# Asymptotic Behavior of the Solutions of the Generalized Globally Modified Navier–Stokes Equations

Authors: Yonghong Duan; Xiaojuan Chai
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

The paper is concerned with the existence and the asymptotic behavior of solutions to a class of generalized Navier–Stokes equations, which generalises the so-called globally modified Navier–Stokes equations. The existence and uniqueness of solutions are proved under different assumptions on the dissipation and modification factors. For the asymptotic behavior of solutions, we prove the existence of global attractors in proper spaces. The results generalize some results derived in our previous work Ann. Polon. Math. 122(2):101–128(2019).


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## Body

## 1. Introduction

Well-posedness of 3D Navier–Stokes equation is one of the most challenging problems in modern mathematics [1, 2]. To understand or approximate Navier–Stokes equations, various kinds of modified models were introduced in different contexts, such as the Navier–Stokes-α models introduced by Chen et al. and Ilyin and Titi [3, 4], the Leray-α, Clark-α, and simplified Bardina models introduced by Titi et al. [5–7], and some other modified Navier–Stokes equations introduced and studied, respectively, by Caraballo and Kloeden, Constantin, Sohr, and Flandoli et al., see [8–12].In [8], the authors proposed the global modified Navier–Stokes equations:(1)∂tu−νΔu+FN∇uu⋅∇u+∇P=f,∇⋅u=0,where FNr=min1,N/r,r∈ℝ+,N∈ℝ+. Since the modifying factor FNΛβu decreases the singularity of the quadratic convection term u⋅∇u, it allows the authors to derive the existence and uniqueness of global solutions [8]. Following [8], the existence results and the asymptotic behaviors of solutions to problem (1) were extensively studied in different contexts, see e.g., [13–21] and the review paper [22].Recently, Dong and Song [23] studied the globally modified Navier–Stokes equations with fractional dissipation in the whole space ℝ3:(2)∂tu+νΛ2αu+FN∇uu⋅∇u+∇P=0,∇⋅u=0.The existence and uniqueness of global solutions was obtained under the assumptionα>3/4, see also [24], for the existence and uniqueness results in a bounded domains. These results review that the modifying factor FNΛβu decreases the singularity of the term u⋅∇u “too much” so that one can control the nonlinear term by using only the fractional dissipation −Δαu,α<1 rather than Δu in (1). This inspires us to weaken the modification term and to investigate that how the dissipation and modification terms interact with each other to determine the existence and uniqueness of the solutions. Precisely speaking, we shall consider the following modified Navier–Stokes equations in Ω=0,L3:(3)∂tu+νΛ2αu+FNΛβuu⋅∇u+∇P=f,∇⋅u=0,u0,x=u0x,with periodic boundary conditions, where the constants ν>0 and α,β≥0.Assume that the initial data and the forcing term are mean-free functions, i.e.,(4)∫Ωu0dx=∫Ωfdx=0.Then, the solution is also a mean-free function, and the Poincaré inequality holds. We prove that system (3) admits at least one global weak solution when 4α+2β>5,u0,f∈ℍ. Moreover, if α>βand4α2−5α+2β2≥0or2α+4β>5, the weak solution is unique. On the contrary, we prove that when 4α+2β>5,u0∈ℍs,f∈ℍs−α,ands≥β, the system possesses a unique global strong solution and the existence of global attractor A, for the solution semigroup in ℍs can be proved when s≥β. Furthermore, if s≥max1,β, we can give explicit upper bound for the fractal dimension of the attractor A. When β=1, we prove that the system admits at least one weak solution when 3/4<α≤1,u0,f∈ℍ, and a unique global strong solution when 3/4<α≤1,u0∈ℍs,f∈ℍs−α,ands≥1. The existence and uniqueness in [24] are consistent with the results in this study. If β=0, the modifying factor FNΛβu is constant 1, and system (3) becomes the well-known generalized Navier–Stokes equation. The standard existence result shows that the system has global regularity when α≥5/4 [25], which are consistent with our result. These results extend the previous results in [8, 23, 24] to more general settings.The rest of the paper is organized as follows. In Section2, we provide some preliminaries about the function spaces and several useful lemmas. Then, in Section 3, we prove the existence and uniqueness results of solutions, while in Section 4 and Section 5, we discuss the existence of a global attractor and the upper bound of its fractal dimension.


## 2. Preliminaries and Inequalities

LetΩ=0,L3. The fractional operator Λ2α=−Δα for any α∈ℝ can be defined as(5)Λ2αfξ^=∑ξ∈ℤ3ξ2αf^ξeiξ⋅x,for any tempered distribution f, where f^ξ is the Fourier transform of fx. Especially, Λ=−Δ1/2. Let C˙p∞Ω be the space of restrictions to Ω of infinitely differentiable functions that are L-periodic in each direction and with zero mean in Ω. For s∈ℝ, we denote by HsΩ the closure of C˙p∞Ω under the norm(6)fHs=ΛsfL2=∑ξ∈Z3ξ2sf^ξ21/2,that is, the space of periodic functions with zero mean such that ‖f‖Hs<∞. It is obvious that Hs1Ω⟶Hs2Ω(compact imbedding), for any s1>s2. Moreover, for p∈1,∞, we denote by Hs,pΩ the space of periodic mean-free LpΩ functions φ, which can be written as φ=Λ−sψ, with ψ∈Lp. This is normed by φHs,p=ΛsφLp.Fors∈ℝ, we denote(7)ℍs=u∈HsΩ3,div,u=0,ℍs,p=u∈Hs,pΩ3,div,u=0.Particularly, whens=0, we denote ℍ0 by ℍ for short. In this study, for any Banach space X, we denote its norm as ⋅X; particularly, ⋅L2 will be abbreviated as ⋅.Now, we recall the definitions of the global attractor and the fractal dimension, see [26, 27].Definition 1.
LetStt≥0 be a semigroup on a Banach space X. A subset A⊂X is called a global attractor for the semigroup if A enjoys the following properties:(i)
A is compact in X.(ii)
A is invariant, i.e., StA=A, for any t≥0.(iii)
A attracts every bounded subset of X, i.e., ∀B⊂Xbounded,limt⟶∞distStB,A=0, where dist is the Hausdorff semidistance between sets in X, defined as(8)distA,B=supa∈Ainfb∈Ba−bX,∀A,B⊂X.Definition 2.
The fractal dimension of a compact setK in a Banach space X is defined as(9)dfK=limsupϵ⟶0logNϵK−logϵ,where NεK is the minimal number of balls of radius ε in X needed to cover K.The following inequalities may be found in [26, 28].Lemma 1.
(Young’s inequality). For any positive constantsa,b,andε and any 1<p<∞, it holds that(10)ab≤εpap+p−1pε1/p−1bp/p−1.Lemma 2.
(Poincare’s inequality). LetΩ be a bounded Lipschitz domain in Rd and let p be a continuous seminorm on H1Ω, which is a norm on the constants. Then, there exists a constant c depending only on Ω such that(11)uL2Ω≤CΩ∇uL2Ω+pu,∀u∈H1Ω.Lemma 3.
(Gagliardo–Nirenberg inequality). Let1<p,q,r≤∞,0≤j<m,j/m≤λ≤1. For any u∈Wm,pΩ∩LqΩ, there exists a constant C such that(12)DjuLr≤CDmuLpλuLq1−λ,where p,q,r,n,m,j,andλ satisfy(13)1r−jn=λ1p−mn+1−λ1q.The following product estimates play an essential role in our analysis (see [29]).Lemma 4.
Suppose thatf,g∈S the Schwartz class. Then, for s>0,1<p<+∞, there exist a positive constant C such that(14)ΛsfgLp≤CfLp1ΛsgLq1+ΛsfLp2gLq2,with q1,p2∈1,+∞ satisfying(15)1p=1p1+1q1=1p2+1q2.The following lemma will play an important role in the proof of our result. It was first proved by Romito in [30] for the case β=1. The general case can be proved similarly.Lemma 5.
For everyu,v∈ℍ1 and each N>0, we have(16)0≤FNΛβuΛβu≤N,FNΛβu−FNΛβv≤FNΛβuFNΛβvNΛβu−Λβv.


