# The Unique Existence of Weak Solution and the Optimal Control for Time-Fractional Third Grade Fluid System

Authors: Guangming Shao; Biao Liu; Yueying Liu
Journal: Complexity
 (2018)

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## Abstract

The paper concerns the third grade fluid system with the time-fractional derivative of the orderα∈(0,1). We first establish unique existence criterion of weak solutions in the case that the dimension n=3. Then we prove the sufficient condition of optimal pairs.


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## Body

## 1. Introduction

We consider the significant case of the fluids of graden introduced by Rivlin-Ericksen [1] in a bounded domain Ω⊂R3(1)∂v∂t-νΔu+u·∇v+∑jvj∇uj-σ1+σ2div⁡A2u-βdiv⁡Au2Au+∇p=f,v=u-σ1Δu,where ν>0 is the constant kinematic viscosity, σ1, σ2, and β are material constants, and β>0. The unknown functions u,p are the fluid velocity and the scalar pressure, respectively. The given functions u0,f represent the initial fluid velocity and the forcing term, respectively. In fact, A is the tensor whose component is (2)Au=∇u+∇uT.Here, (∇u)T is the transposition of the Jacobian matrix ∇u, and |A(u)|2 denotes tr(A2(u)).In [2], Ladyzhenskaya studied a particular case of the third grade fluids (1) in R3, where they assumed σ1=0. In this case, system (1) becomes(3)∂tu-νΔu+u·∇u-σdiv⁡A2u-βdiv⁡Au2Au+∇p=f,div⁡u=0,u=0,x∈∂Ω,t>0,ux,0=u0,x∈Ω.In the past years, some nice works on the third grade fluid system have already been available. We refer to Fosdick and Rajagopal [3], Amrouche and Cioranescu [4], Sequeira and Videman [5], Busuioc and Iftimie [6], Paicu [7], Hamza and Paicu [8], Zhao et al. [9], and Chai et al. [10]; much is yet to be carried out.Unlike the boundary value problems of the classical third grade fluid system, the theoretical analysis of time-fractional third grade fluid system is not so rich, so we need to develop many aspects of these problems. It is significant to study this type of system when it is recognized that the time-fractional third grade fluid system can be used to describe diffusion phenomenon in fractal media. So the scholars have been more and more interested in investigating the fractional calculus.The fractional calculus was introduced by Herrmann [11] and Hilfer [12], especially for the classical time-fractional Navier-Stokes equations (4)∂tαu-νΔu+u·∇u+∇p=f,div⁡u=0,where ∂tα is the Caputo fractional derivative of order α∈(0,1).For some recent work on analytical solutions of these systems, we refer to El-Shahed et al. [13], Ganji et al. [14], Zhou [15], Momani and Zaid [16], Carvalho-Neto and Gabriela [17], Zhou and Peng [18, 19], Zhou et al. [20], Al-Mdallal et al. [21], Jarad [22–25], and further papers cited therein.In this paper, we consider mainly the weak solutions of the time-fractional third grade fluid system and optimal control in an open setΩ⊂R3 with smooth boundary ∂Ω(5)∂tαu-νΔu+u·∇u-σdiv⁡A2u-βdiv⁡Au2Au+∇p=f,div⁡u=0,with the initial and boundary value conditions (6)u=0,x∈∂Ω,t>0,ux,0=u0,x∈Ω.Noting that the system converges formally to the well-known time-fractional Navier-Stokes equations when the material constantsσ,β in (5) tend to zero. So it is reasonable to propose the time-fractional third grade fluid system.In this paper, we firstly investigate the existence and uniqueness of solutions for the time-fractional third grade fluid equations (5) (associated with appropriate initial and boundary conditions). Then we proceed the systems with control in Ω∈R3(7)∂tαu-νΔu+u·∇u-σdiv⁡A2u-βdiv⁡Au2Au=-∇p+c0ω+f,∇·u=0,u=0,x∈∂Ω;ux,0=u0,x∈Ω,where U is a real Hilbert space, ω:[0,T]→U, and the operator c0:U→H is linear continuous.Now let us talk about the organization of this paper. In Section2, we provide some preliminaries about the function spaces. In Section 3, we prove the existence of solutions for system (5). The uniqueness of solutions is established in Section 4, while in Section 5 we show the existence of the optimal control of system (7).


