0) = ujand ξ(R)⊂ ANj. Hence SNj(j)ξj(−j) = ξj(0) = uj j→∞ −→ u0weakly in H. This shows that u0∈ A. Hence, distw(AN,A)N→∞ −→ 0.  Declarations Data Availability. Data sharing is not applicable to this article as no new data were created or analyzed in this study. Competing Interests. The authors have not disclosed any competing interests. References [1] H. Amann. Linear and quasilinear parabolic problems. Vol. I, volume 89 of Monographs in Mathematics. Birkh¨auser Boston, Inc., Boston, MA, 1995. Abstract linear theory. [2] J. M. Arrieta and A. N. Carvalho. 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Email address:[email protected] (A. N. Carvalho) Instituto de Ciˆ encias Matem´ aticas e de Computac¸˜ ao, Universidade de S˜ ao Paulo, Campus de S˜ ao Carlos, Caixa Postal 668, S˜ ao Carlos SP, Brazil. Email address:[email protected] (P. Mar´ın-Rubio) Departamento de Ecuaciones Diferenciales y An´ alisis Num´ erico, Universidad de Sevilla, C/Tarifa s/n, 41012-Sevilla, Espa˜ na. Email address:[email protected] (J. Valero) Departamento de Estad´ ıstica, Matem´ aticas e Inform´ atica, Universidad Miguel Hernandez de Elche, Elche - Alicante, Espa˜ na. Email address:[email protected]