## 3. Existence and Uniqueness Results

We now give the definition of weak solutions to system (3).Definition 3.
Letu0,f∈ℍ. A function u is called a weak solution to system (3) if(17)u∈L∞0,T;ℍ∩L20,T;ℍα,∂tu∈L20,T;ℍ−α,for allT>0,and for any function φ∈ℍα and any T>0, it holds that(18)∫ΩuTφdx+∫0T∫ΩFNΛβuu⋅∇uφdxds+ν∫0T∫ΩΛαuΛαφdxds=∫0T∫Ωfφdxds+∫Ωu0φdx.Remark 1.
Obviously, ifut is a weak solution of system (3), then u∈C0,T;ℍ, see [26, 27].Theorem 1.
Letαandβ be two constants such that 4α+2β>5,0≤α<5/4,0≤β<3/2. (i) If u0,f∈ℍ, there exists at least one weak solution ut to problem (3) with(19)u∈L∞0,T;ℍ∩L20,T;ℍαfor allT>0.If in addition4α2−5α+2β2≥0or2α+4β>5, the weak solution is unique. (ii) On the other hand, if u0∈ℍs,f∈ℍs−α,ands≥β, then problem (3) admits a unique global solution u satisfying(20)ut∈L∞0,T;ℍs∩L20,T;ℍs+α∩C0,T;ℍs,for allT≥0.Remark 2.
The standard existence result for the Navier–Stokes equations shows that system (3) possess a unique global solution, for all β=0, when α≥5/4, so we only consider the case α<5/4. However, when α≤1/2, we cannot use the dissipation term of the equations to control the nonlinear term, and the existence results is difficult to prove in this case.Proof.
Let us divide the proof into several steps.Step 1:
we prove the existence of the weak solution by the Galerkin approximation method. Letϕjj=1∞ be an orthonormal basis of ℍ consisting of eigenfunctions of the Stokes operator Aandλj are the corresponding eigenvalues which are increasing with j. Consider the following ordinary differential system:(21)dumdt+νΛ2αum+PmFNΛβumum⋅∇um=Pmf,um0=Pmu0,where um=∑j=1mcjmtϕj,Λ2αum=∑j=1mλjαcjmtϕj,Λβum=∑j=1mλjβ/2cjmtϕj, and Pm is the orthogonal projection form ℍ onto the space spanned by ϕ1,ϕ2,…ϕm. By the standard existence theorem for ordinary differential equations, for each m, there exists a local solution um to system (21) in the interval 0,Tm.
Multiplying (21) by umt, using the Poincaré inequality λ1αum2≤Λαum2, we can deduce that(22)ddtum2+νΛαum2≤1νλ1αf2,and integrating from 0 to t, we obtain(23)umt2+ν∫0tΛαums2ds≤tνλ1αf2+um02,∀t≥0.
Using Gronwall’s inequality, we obtain(24)umt2≤um02e−νλ1αt+f2ν2λ12α1−e−νλ1αt≤u02+f2ν2λ12α,which implies that(25)umis bounded inL∞0,T;ℍ∩L20,T;ℍα.
Let us perform the estimates for∂um/∂t. For any φ∈ℍα, using Hölder’s inequality, the product estimates (see Lemma 1), the Gagliardo–Nirenberg inequality, and Young’s inequality, we deduce that(26)FNΛβum∫Ωum⋅∇umφdx≤FNΛβumΛ−αum⋅∇umΛαφ≤FNΛβumΛ1−αumumΛαφ≤CFNΛβumumL6/3−2βΛ1−αumL3/βΛαφ≤CΛαumΛαφ.
The last inequality holds since(27)12−β3=3−2β6,12−α3≤β3−1−α3.
In view of (25), the sequence FNΛβumum⋅∇um is bounded in L20,T;ℍ−α. Obviously, −νΛ2αum and Pmf are bounded in L20,T;ℍ−α. Hence, from (21), we conclude that(28)∂um∂tis bounded inL20,T;ℍ−α.
Using the standard Aubin-Simon-type compactness results [26, 27], there exists an element(29)u∈L20,T;ℍα∩L∞0,T;ℍ,for allT>0,such that up to subsequences,(30)um⟶ustrongly inL20,T;ℍ,um⟶ua.ein0,T×Ω,um⟶uweakly inL20,T;ℍα,∂um∂t⟶∂u∂tweakly∗inL20,T;ℍ−α.
Now, it remains to verify thatu is a weak solution to problem (3). We treat the case α>β and α≤β separately.Case 1.
α>β≥0. By the Gagliardo–Nirenberg inequality and Hölder’s inequality [27], we deduce that(31)∫0TΛβum−u2ds≤C∫0Tum−u2θ1Λαum−u2−2θ1ds≤C∫0Tum−u2dsθ1∫0TΛαum−u2ds1−θ1,where θ1=α−β/α. This, together with (25) and (30), implies that(32)um⟶ustrongly inL20,T;ℍβ.
Thus, up to subsequences,(33)Λβum⟶Λβua.e in0,Tfor anyT>0.
And, hence,(34)FNΛβum⟶FNΛβua.e.in0,Tfor anyT>0.