## 2. Preliminaries

In this section, for convenience, we mainly introduce some notions, operators, definitions, and lemmas, which are useful in the sequel.LetΩ⊂R3 be an open bounded smooth domain; we define (8)V=u∈C0∞Ω×C0∞Ω×C0∞Ω,div⁡u=0},H=VL2Ω,withnorm·=·L2theusualL2norm,H0s=VH2Ω,withnorm·=·H2theusualH2norm,V=VH01Ω,withnorm·V=∇·,andthedualspaceV∗,W=VW01,4Ω,withnorm·W=∇·L4,andthedualspaceW∗.Define the linear “Stokes operator” A=-△ from V to V∗ by(9)Au,vV∗,V=∑i,j=13∫Ω∂jui∂jvidx,∀u,v∈V,and the bilinear operator B(u,v) from V×V into V∗ as(10)Bu,v,wV∗,V=∑i,j=13∫Ωui∂vj∂xiwjdx,∀u,v,w∈V.We also introduce the operators K(u), J(u) from W into W∗ as(11)Ku,ϕW∗,W=α∫ΩA2u·∇ϕdx≤CAuL42∇ϕL4,∀u,ϕ∈W,Ju,ϕW∗,W=β∫ΩAu2Au·∇ϕdx,∀u,ϕ∈W.Let |α|<2νβ. We denote T(u) from W into W∗ as(12)Tu=ν-νδ0Au+1-δ0Ju+Ku,δ0=1-α22νβ∈0,1.Note that (see, e.g., [26–28])(13)Au=-Δuforu∈DA=u∈H02Ω,div⁡u=0.By the Hilbert-Schmidt theorem, one can deduce that A has a sequence of orthonormal eigenfunctions wj, belonging to Cp∞(Ω) with zero mean in Ω. Since A is a self-adjoint positive operator with compact inverse, {ωj}j=1∞ forms a basis of the space H. Moreover, {ωj}j=1∞ also forms a basis of the space D(As/2)={H0s(Ω),div⁡u=0}, for any positive integer s [28].Next, we will introduce some definitions and lemmas, which are used throughout this paper.Definition 1 (see [18, 29]).
LetX be a Banach space, v:[0,T]→X; the left and right Riemann-Liouville fractional integrals I0tαv(t) and ItTαv(t) of order α∈(0,1] are defined by(14)Itα0vt=∫0tgαt-svsds,ITαtvt=∫tTgαs-tvsds,t>0,provided the integrals are point-wise defined on [0,∞), where gα denotes the Riemann-Liouville kernel(15)gαt=tα-1Γα,t>0.Definition 2 (see [29]).
The left Caputo and right Riemann-Liouville fractional derivativeDtα0cv(t) and DTαtv(t) of order are defined by (16)Dtα0cvt=∫0tg1-αt-sddsvsds,DTαtvt=-ddt∫tTg1-αs-tvsds,t>0.More generally, for v:[0,∞)×Rn→Rn, the left Caputo fractional derivative with respect to time can be defined by(17)∂tαux,t=∫0tg1-αt-s∂∂svx,sds,t>0.For more detail, we can refer to [30].Definition 3 (see [18, 29]).
Letv:R→X; the Liouville-Weyl fractional integral and the Caputo fractional derivative on the real axis are defined, respectively, as follows:(18)Itα-∞vt=∫-∞tgαt-svsds,Dtαc-∞vt=∫-∞tg1-αt-sddsvsds.Lemma 4 (see [8]).
The operatorJ is a monotone operator; that is, for any u,v∈W, we have(19)Ju-Jv,u-vW∗,W≥0.Lemma 5 (see [8]).
The operatorT is a monotone operator; that is, for any u,v∈W, we have(20)Tu-Tv,u-vW∗,W≥0.LetT1:W→W∗ be defined as(21)T1v=Av+Kv+Jv.Thanks to the above lemmas, T1 is also a monotone operator.Lemma 6 (see [31]).
The fractional integration by parts in the formula(22)∫0T∂tαvt,φt=∫0tvt,DTαtφtdt+vt,ItT1-αφt0T.Since limt→TItT1-αφ(t)=0 for φ∈C0∞(0,T;X), then(23)∫0T∂tαvt,φt=∫0Tvt,DtTαφtdt-v0,ItT1-αφt.For more details, we refer to relations (16)-(21) in [31].Lemma 7 (see [19]).
∂ t α ( u ( t ) , v ) = 〈 ∂ t α u ( t ) , v 〉 in L(C0∞(0,T;Ω)) for u∈Wα([0,T],V,V′), v∈V, where Wα([[0,T],V,V′)≔{u∈L2([0,T],V):∂tαu∈L2([0,T],V′)}.Lemma 8 (see [32]).
Suppose thatX is a real Hilbert space and v:[0,T]→X have a derivative, then there holds (24)vt,Dtα0cvt≥12Dtα0cvt2.Lemma 9 (see [19]).
Suppose that a nonnegative functionv(t) satisfies(25)Dtα0cvt+c1vt≤c2tfor almost all t∈[0,T], where c1>0 and the function c2 is nonnegative and integrable for t∈[0,T]. Then(26)vt≤v0+1Γα∫0tt-sα-1c2sds.We need the compactness theorem [26] involving fractional derivative.AssumeX0,X,X1 are Hilbert spaces with(27)X0↪X↪X1,being continuous, and(28)X0↪Xiscompact.Let ψ(t) be a function from R to X1; we denote by ψ^(τ) its Fourier transform(29)ψ^τ=∫-∞+∞exp⁡-2πitτψtdt.The derivative in t of order γ is the inverse Fourier transform of (2iπτ)γψ^(τ); that is,(30)Dtγψt^=2iπτγψ^τ.For given γ>0, define the space(31)Mγ=MγR;X0,X1=ψ∈L2R;X0,Dtγψ∈L2R;X1.Then, Mγ is a Hilbert space with the norm(32)ψMγ=ψL2R;X02+τγψ^τL2R;X121/2.For any setK⊂R, the subspace MKγ of Mγ is defined as the set of functions u∈Mγ with support contained in K:(33)MKγ=MKγR;X0,X1=ψ∈Mγ,supp⁡ψ⊆K.Lemma 10 (see [26]).
AssumeX0,X,X1 are Hilbert spaces satisfying (27) and (28). Then, for any bounded set K and ∀γ>0, we have following compact embedding:(34)MKγR;X0,X1↪L2R;X.The following Korn’s inequality plays an essential role in our analysis.Lemma 11 (see [33]).
Assume that1<q<∞ and Ω∈Rn, n=2,3. Let v∈W01,q(Ω). Then there exist two positive constants c0=c0(q,Ω), c1=c1(q,Ω) such that(35)c0∇vLq≤AvLq≤c1∇vLq.Using the notation and operators introduced earlier, we can express the weak formulation of the time-fractional third grade fluid system (5) in the solenoidal vector fields as follows.Definition 12.
Letu0∈H and f∈L2/α1(0,T;V∗) (α1∈(0,α)) be given. A weak solution of system (5) is a function u∈L2(0,T;V)∩L∞(0,T;H)∩L4(0,T;W) such that for any v∈W(36)∂tαu,v+νu,v+bu,u,v-σdiv⁡A2u+βdiv⁡Au2Au,v=f,v,ux,0=u0.Now we give the equivalent form of the first equality of (36) as follows:(37)∂tαu,v=f-νAu-Bu,u+σdiv⁡A2u+βdiv⁡Au2Au,v,∀v∈W.Since Au∈L2(0,T;V∗)↪L4/3(0,T;W∗), then f-νAu-Bu,u+σdiv⁡(A2(u))+βdiv⁡(|A(u)|2A(u))∈W∗. Thus(38)∂tαu=f-νAu-Bu,u+σdiv⁡A2u+βdiv⁡Au2Au.An equivalent form of (36) is as follows:(39)Dtα0cu+νAu+Bu,u-σdiv⁡A2u-βdiv⁡Au2Au=f,ux,0=u0.