Thanks to (30) and (34), taking φ∈ℍα as a test function in (21) and passing to the limits, we obtain that u is a weak solution to system (3). As the calculations are rather similar to those in [8], we omit the details for concision.Case 2.
β≥α. Assume that there exist a positive integer N0 such that N0α≤β<N0+1α, without lose of generality, we set N0=1. After multiplying equation (21) with Λ2αum and integrating, we obtain(35)12ddtΛαum2+νΛ2αum2≤FNΛβumΛumumΛ2αum+1νf2+ν4Λ2αum2.
We can estimate the first term of the right side as follows. Sinceβ+2α>5/2,α>1/2, using the product estimates (see Lemma 1), the Gagliardo–Nirenberg inequality, and Young’s inequality, we have the estimate(36)FNΛβumΛumumΛ2αum≤CFNΛβumumL6/3−2βΛumL3/βΛ2αum≤Cumθ4Λ2αum2−θ4≤ν4Λ2αum2+Cum2,where θ4=min1,4α+2β−5/4α. Combining with (24), (35), and (36), we obtain(37)ddtΛαum2+νΛ2αum2≤2νf2+Cum2≤2ν+Cν2λ12αf2+Cu02.
For anyt≥τ≥0, integrating (22) between t and t+τ and using (24), we obtain(38)ν∫tt+τΛαum2ds≤u02+f2νλ1ατ+1νλ1α.
Set(39)a2=2ντu02+f2νλ1ατ+1νλ1α,Ωm=s∈t,t+τ:Λαum≥a,and denote by Ωm the Lebesgue measure of Ωm. We have(40)a2Ωm≤∫ΩmΛαum2ds≤∫tt+τΛαum2ds≤1νu02+f2νλ1ατ+1νλ1α=τa22,which implies that Ωm≤τ/2. Therefore, for any given ε>0, there exist a t0∈0,ε such that(41)Λαumt02≤2νεu02+f2νλ1αε+1νλ1α.
By using the Gronwall inequality, we obtain, for allt≥ε,(42)Λαumt2≤Λαumt02e−νλ1αt−t0+2ν2λ1α+Cν3λ13αf2+Cνλ1αu02.
Integrating (37) from ε to T and taking (42) into consideration, we deduce that(43)Λαumt2+∫εTΛ2αumt2ds≤∫εT1ν+Cν2λ12αf2+Cu02ds+Λαumε2≤2ν+Cν2λ12αf2+Cu02T−ε+2νεu02+f2νλ1αε+1νλ1α+2νλ1αCN,ν,α,βu02+f2νλ1α2+2f2ν,for all ε≤t≤T. Thus, we have(44)umis bounded inL∞ε,T;ℍα∩L2ε,T;ℍ2α,for anyT>ε.
TakingΛ4αum,Λ6αum,…,ΛN0αum as test functions and performing similar analysis, we may prove that(45)umis bounded inL∞ε,T;ℍN0α∩L2ε,T;ℍN0+1α.
Denotesα1=N0+1α; since β<α1, we deduce that(46)∫ϵTΛβum−u2ds≤C∫ϵTum−u2δ1Λα1um−u2−2δ1ds≤C∫ϵTum−u2dsδ1∫ϵTΛα1um−u2ds1−δ1,with δ1=α1−β/α1. Therefore,(47)um⟶ustrongly inL2ε,T;ℍβ.
Thus, up to subsequences,(48)Λβum⟶Λβua.einε,Tfor any T>ε>0.
By the standard diagonal process, we can extract a subsequence ofum (still labeled by um) such that(49)Λβum⟶Λβua.e.in0,Tfor anyT>0.
Then, using (30) and (34), we can take limits in (21) as in Case 1 to obtain that u is a solution of (3).Step 2:
now, we prove that if4α2−5α+2β2≥0or2α+4β>5, the weak solution is unique. Let u and v be two solutions to system (3) corresponding to the initial condition u0,v0, respectively. Set w=u−v and let P be the Helmholtz-Leray project operator [27]. It is easy to check that w satisfies(50)wt+νΛ2αw+FNΛβuPv⋅∇w+FNΛβuPw⋅∇u+FNΛβu−FNΛβvPv⋅∇v=0.
Multiplying (50) by w and integrating, we obtain(51)12ddtw2+νΛαw2≤FNΛβu∫0Tw⋅∇uwdx|+FNΛβu−FNΛβv∫0Tv⋅∇vwdx,where we have used the fact ∫0Tv⋅∇wwdx=0. Since 4α+2β>5 and α>1/2,β≥0, we have α+β>1. Using Hölder’s inequality and the product estimates (see Lemma 1) and the Gagliardo–Nirenberg inequality, we have the first term of the right side(52)FNΛβu|∫0Tw⋅∇uwdx|≤FNΛβuΛβuΛ1−βww≤CΛ1−βwL3/αwL6/32α≤Cwθ2Λαw2−θ2≤Cw2+ν4Λαw2,where θ2=min1,4α+2β−5/2α. For the second term of the right side in (51), using Lemma 2 and Gagliardo–Nirenberg inequality, we have(53)FNΛβu−FNΛβv≤FNΛβuFNΛβvNΛβw≤FNΛβuFNΛβvNΛβw≤FNΛβuFNΛβvNwθ1Λαw1−θ1,with θ1=α−β/α. When 2α+4β≥5, we can always find p,q=3/β,6/3−2β such that 1/p+1/q=1/2 and(54)1p−1−α3≥12−β3,1q=12−β3.
Thanks to the product estimates (see Lemma1) and the fractional Sobolev inequality, we have(55)∫0Tv⋅∇vwdx=∫Ωv⋅∇uwdx≤CΛαwΛ1−αuLpvLq+Λ1−αvLpuLq≤CΛαwΛβuΛβv+ΛβvΛβu.