## 3. Existence for Time-Fractional Third Grade Fluid System

In this section, we prove the existence of solutions for time-fractional third grade fluid system.Theorem 13.
Letu0∈H and f∈L2/α1(0,T;V∗), β>0,|σ|<2νβ. Then there exists at least one solution u∈L2(0,T;V)∩L∞(0,T;H)∩L4(0,T;W).Proof.
In order to apply the Galerkin procedure. We consider a basis ofH01(Ω) constituted of elements ωi of D(Ω) and set Pm:H→Hm be the corresponding projection operators. For each m, we define um by(40)umt=∑i=1mgimtωi.Applying Pm to (39), we can obtain(41)Dtα0cum+νAum+PmBum,um-σPmdiv⁡A2um-βPmdiv⁡Aum2Aum=fm,um0=u0m,where u0m is the orthogonal projection of u0 onto the space spanned by ω1,…,ωm in H.
Taking the scalar product of (41)1 with ωk, we have(42)Dtα0cum,ωk+νum,ωk+PmBum,um,ωk-σPmdiv⁡A2um+βPmdiv⁡Aum2Aum,ωk=fm,ωk.Equations (42) form a nonlinear differential system for the functions g1m,…,gmm. By the standard theory of ODE, we have the existence of a solution defined at least on some interval [0,tm), 0<tm≤T. Moreover, if tm<T, then limt→tmum→+∞. Otherwise, the following a priori estimates show that in fact tm=T.
Indeed, if we multiply (42) by gkm(t) and add all these equations for k=1,…,m, (43)Dtα0cum,um+νum,um+bum,um,um-σdiv⁡A2um+βdiv⁡Aum2Aum,um=fm,um.Since b(um,um,um)=0, according to Lemma 5 and Young’s inequality, we have(44)Dtα0cum2+2νum2+2∫ΩσA2um+βAum2Aum·∇umdx≤1νf2+νum2.Note that A(um) is symmetric; we get (45)∫ΩAum2Aum·∇umdx=12AumL44.From the Cauchy-Schwartz inequality and the fact that (46)A2um=AumAum≤Aum2,we deduce that (47)σ∫ΩA2um·∇umdx≤σδ2AumL44+σ2δum2.Since |σ|<2βν, we can choose δ=2βν+σ2/4|σ|ν, such that (48)σδ2<β2andσ2δ<ν.We deduce that(49)Dtα0cum2+C0um2+C1AumL44≤1νf2.Integration of (49) from 0 to t shows that (50)umt2+C0∫0tt-sα-1um2ds+C1∫0tt-sα-1AumL44ds≤u0m2+1ν∫0tt-sα-1f2ds≤u0m2+1ν∫0tf2/α1ds+1ν∫0tt-sα-1/1-α1ds≤u0m2+1ν∫0tft2/α1ds+1-α1να-α1Tα-1/1-α1≤u02+1ν∫0Tft2/α1ds+1-α1να-α1Tα-1/1-α1.Hence tm=T and(51)supt∈0,Tumt≤d3,(52)∫0tt-sα-1um2ds≤d3C0,(53)∫0tt-sα-1AumL44ds≤d3C1,where (54)d3=u02+1ν∫0Tft2/α1ds+1-α1να-α1Tα-1/1-α1.Moreover(55)Tα-1∫0Tum2ds≤∫0tt-sα-1um2ds≤d3C0,(56)Tα-1∫0TAumL44ds≤∫0tt-sα-1AumL44ds≤d3C1.Hence, {um} is bounded in L2(0,T;V)∩L∞(0,T;H)∩L4(0,T;W).
As done several times before, we extend all functions by 0 outside the interval[0,T] and consider the Fourier transform of the different equations. The following relations then hold on R.(57)Dtα0cu~mt,ωk=ϕ~mt,ωk+u0m,ωkIt1-α-∞δ0-umT,ωkIt1-α-∞δT,where (58)ϕm=f-νAum-Bum+σdiv⁡A2um+βdiv⁡Aum2Aum.After taking Fourier transforms, (57) yields(59)2iπταu^m,ωk=ϕ^m,ωk+u0m,ωk2iπτα-1-umT,ωk2iπτα-1exp⁡-2iπτT,where u^m and ϕ^m represent, respectively, the Fourier transforms of u~m and ϕ~m.
We multiply (59) by g^k(τ) (g^k= Fourier transform of g~k) and then add these relations for k=1,…,m; we obtain(60)2iπταu^mτ2=ϕ^mτ,u^mτ+u0m,u^mτ2iπτα-1-umT,u^mτ2iπτα-1exp⁡-2iπτT.In view of the inequality (61)BuV∗≤Cu2,forallu∈V.We can obtain (62)∫0TϕmV∗dt≤∫0TfV∗+C0um+C1um2+C2AumL44dt≤∫0TfV∗+C0um+C1um2+C2′umW4dt≤C.Therefore (63)supτ∈R⁡ϕ^mτV∗≤C,u0m≤C,umT≤C,forallm.And we can conclude from (60) that (64)ταu^mτ2≤C3u^mτ+C4τα-1u^mτ;namely (65)ταu^mτ2≤C5max⁡1,τα-1u^mτ.For some fixed γ∈(0,1/4), we have |τ|2γ≤C6(γ)1+|τ|α/1+|τ|α-2γ,∀τ∈R. Thus(66)∫-∞+∞τ2γu^mτ2dτ≤C6γ∫-∞+∞1+τα1+τα-2γu^mτ2dτ=C6γ∫-∞+∞u^mτ21+τα-2γdτ+C6γ∫-∞+∞ταu^mτ21+τα-2γdτ≤C6γ∫-∞+∞u^mτ2dτ+C6γ∫-∞+∞ταu^mτ21+τα-2γdτ.By Parseval equality and Poincaré inequality, we have(67)C6γ∫-∞+∞u^mτ2dτ=C6γ∫-∞+∞u~mt2dt=C6γ∫0Tumt2dt≤λ1-2C6γ∫0Tumt2dt≤CC0,γ,d3,and (68)
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.We have also used the convergence of the infinite integral in the above integral (69)∫-∞+∞dτ1+τα-2γ21/2,forsomeγ∈0,14.We can conclude that(70)∫-∞+∞τ2γu^mτ2dτ≤C.forsomeγ∈0,14.
Next, we want to pass to the limit asm→∞ in (42) using the estimates (51), (55)-(56), and (70). We are only concerned with a passage to the limit as m→∞.
Considering the Lemma7, there exist a sequence m′→∞, and one function {u} such that (71)um′⇀uinL20,T;Vweakly,um′→uinL∞0,T;Hweak-star,um′→uinL20,T;Hstrongly,um′⇀uinL40,T;Wweakly.Taking v∈V, we multiply (41) by v, integrate over [t0,t], and we have(72)um′t,v-um′t0,v=∫0t0t0-sα-1-t-sα-1νum′s,v+bum′s,um′s,Pm′v-σdiv⁡A2um′+βdiv⁡Aum′2Aum′,Pm′v-fm′,vds-∫t0tt-sα-1νum′s,v+bum′s,um′s,Pm′v-σdiv⁡A2um′+βdiv⁡Aum′2Aum′,Pm′v-fm′,vds.In view of um′⇀uinL2(0,T;V), we can assume that um′(t0)⇀u(t0) in V for all t0∈[0,T]\K, where mes(K)=0. Therefore, limm′→∞um′t0=ut0 in H for t0∉K.