Combining (53) and (55) and using Young inequality, we obtain(56)FNΛβu−FNΛβv∫0Tv⋅∇vwdx≤CFNΛβuFNΛβvNwθ1Λαw2−θ1ΛβuΛβv+ΛβvΛβu≤Cw2+ν4Λαw2.
Hence, if2α+4β≥5, from (56), (52), and (51), we obtain that(57)ddtw2+Λαw2≤Cw2,from which the uniqueness result follows easily.
On the contrary, if2α+4β<5, setting p=6/3−2βandq=3/β, we have(58)∫0Tv⋅∇vwdx=∫Ωv⋅∇uwdx≤CΛαwΛ1−αuL3/βvL6/3−2β+Λ1−αvL3/βuL6/3−2β≤CΛαwΛαuθ3Λβu1−θ3Λβv+Λαvθ3Λβv1−θ3Λβu≤CΛαwΛαuθ3+Λαvθ3Λβu1−θ3Λβv+Λβv1−θ3Λβu,where θ3 satisfies(59)12−α3θ3+12−β31−θ3=β3−1−α3i.e.θ3=−4β−2α+52α−2β>0.
Note that when4α+2β>5 and α>β≥0, we have θ1=α−β/α≥θ3=5−2α−4β/2α−2β iff 4α2−5α+2β2≥0. Combining (53) and (58), we have(60)FNΛβu−FNΛβv∫0Tv⋅∇vwdx≤Cwθ1Λαw2−θ1Λαuθ3+Λαvθ3≤ν4Λαw2+CΛαu2θ3/θ1+Λαv2θ3/θ1/θ1w2,which, combined with (51) and (52), implies that when 2α+4β<5 and 4α2−5α+2β2≥0:(61)ddtw2+Λαw2≤Cw2Λαu2+Λαv2+1.
By using Gronwall’s inequality, we obtain(62)wt2≤w02expC∫0tΛαu2+Λαv2+1ds.
The uniqueness result follows easily.Step 3:
we now prove the second part of the theorem. Ifu0∈ℍs,f∈ℍs−α,ands≥β, we multiply (21) by Λ2sum to deduce that(63)12ddtΛsum2+νΛs+αum2≤fℍs−αΛs+αum+FNΛβum∫Ωum⋅∇umΛ2sumdx≤1νfℍs−α2+ν4Λs+αum2+FNΛβumΛs+1−αumumΛs+αum.
Using the product estimates and the Gagliardo–Nirenberg inequality, we deduce that(64)FNΛβumΛs+1−αumumΛs+αum≤CFNΛβumumL6/3−2βΛs+1−αumL3/βΛs+αum≤CNumθ5Λs+αum2−θ5≤12C0N,ν,α,βum2+ν4Λs+αum2,where θ5=4α+2β−5/2s+2αandC0N,ν,α,β=CN2/θ5ν/4−2θ5θ5−2/θ5θ5. This, combined with (63), yields that(65)ddtΛsum2+νΛs+αum2≤C0N,ν,α,βum2+2νfℍs−α2.
Hence, for allt≥τ≥0,(66)Λsumt2+ν∫τtΛs+αum2dς≤Λsuτ2+C0N,ν,α,βt−τu02+f2νλ1α2+2t−τνfℍs−α2.
Thus,um is bounded in L∞0,T;ℍs∩L20,T;ℍs+α,∀T>0. Passing to the limit, we obtain (20) immediately.
Next, we prove the uniqueness result. Letuandv be two solutions in L∞0,T;ℍs∩L20,T;ℍs+α. Taking the inner product in (50) with Λ2sw, we obtain that(67)12ddtΛsw2+νΛs+αw2≤FNΛβu∫Ωu⋅∇wΛ2swdx+FNΛβu∫Ωw⋅∇vΛ2swdx+FNΛβu−FNΛβv∫Ωv⋅∇vΛ2swdx≐I1+I2+I3.
ForI1, using Hölder’s inequality, the product estimates, Gagliardo–Nirenberg inequality, and Young’s inequality, we deduce that(68)I1≤FNΛβuΛs+1−αuwΛs+αw≤CFNΛβuuL6/3−2βΛs+1−αwL3/β+wL6/3−2sΛs+1−αuL3/sΛs+αw≤CwHsθ2wHs+α2−θ2+wHsuHs+αwHs+α≤C1+uHs+α2wHs2+ν6Λs+αw2,where θ2=min1,2β+4α−5/2α (here, we may assume that s<3/2. If s=3/2, we can choose p,q satisfies 1/p+1/q=1/p and 2<q<6/5−4α, and the Sobolev inequality implies that wLpΛs+1−αuLq≤CwH3/2uHs+α. If s>3/2, we may choose p,q=∞,2 to get wL∞Λs+1−αuL2≤CwHsuHs+α). Similarly, we have(69)I2≤C1+vHs+α2wHs2+ν6Λs+αw2.
Moreover,(70)I3≤FNΛβu−FNΛβvΛs+1−αvvΛs+αw≤C3FNΛβuFNΛβvNΛβwvL6/3−2βΛs+1−αvL3/βΛs+αw≤CΛswΛs+αvΛs+αw≤ν6Λs+αw2+CΛs+αv2Λsw2.
By (67)–(70), we obtain that(71)ddtΛsw2≤CΛs+αu2+Λs+αv2+1Λsw2.
Gronwall’s inequality then implies that(72)Λsw2≤expC∫0tΛs+αu2+Λs+αv2+1dτΛsw02,from which the uniqueness of the solution and the continuity of the solution semigroup in ℍs follow immediately.
Finally, let us verify thatu∈C0,T;ℍs. Note that (66) implies that(73)u∈L20,T;ℍs+α,∀T>0,i.e.,Λsu∈L20,T;ℍα,∀T>0.
Hence, according to the standard Sobolev embedding result [26, 27], we need only to show that(74)Λsut∈L20,T;ℍ−α.
Indeed, for anyφ∈ℍα, we have(75)Λsut,φ=−FNΛβuΛsu⋅∇u,φ−Λs+2αu,φ+Λsf,φ.
Therefore,(76)Λsut,φ≤FNΛβuΛs−αu⋅∇u+Λs+αu+Λs−αfΛαφ,which implies that(77)Λsutℍ−α≤FNΛβuΛs−αu⋅∇u+Λs+αu+Λs−αf.
Applying the product estimates and the imbedding of fractional Sobolev spaces, we have(78)FNΛβuΛs−αu⋅∇u≤CFNΛβuuL6/3−2βΛs+1−αuL3/β≤CNΛs+αu.
Therefore,(79)Λsutℍ−α≤CΛs+αu+Λs−αf.
Combining (73) and the assumption f∈ℍs−α, we know that Λsut∈L20,T;ℍ−α. The proof is thus complete.