Finally, we look at the convergence of the nonlinear terms in (72).
By (55)-(56) and using Lebesgue’s dominated convergence theorem, we have (73)limm′→∞⁡∫t0tt-sα-1bum′s,um′s,Pm′vds=∫t0tt-sα-1bus,us,Pvds,limm′→∞⁡∫0t0t0-sα-1-t-sα-1bum′s,um′s,Pm′vds=∫0t0t0-sα-1-t-sα-1bus,us,vds.
Next, letT1:W→W∗ be defined as(74)T1v=-ν△v-σdiv⁡A2v-βdiv⁡Av2Av.
It is clear thatT1 is a monotone operator. We show that (75)Ilimm′→∞⁡∫0t0t0-sα-1-t-sα-1T1um′t,vds=∫0t0t0-sα-1-t-sα-1T1us,vds.
This can be achieved by the monotony method as in [8, 34]; see also [35, 36]. Indeed, since T1(um′) is uniformly (in m′) bounded in L4/3(0,T;W∗), we may assume that (76)T1um′→Ξweak-starinL4/30,T;W∗.Due to (43) and taking a sequence {m′∣m′→∞}, we have(77)Dtα0cum′,um′+T1um′,um′=f,um′,and integration of (77) from 0 to t0 red shows that(78)∫0t0t-sα-1T1um′,um′1ds≤12um′02-12um′t02+∫0t0t-sα-1f,um′1ds,where 〈,〉1 is the duality product between the spaces L4/3(0,T;W∗) and L4(0,T;W).
By (42), we have(79)Dtα0cum′,ωψ+Bum′,um′,ωψ-T1um′,ωψ=f,ωψ.Passing to the limit in (79),(80)Dtα0cu,ωψ+Bu,u,ωψ-Ξ,ωψ=f,ωψ.Taking ωψ=u, we can obtain(81)∫0t0t-sα-1Ξ,u1ds≤12u02-12ut02+∫0t0t-sα-1f,um′1ds.Due to the monotonicity of T1, we have for any φ∈L4(0,T;W)(82)T1um′,um′1≥T1um′,φ1+T1φ,um′-φ1.Combining (78), (81), and (82), we obtain by passing to the limit in m′(83)∫0t0t-sα-1Ξ-T1φ,uϵ-φ1ds≥0,foranyφ∈L40,T;W.Take φ=u-δϕ for δ>0 and ϕ∈L4(0,T;W). Then after division by δ and tending δ to 0, we deduce that (84)∫0t0t-sα-1Ξ-T1u,ϕ1ds≥0,foranyϕ∈L40,T;W.Hence, Ξ=T1(u). Then (85)limm′→∞⁡∫0t0t0-sα-1-t-sα-1T1um′t,vds=∫0t0t0-sα-1-t-sα-1T1us,vds.A simple argument shows that (86)limm′→∞⁡∫t0tt-sα-1T1um′t,vds=∫t0tt-sα-1T1us,vds.Taking the limit of (72) as m′→∞, we find that(87)ut,v-ut0,v=∫0t0t0-sα-1-t-sα-1νus,v+bus,us,v-σdiv⁡A2u+βdiv⁡Au2Au,v-f,vds-∫t0tt-sα-1νus,v+bus,us,v-σdiv⁡A2u+βdiv⁡Au2Au,v-f,vds.It is clear that (u(t),v)-(u(t0),v)→0, as t→t0.
Sett0=0 in (87); we can obtain (36). The proof is complete.