## 4. Long Time Behaviors

### 4.1. Attractor for Strong Solution

In this section, we prove the existence of a global attractor for system (3).Theorem 2.
Assume that4α+2β>5,0≤α<5/4,0≤β<3/2, and f∈ℍs−α,u0∈ℍs,s≥β. Then, system (3) generates a continuous semigroup Stt≥0 in ℍs, and the semigroup possesses a global attractor A, which is compact, invariant, and connected in ℍs and attracts all the bounded subsets of ℍs in the ℍs-norm. Moreover, if s≥maxβ,1,f∈ℍs−1+α, the global attractor is bounded in ℍs−1+2α.Proof.
Thanks to Theorem1, we know that the semigroup is continuous. It remains to prove the existence of an absorbing set and the compactness of the semigroup in ℍs.
Absorbing set: letut be the solution of system (1). Similar to (24), we have(80)ut2≤u02e−νλ1αt+f2ν2λ12α.
From the above inequality, we can deduce that there exists aT0=tu0 such that(81)ut2≤2f2ν2λ12α,∀t>T0.
Multiplying (3) by Λ2su and integrating, we have(82)12ddtΛsu2+νΛs+αu2≤fℍs−αΛs+αu+FNΛβu∫Ωu⋅∇uΛ2sudx,for all t>0. Similar to (65), we deduce that(83)ddtΛsu2+νΛs+αu2≤C0N,ν,α,βu2+2νfℍs−α2,for allt>0,where C0N,ν,α,β=CN2/θ5ν/4−2θ5θ5−2/θ5θ5,θ5=4α+2β−5/2s+2α. Integrating the both sides from 0 to T0 and taking (24) into consideration, we obtain(84)ΛsuT02≤Λsu02+∫0T0C0N,ν,α,βu2+2νfℍs−α2dτ≤Λsu02+C0N,ν,α,βT0u02+f2νλ1α2+2T0νfℍs−α2≐K.
Using the Poincaré inequality and (81), we obtain from (83) that(85)ddtΛsu2+νλ1αΛsu2≤C0N,ν,α,βf2νλ1α2+2νfℍs−α2,for allt≥tu0.
Gronwall’s inequality then implies that(86)Λsut2≤ΛsuT02e−νλ1αt−T0+1νλ1αC0N,ν,α,βf2νλ1α2+2νfℍs−α2.
Thanks to (96), we know that if(87)t≥maxT0,1νλ1αlnK+T0,then(88)Λsut2≤1νλ1αC0N,ν,α,βf2νλ1α2+2νfℍs−α2+1=ρ02.
Therefore, there is an absorbing setB1 for the semigroup Stt≥0 in ℍs.
Compactness of the semigroup: we show that, for any bounded sequencev0n in ℍs any t>0, the sequence Stv0n=vnt has a convergent subsequence in ℍs. Similar to (66) and (79), we can prove that(89)Λsvntis bounded inL20,1;ℍα,Λsvtntis bounded inL20,1;ℍ−α.
Using the Aubin-Simon type compactness results [26, 27], there exists an element v with(90)Λsv∈L20,1;ℍα,Λsvt∈L20,1;ℍ−α,such that up to subsequences,(91)Λsvnt⟶Λsvtstrongly inL20,1;ℍ,i.e.,(92)vnt⟶vtstrongly inL20,1;ℍs.
In particular, there exists aτ∈0,1 such that(93)vnτ⟶vτ,inℍs.
Recall that the mapSt:ℍs⟶ℍs is continuous, and we obtain that(94)Stv0n=St−τSτv0n=St−τvnτ⟶St−τvτinℍs,for allt≥1.
Thus, the semigroupSt is compact, for any t≥1. Thanks to the standard existence results on global attractors, we may obtain a global attractor in ℍs for the solution semigroup St.
Regularity of the attractor: now, we prove thatA is bounded in ℍs0+α if f∈ℍs0,s0=s−1+α. Take the inner product of (3) with Λ2s0+2α, and we have(95)12ddtΛs0+αu2+νΛs0+2αu2≤fℍs0Λs0+2αu+FNΛβu∫Ωu⋅∇uΛ2s0+2αudx,∀t>0.
Similar to (65) and (83), we have(96)ddtΛs0+αu2+νΛs0+2αu2≤C1N,ν,α,βu2+2νfℍs02≤C1N,ν,α,βu02+f2ν2λ12α+2νfℍs02,∀t>0.
Ifs0≤s, we have u0∈ℍs0. By standard regularity result, we know that utis bounded inL∞0,1;ℍs0∩L20,1;ℍs0+α. Thus, there exists a time t0∈0,1 and a positive constant M1 such that Λs0+αut02<M1. By using Gronwall’s inequality, we deduce from (96) that(97)Λs0+αut2≤M1e−νλ1αt−t0+1νλ1αC0N,ν,α,βu02+f2ν2λ12α+2νfℍs02,∀t>t0.
Ifs0>s, since u0∈ℍsandf∈ℍs0, we know that ut is bounded in L∞0,1;ℍs∩L20,1;ℍs+α. Since s0<s+α, we conclude that there exist a time t1∈0,1 and a positive constant M2 such that Λs0ut12<M2. Let vt=Stv0=StSt1u0=ut+t1. We know that vt is bounded in L∞0,1;ℍs0∩L20,1;ℍs0+α. Then, there exists a time t2∈0,1 and a positive constant M2, which does not depend on v, such that Λs0+αvt22<M2,i.e.,Λs0+αut2+t12<M2. Denote t3=t2+t1,t3∈0,2; then, ut3∈ℍs0+α. Similar to (97), we have(98)Λs0+αut2≤M3e−νλ1αt−t3+1νλ1αC0N,ν,α,βu02+f2ν2λ12α+2νfℍs02.
Since the attractorA is invariant, for any t>0 and any χ∈A, there exists a u0∈A such that Stu0=St−t0St0u0=ut=χ (or Stu0=St−t3St3u0=ut=χ). We assume t is large enough and takes (81) into consideration to obtain(99)Λs0+αχ2=Λs0+αut2≤2νλ1α3C1N,ν,α,βf2ν2λ12α+2νfℍs02=CA2.
Therefore,A is bounded in ℍs0+α by CA, which may depend on f,fℍs−α,N,α,β,andν and the bound in ℍs of the attractor A.


## 4.1. Attractor for Strong Solution

In this section, we prove the existence of a global attractor for system (3).Theorem 2.
Assume that4α+2β>5,0≤α<5/4,0≤β<3/2, and f∈ℍs−α,u0∈ℍs,s≥β. Then, system (3) generates a continuous semigroup Stt≥0 in ℍs, and the semigroup possesses a global attractor A, which is compact, invariant, and connected in ℍs and attracts all the bounded subsets of ℍs in the ℍs-norm. Moreover, if s≥maxβ,1,f∈ℍs−1+α, the global attractor is bounded in ℍs−1+2α.Proof.
Thanks to Theorem1, we know that the semigroup is continuous. It remains to prove the existence of an absorbing set and the compactness of the semigroup in ℍs.
Absorbing set: letut be the solution of system (1). Similar to (24), we have(80)ut2≤u02e−νλ1αt+f2ν2λ12α.
From the above inequality, we can deduce that there exists aT0=tu0 such that(81)ut2≤2f2ν2λ12α,∀t>T0.
Multiplying (3) by Λ2su and integrating, we have(82)12ddtΛsu2+νΛs+αu2≤fℍs−αΛs+αu+FNΛβu∫Ωu⋅∇uΛ2sudx,for all t>0. Similar to (65), we deduce that(83)ddtΛsu2+νΛs+αu2≤C0N,ν,α,βu2+2νfℍs−α2,for allt>0,where C0N,ν,α,β=CN2/θ5ν/4−2θ5θ5−2/θ5θ5,θ5=4α+2β−5/2s+2α. Integrating the both sides from 0 to T0 and taking (24) into consideration, we obtain(84)ΛsuT02≤Λsu02+∫0T0C0N,ν,α,βu2+2νfℍs−α2dτ≤Λsu02+C0N,ν,α,βT0u02+f2νλ1α2+2T0νfℍs−α2≐K.
Using the Poincaré inequality and (81), we obtain from (83) that(85)ddtΛsu2+νλ1αΛsu2≤C0N,ν,α,βf2νλ1α2+2νfℍs−α2,for allt≥tu0.
Gronwall’s inequality then implies that(86)Λsut2≤ΛsuT02e−νλ1αt−T0+1νλ1αC0N,ν,α,βf2νλ1α2+2νfℍs−α2.
Thanks to (96), we know that if(87)t≥maxT0,1νλ1αlnK+T0,then(88)Λsut2≤1νλ1αC0N,ν,α,βf2νλ1α2+2νfℍs−α2+1=ρ02.
Therefore, there is an absorbing setB1 for the semigroup Stt≥0 in ℍs.
Compactness of the semigroup: we show that, for any bounded sequencev0n in ℍs any t>0, the sequence Stv0n=vnt has a convergent subsequence in ℍs. Similar to (66) and (79), we can prove that(89)Λsvntis bounded inL20,1;ℍα,Λsvtntis bounded inL20,1;ℍ−α.
Using the Aubin-Simon type compactness results [26, 27], there exists an element v with(90)Λsv∈L20,1;ℍα,Λsvt∈L20,1;ℍ−α,such that up to subsequences,(91)Λsvnt⟶Λsvtstrongly inL20,1;ℍ,i.e.,(92)vnt⟶vtstrongly inL20,1;ℍs.
In particular, there exists aτ∈0,1 such that(93)vnτ⟶vτ,inℍs.
Recall that the mapSt:ℍs⟶ℍs is continuous, and we obtain that(94)Stv0n=St−τSτv0n=St−τvnτ⟶St−τvτinℍs,for allt≥1.
Thus, the semigroupSt is compact, for any t≥1. Thanks to the standard existence results on global attractors, we may obtain a global attractor in ℍs for the solution semigroup St.
Regularity of the attractor: now, we prove thatA is bounded in ℍs0+α if f∈ℍs0,s0=s−1+α. Take the inner product of (3) with Λ2s0+2α, and we have(95)12ddtΛs0+αu2+νΛs0+2αu2≤fℍs0Λs0+2αu+FNΛβu∫Ωu⋅∇uΛ2s0+2αudx,∀t>0.
Similar to (65) and (83), we have(96)ddtΛs0+αu2+νΛs0+2αu2≤C1N,ν,α,βu2+2νfℍs02≤C1N,ν,α,βu02+f2ν2λ12α+2νfℍs02,∀t>0.
Ifs0≤s, we have u0∈ℍs0. By standard regularity result, we know that utis bounded inL∞0,1;ℍs0∩L20,1;ℍs0+α. Thus, there exists a time t0∈0,1 and a positive constant M1 such that Λs0+αut02<M1. By using Gronwall’s inequality, we deduce from (96) that(97)Λs0+αut2≤M1e−νλ1αt−t0+1νλ1αC0N,ν,α,βu02+f2ν2λ12α+2νfℍs02,∀t>t0.
Ifs0>s, since u0∈ℍsandf∈ℍs0, we know that ut is bounded in L∞0,1;ℍs∩L20,1;ℍs+α. Since s0<s+α, we conclude that there exist a time t1∈0,1 and a positive constant M2 such that Λs0ut12<M2. Let vt=Stv0=StSt1u0=ut+t1. We know that vt is bounded in L∞0,1;ℍs0∩L20,1;ℍs0+α. Then, there exists a time t2∈0,1 and a positive constant M2, which does not depend on v, such that Λs0+αvt22<M2,i.e.,Λs0+αut2+t12<M2. Denote t3=t2+t1,t3∈0,2; then, ut3∈ℍs0+α. Similar to (97), we have(98)Λs0+αut2≤M3e−νλ1αt−t3+1νλ1αC0N,ν,α,βu02+f2ν2λ12α+2νfℍs02.
Since the attractorA is invariant, for any t>0 and any χ∈A, there exists a u0∈A such that Stu0=St−t0St0u0=ut=χ (or Stu0=St−t3St3u0=ut=χ). We assume t is large enough and takes (81) into consideration to obtain(99)Λs0+αχ2=Λs0+αut2≤2νλ1α3C1N,ν,α,βf2ν2λ12α+2νfℍs02=CA2.
Therefore,A is bounded in ℍs0+α by CA, which may depend on f,fℍs−α,N,α,β,andν and the bound in ℍs of the attractor A.