## 4. Uniqueness of Solutions for the Time-Fractional Third Grade Fluid System

In this section, we prove the uniqueness of solutions of problem (39).Theorem 14.
The solutionu of problems (39) given by Theorem 13 is unique.Proof.
We denote byu1,u2 two solutions of problem (39) and set u=u1-u2, by subtracting the relation (39) satisfied by u1 and u2; we obtain(88)Dtα0cu-ν△u+Ju1-Ju2+Ku1-Ku2=Bu2-Bu1,∇·u=0.Taking the inner product of (88) with u, it yields (89)Dtα0cu,u+νut2=-Ju1-Ju2,u-Ku1-Ku2,u-bu,u2,u.
Sinceb(v,w,w)=0, then (90)Dtα0cut2+2νut2=-2Ju1-Ju2,u-2Ku1-Ku2,u-2bu,u2,u.Bear in mind that (91)Tu=-ν-νδ0△u+1-δ0Ju+Ku,δ0=1-α22νβ∈0,1.We deduce that (92)Dtα0cut2+2νδ0ut2≤2bu,u2,u.Noting that H01↪L6 and (93)bu,u2,u≤CuL6∇u2L3uimplies that (94)2bu,u2,u≤Cu∇u2L3u≤2νδ0u2+C9u2∇u2L32.Thus (95)Dtα0cut2≤C9u2∇u2L32.By Gronwall inequality and (51), (53), we have (96)ut2≤u02exp⁡C9Γα∫0tt-sα-1∇u2L32ds≤u02exp⁡C9Γα∫0tt-sα-1Au2L44ds≤u02exp⁡C9d3ΓαC1.Since u(0)=0, the last relation can imply that (97)ut2=0,0≤t≤T,and the uniqueness is proved.


## 5. Optimal Control

In this section, we will consider the optimal control problems(98)minimize12∫0T∫Ωux,t-zx,t2dxdt+∫0Thωtdt,where ω∈L2/α1(0,T;U).We will consider the systems with control inΩ⊂R3(99)∂tαu-νΔu+u·∇u-σdiv⁡A2u-βdiv⁡Au2Au=-∇p+c0ω+f,∇·u=0,u=0,x∈∂Ω;ux,0=u0,x∈Ω.Definition 15.
Letu0∈H,f∈L2/α1(0,T;V∗) and ω∈L2/α1(0,T;U); a function u∈L2(0,T;V)∩L∞(0,T;H)∩L4(0,T;W) is called a weak solution of system (7) if (100)∂tαu,v+νu,v+bu,u,v-σdiv⁡A2u+βdiv⁡Au2Au,v=c0ω+f,v,u0=u0.Similarly we give the equivalent form as follows:(101)Dtα0cu+νAu+Bu,u+Ku+Ju=Cω+f,u0=u0.If z∈L2(0,T;H), rewriting problem (98) in the abstract form, we have (102)PminimizeJu,ω=12∫0Tux,t-zx,t2dt+∫0Thωtdtover (u,ω)∈(L2(0,T;V)∩L∞(0,T;H)∩L4(0,T;W))×L2/α1(0,T;U) subject to (101).We assume that(i)
z∈L2(0,T;H);(ii)
the functionh:U→R is convex and lower semicontinuous and satisfies (103)hω≥b1ωU2/α1+b2for some b1>0, b2∈R.Theorem 16.
If hypotheses (i) and (ii) hold, then problem (P) has at least one solution(u∗,ω∗)∈(L2(0,T;V)∩L∞(0,T;H)∩L4(0,T;W))×L2/α1(0,T;U).Proof.
Replacingf with Cω+f in Theorems 13 and 14, we show that system (101) has an weak solution corresponding to ω. Suppose that (um,ωm) are a minimizing sequence of problem (P); namely,(104)Jum,ωm→inf⁡Ju,ω,Dtα0cum+νAum+Bum,um+Kum+Jum=Cωm+f,um0=u0.According to hypothesis (ii), we can conclude that {ωm} is bounded in L2/α1(0,T;U). Thus, there exists ω∗∈L2/α1(0,T;U) such that (105)ωm⇀ω∗inL2/α10,T;Uweakly.Similar to (49), we obtain (106)Dtα0cum2+C0um2+C1AumL44≤1νf+Cωm2.This yields (107)umt2+C0∫0tt-sα-1um2ds+C1∫0tt-sα-1AumL44ds≤um02+1ν∫0Tft+Cωm2/α1ds+1-α1να-α1Tα-1/1-α1.Hence (108)C0Tα-1∫0tums2ds≤C0∫0tt-sα-1ums2ds≤um02+1ν∫0Tft+Cωms2/α1ds+1-α1να-α1Tα-1/1-α1,(109)C1Tα-1∫0tAumsL44ds≤C1∫0tt-sα-1AumsL44ds≤um02+1ν∫0Tft+Cωms2/α1ds+1-α1να-α1Tα-1/1-α1.Therefore, {um} is bounded in L∞(0,T;H)∩L2(0,T;V)∩L4(0,T;W). Thus, there exists an element u∗∈L2(0,T;V)∩L∞(0,T;H)∩L4(0,T;W) such that (110)um→u∗inL∞0,T;Hweak-star,um⇀u∗inL20,T;Vweakly,um→u∗inL20,T;Hstrongly,um⇀u∗inL40,T;Wweakly.We need to have some uniform bound on Aum,B(um,um), K(um), and J(um).
We firstly estimate the termB(um,um). In fact (111)Bum,um,v=Cumumv,∀v∈V.Then (112)Bum,umV∗≤Cumum.Therefore (113)∫0tBums,umsV∗2ds≤C∫0tums2ums2ds≤Csup⁡ums2∫0tums2ds<∞.This implies that there exists η∗ such that (114)Bum,um⇀Bu∗,u∗inL20,T;V∗weakly.For the similar derivations of relation (I), we deduce that (115)νAum+Kum+Jum=T1um⇀T1u∗=νAu∗+Ku∗+Ju∗inL4/30,T;W∗.Now let v∈V; take the scalar product of (104) with v and integrate; it follows (116)umt,v-umt0,v=∫0t0t0-sα-1-t-sα-1bums,ums,v+νAum+Kum+Jum,v-Cωms+fs,vds-∫t0tt-sα-1bums,ums,v+νAum+Kum+Jum,v-Cωms+fs,vds.By similar arguments in Theorem 13 and taking the limits as m→∞, we can conclude (117)u∗t,v-u∗t0,v=∫0t0t0-sα-1-t-sα-1bu∗s,u∗s,v+νAu∗+Ku∗+Ju∗,v-Cω∗s+fs,vds-∫t0tt-sα-1bu∗s,u∗s,v+νAu∗+Ku∗+Ju∗,v-Cω∗s+fs,vds.Applying the Caputo fractional derivative of order α on the sides of the above equality for t0=0, we obtain (118)Dtα0cu∗,v+νAu∗+Ku∗+Ju∗,v+bu∗,u∗,v=Cω∗+f,v,u∗0=u0.Namely, an equivalent form of the last equality is (119)Dtα0cu∗+νAu∗+Ku∗+Ju∗+Bu∗,u∗=Cω∗+f,u∗0=u0.Hence, we can see that (u∗,ω∗) satisfies system (104).
From weak lower semicontinuity of function(u,ω)→J(u,ω), we deduce (120)Ju∗,ω∗=inf⁡Ju,ω.Then (u∗,ω∗) is an optimal pair for problem (P). The proof of Theorem 16 is complete.



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*Source: 7941012-2018-11-13.xml*