## 5. Finite Dimensionality of the Attractor

In this section, we provide the upper bound for the fractal dimension of the attractor derived in Section4.Theorem 3.
Assume that4α+2β>5,0≤α<5/4,0≤β<3/2, u0∈ℍs,andf∈ℍs−1+α,s≥max1,β. Then, the fractal dimension of the global attractor A derived in Theorem 2 is finite.To prove Theorem3, we use the following abstract results derived in [31–33].Lemma 6.
LetH0 be a separable Hilbert space and let M be a bounded closed set in H0. Assume that there exists a mapping S0:M↦H0 such that M⊆S0M:(i)
S0 is Lipschitz on M, i.e., there exists L>0 such that(100)S0v1−S0v2H0≤Lv1−v2H0,v1,v2∈M.(ii)
There exist finite dimension orthoprojectorsP1 and P2 on H0 such that(101)S0v1−S0v2H0≤ηv1−v2H0+KP1v1−v2H0+P2v1−v2H0,∀v1,v2∈M,where 0<η<1 and K>0 are constants.
Then,(102)dimfM≤dimP1+dimP2ln1+821+LK1−ηln21+η−1.DenoteZm=spanej−kjk/k2eik⋅x:j=1,2,3,k=k12+k22+k32≤m, where k=k1,k2,k3∈ℤ3,k≠0, and e1,e2,ande3 represent the canonical basis of ℝ3. Let Pm:L2Ω↦Zm be the projection operator. Similar to Lemma 3.4 in [34] (see also Lemma 2.12 in [31]), we have the following lemma.Lemma 7.
Letη≥0andι>0. For any ε>0, there exists a positive integer mε such that for m≥mε, and we have(103)φHη≤εφHη+ι+PmφHη,∀φ∈Hη+ι,where mε=ε−1/t, the integer part of number ε−1/t.Thanks to Lemma8 in [24], we have the following.Lemma 8.
The projection operatorPm:L2Ω↦Zm has a finite range with(104)dimPm≤84m3+6m2+8m+3.Lemma 9.
Assume that4α+2β>5,0≤α<5/4,0≤β<3/2, u0∈ℍs,andf∈ℍs0,s≥min1,β. Let A be the global attractor of system (3) derived in Theorem 2 for the smooth solution. Let utandvt be two solutions of system (3) corresponding to the initial data u0,v0∈A, respectively. Let wt=ut−vt, and let ϑ be a positive constant such that maxs,3/2<s−1+2α−ϑ. Then, for any s1∈s−1+2α−ϑ,s−1+2α, we have(105)Λs1wt2≤expCN,α,β,ν∫0tΛs+αu2+Λs+αv2+1dτΛs1w02,for some positive constant.Proof.
Take the inner product of (50) with Λ2s1w, and we know that w satisfies, for any s−1+2α−ϑ≤s1≤s−1+2α:(106)12ddtΛs1w2+νΛs1+αw2≤FNΛβu∫Ωu⋅∇wΛ2s1wdx+FNΛβu∫Ωw⋅∇vΛ2s1wdx+FNΛβu−FNΛβv∫Ωv⋅∇vΛ2s1wdx≐L1+L2+L3.
SinceA is bounded in ℍs0+α, we have (107)Λs0+αut≤CA,Λs0+αvt≤CA,∀t≥0.
ForL1, using Hölder’s inequality, the product estimates, Gagliardo–Nirenberg inequality, and Young’s inequality, we deduce that(108)L1≤FNΛβuΛs1+1−αuwΛs1+αw≤CFNΛβuuL∞Λs1+1−αw+wL∞Λs1+1−αuΛs1+αw≤C1Λs0+αuΛs1wθ6Λs1+αw2−θ6+C2Λs+αuΛs1wΛs1+αw≤C1CAΛs1wθ6Λs1+αw2−θ6+C2Λs+αuΛs1wΛs1+αw≤ν6Λs1+αw2+C1N,α,,β,ν1+Λs+αu2Λs1w2,where θ6=min2α−1/α,1. Similarly, we have(109)L2≤ν6Λs1+αw2+C1N,α,β,ν1+Λs+αv2Λs1w2.
Finally, forL3, we use Hölder’s inequality, Lemma 2, product estimates, and imbedding of fractional Sobolev spaces to deduce that(110)L3≤FNΛβu−FNΛβvΛs1+1−αvvΛs1+αw≤C3FNΛβuFNΛβvNΛβwvL∞Λs1+1−αvΛs1+αw≤C3NΛs1wΛs0+αvΛs+αvΛs1+αw≤C3CANΛs1wΛs+αvΛs1+αw≤ν6Λs1+αw2+3C32CA22N2νΛs+αv2Λs1w2.
Combining (106)–(110), we obtain that(111)ddtΛs1w2≤CN,α,β,νΛs+αu2+Λs+αv2+1Λs1w2,with(112)CN,α,β,ν=maxC1CA2/θ6ν12−6θ6θ6−2/θ6θ62,3C22ν,3C32CA22N2ν.
Gronwall’s inequality then implies (105):(113)Λs1wt2≤expCN,α,β,ν∫0tΛs+αu2+Λs+αv2+1dτΛs1w02≐C˜tΛs1w02.
Obviously,C˜t is a monotone function with respect to t, and it is finite for finite time t.Lemma 10.
Assume that4α+2β>5,α>1/2,β≥0,u0∈ℍs,andf∈ℍs0,s≥max1,β. Let uandv be two solutions of system (3) with initial condition u0,v0∈A, respectively. Setting w=u−v, then there exists a positive constant C such that ∀t≥0:(114)wtℍγ≤Ce−νλ1αt/2w0ℍγ+ν−ϵ0CCA2+2CAC˜tw0ℍγ−ϵ0,where ϵ0=1/2min2α−1/2α+1,2ϑ,γ=s−1+2α, C˜t is from (113), and CA is from (99).Proof.
Similar to (50), we know that w satisfies(115)wt+νΛ2αw+FNΛβuPu⋅∇w+FNΛβuPw⋅∇v+FNΛβu−FNΛβvPv⋅∇v=0.
By using the Duhamel principle [35], the solution of (114) can be given by(116)wt=e−ν−Δαtw0−∫0te−ν−Δαt−τFNΛβuPu⋅∇w+FNΛβuPw⋅∇v+FNΛβu−FNΛβvPv⋅∇vdτ.
And moreover, we have the following estimate of the semigroupe−−Δαt:(117)e−−Δαtℍβ″ℍβ′≤Ce−λ1αt/2t−β′−β″/2α,β′≥β″≥0,t>0.
Denote(118)Ψτ=FNΛβuu⋅∇w+FNΛβuw⋅∇v+FNΛβu−FNΛβvv⋅∇vℍγ−21−εα,and then,(119)wtℍγ≤Ce−νλ1αt/2w0ℍγ+C∫0te−νλ1αt−τ/2t−τ−1+ϵΨτdτ,and 0<ϵ<1 will be determined later. Setting σ=γ−21−ϵα, we have(120)Ψτ≤FNΛβuu⋅∇wℍσ+FNΛβuw⋅∇vℍσ+FNΛβu−FNΛβvv⋅∇vℍσ≐S1+S2+S3.
Next, we estimateS1,S2,andS3 one by one. By the product estimates of Sobolev spaces, we have(121)S1≤CFNΛβuuLp1∇wℍσ,q1+uℍσ,p2∇wLq2,for positive integers pi,qi satisfying 1/pi+1/qi=1/2,i=1,2. Let ε<2α−1/2α+1; since 4α+2β>5,α>1/2ands≥max1,β, we have(122)2γ+4α−54α+2≥2β−1+2α+4α−54α+2>2α−12α+1>ε>0.
Hence, it is easy to check that there exists a pair of positive integersp1,q1 such that 1/p1+1/q1=1/2, and(123)1p1≥12−γ3,1q1≥12−γ−ε−1−σ3,σ+1≤γ−ε.
By the Sobolev inequality, we have(124)uLp1≤Cuℍγ,∇wℍσ,q1≤Cwℍγ−ε,and thus,(125)uLp1∇wℍσ,q1≤Cuℍγwℍγ−ε.
On the contrary, since(126)γ−1=s−1+2α−1≥2α−1>2α−12α+1>ε>0,rhen we can check that there exists a pair of positive integers p2,q2 such that 1/p2+1/q2=1/2, and(127)1p2≥12−γ−σ3,1q2≥12−γ−ε−13,γ−ε>1.
By the Sobolev inequality, we have(128)uℍσ,p2≤Cuℍγ,∇wLq2≤Cwℍγ−ε,and hence,(129)uℍσ,p2∇wLq2≤Cuℍγwℍγ−ε.
Combining (121)–(129), we obtain(130)S1≤Cuℍγwℍγ−ε,when ε∈0,2α−1/2α+1.
Similarly, forS2, we have(131)S2≤Cvℍγwℍγ−ε.
Sinceε∈0,2α−1/2α+1,s≥max1,β, we have that β<γ−ε. For S3, using Lemma 2, the product estimate, and the Sobolev inequality, it is easy to deduce that(132)S3≤CFNΛβuFNΛβvNΛβwvℍγvℍγ−ε≤Cvℍγ2wℍγ−ε.
Now, combining (99) and (130)–(132), we obtain(133)S1+S2+S3≤Cuℍγ+vℍγ+vℍγ2wℍγ−ε≤CCA2+2CAwℍγ−ε.
Now, setε0=1/2min2α−1/2α+1,2ϑ. Taking (132) into (119) and using (99) and (105) (note that s−1+2α−ϑ≤γ−ε0≤s−1+2α), we obtain that(134)wtℍγ≤Cexp−νλ1αt2w0ℍγ+CCA2+2CA∫0texp−νλ1αt−τ2t−τ−1+ϵ0wτℍγ−ϵ0dτ≤Cexp−νλ1αt2w0ℍγ+CCA2+2CAC˜tw0ℍγ−ϵ0∫0texp−νλ1αt−τ2t−τ−1+ϵ0dτ≤Cexp−νλ1αt2w0ℍγ+Γϵ0ν−ϵ0CCA2+2CAC˜tw0ℍγ−ϵ0,where Γ⋅ is the Gamma function. This completes the proof of Lemma 10.Proof.
of Theorem3. Combining with Lemmas 6–10, we can give an upper bound on the fractal dimension of the attractor derived in Theorem 2 as follows. Let w be a solution of (115). It follows from Lemma 7 and Lemma 10 that(135)wtℍγ≤Cexp−νλ1αt2w0ℍγ+Cν−ϵ0CA2+2CAC˜tw0ℍγ−ϵ0,≤Cexp−νλ1αt2+εν−ϵ0CCA2+2CAC˜tw0ℍγ+ν−ϵ0CCA2+2CAC˜tPmw0ℍγ−ϵ0,with m≥mε=ε−1/ε0andε0=1/2min2α−1/2α+1,2ϑ. After some elementary calculations, we can choose(136)t0=2ln4Cνλ1α,ε=14νε0C−1CA2+2CA−1C˜t0−1,such that(137)Cexp−νλ1αt02=14,εν−ε0CCA2+2CAC˜t0=14,with(138)C˜t0=expC2N,α,β,ν∫0t0Λs+αu2+Λs+αv2+1dτ.
Combining with (135) and (137), we have(139)wt0ℍγ≤12w0ℍγ+ν−ε0CCA2+2CAC˜t0Pmw0ℍγ−ε0.
Meanwhile, it follows from Lemma9 that(140)Λs1wt02≤expCN,α,β,ν∫0t0Λs+αu2+Λs+αv2+1dτΛs1w02≐C˜t0Λs1w02.
DefineS0=St0:w0↦wt0, and we can check that S0 satisfies conditions (i) and (ii) of Lemma 6, and(141)M=A,H0=ℍγ=ℍs−1+2α,S0=St0:w0↦wt0,t0=2ln4Cνλ1α,η=12,K=ν−ε0CCA2+2CAC˜t0,L=C˜t0,m=Cν−1CA2+2CA1/ε0C˜t01/ε0,ε0=12min2α−12α+1,2ϑ,dimP2=0,P1=Pm,dimPm≤84m3+6m2+8m+3,where CA is from (99), C˜t0 is from (140), ϑ is the constant in the Lemma 9, C is a universal constant, and x denotes the integer part of the number x.
Thus, we have(142)dimfA≤84m3+6m2+8m+3ln1+1621+C˜t0ν−ε0CCA2+2CAC˜t0ln43−1.


## 6. Conclusions

The authors established the existence of weak solutions under proper assumptions onαandβ. The existence of finite-dimensional global attractors is also obtained. It is possible to study the global modified MHD equations by using similar ideas herein.



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*Source: 9924912-2021-07-31.xml*