# A New Kind of Weak Solution of Non-Newtonian Fluid Equation

Authors: Huashui Zhan; Bifen Xu
Journal: Journal of Function Spaces
 (2017)

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## Abstract

If the non-Newtonian fluid equation with a diffusion coefficient is degenerate on the boundary, the weak solution lacks the regularity to define the trace on the boundary. By introducing a new kind of weak solutions, the stability of the solutions is established without any boundary condition.


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## Body

## 1. Introduction and the Main Results

The quasilinear parabolic equation(1)∂u∂t-divax∇up-2∇u-bixDiu+cx,tu=fx,t,x,t∈QT=Ω×0,Tcomes from a host of applied fields such as the theory of non-Newtonian fluid, the study of water infiltration through porous media, and combustion theory; one can refer to [1–4] and the references therein. Here p>1, Di=∂/∂xi, a∈C(Ω¯), Ω⊂RN is a bounded domain with the appropriately smooth boundary ∂Ω. If a(x)>c>0, the equations with the type of (1) have been extensively studied; one can refer to [5–7] and the references therein. If a(x)≥0, one wants to obtain the well-posedness of the equation; the initial value (2)ux,0=u0x,x∈Ωis invariably imposed. But the boundary value condition (3)ux,t=0,x,t∈∂Ω×0,Tmay be overdetermined. Yin and Wang [8] made a more important devoting to the problem; they classified the boundary into three parts: the nondegenerate boundary, the weakly degenerate boundary, and the strongly degenerate boundary, by means of a reasonable integral description. The boundary value condition should be supplemented definitely on the nondegenerate boundary and the weakly degenerate boundary. On the strongly degenerate boundary, they formulated a new approach to prescribe the boundary value condition rather than defining the Fichera function as treating the linear case. Moreover, they formulated the boundary value condition on this strongly degenerate boundary in a much weak sense since the regularity of the solution is much weaker near this boundary. In a word, instead of the whole boundary condition (3), only a partial boundary condition (4)ux,t=0,x,t∈Σp×0,Tis imposed in [8], where Σp⊆∂Ω.In our paper, for simplism, we assume thata(x), bi(x), and c(x,t) are C1 functions, and (5)ax>0,x∈Ω,ax=0,x∈∂Ω;the equation is degenerate on the boundary. In our previous works [9, 10], we have shown that such degeneracy may result in the fact that the weak solution of the equation lacks the regularity to define the trace on the boundary. Accordingly, how to construct a suitable function, which is independent of the boundary value condition, to obtain the stability of the weak solutions, becomes formidable. The main aim of the paper is to solve the corresponding problem by introducing a new kind of the weak solutions.Definition 1.
Functionu(x,t) is said to be a weak solution of (1) with the initial value (2), if (6)u∈L∞QT,ut∈L2QT,ax∇up∈L1QT,and for any function g(s)∈C1(R), g(0)=0, φ1∈C01(Ω)¯, φ2∈L∞(0,T;Wloc1,p(Ω)), (7)∬QTutgφ1φ2+ax∇up-2∇u·∇gφ1φ2+ubixixgφ1φ2+bixgxiφ1φ2-cx,tugφ1φ2+fx,tgφ1φ2dxdt=0.The initial value is satisfied in the sense of that (8)limt→0∫Ωux,t-u0xdx=0.
The existence of the solution can be proved in a similar way as that in [8]; we omit the details here. In our paper, we mainly are concerned about the stability of the weak solutions without any boundary value condition.Theorem 2.
Letu,v be two weak solutions of (1) with the initial values u0(x),v0(x), respectively; suppose p>1 and (9)bix≤cax.If (10)∫Ωa-p-1xdx≤c,then (11)∫Ωux,t-vx,tdx≤c∫Ωu0x-v0xdx,a.e.t∈0,Tis true without any boundary value condition.Theorem 3.
Letu,v be two nonnegative solutions of (1) with the initial values u0, v0, respectively. If 1<p≤2 and (12)∫Ωa-1/p-1xdx<∞,then the stability of the weak solutions is true in the sense of (11).Let us give a comparison between Theorems2 and 3. To see that, we specially assume that (13)Ω=BR0=x∈RN:x<R,ax=R-x+α,α>0.Then it is easy to know that if α<1/p-1, then condition (10) is satisfied; while α<p-1, then condition (12) is true. Thus if 1<p<2, then (14)1p-1>p-1,which implies that when R≤1, R-|x|<1, (15)∫Ωa-p-1xdx<∫Ωa-1/p-1xdx.In this case, Theorem 2 cannot include Theorem 3; Theorem 3 has its independent sense. Certainly, if p≥2, Theorem 2 has its sole important significance.At the same time, instead of condition (10) (or (12)), we have the following results in the stability or the local stability.Theorem 4.
Letu,v be two nonnegative solutions of (1) with the initial values u0, v0, respectively. If p>1, a(x) satisfies (12), and, for small enough λ>0, u(x) and v(x) satisfy that (16)1λ∫Ω∖Ωλax∇updxp-1/p≤c,1λ∫Ω∖Ωλax∇vpdxp-1/p≤c,then stability (11) is true. Here Ωλ={x∈Ω:a(x)>λ}.Theorem 5.
Letu, v be two solutions of (1) with the differential initial values u0(x),v0(x), respectively. Then there exists a positive constant β≥1 such that (17)∫Ωaβux,t-vx,t2dx≤c∫Ωaβu0x-v0x2dx.In particular, for any small enough constant δ>0, (18)∫Ωδux,t-vx,t2dx≤cδ-β∫Ωu0x-v0x2dx.
Ifu0=v0, by the arbitrariness of δ, one can see that u(x,t)=v(x,t),a.e.(x,t)∈QT; the uniqueness of the solution is true.We have used some techniques in [9]. But there are many essential improvements in our paper. The main results of my previous work [9] were established on the assumption of that(19)ux≤cdx,vx≤cdx,where d(x) is the distance function from the boundary. Condition (19) is much stronger than the usual homogeneous boundary value condition (3), so the conclusions in [9] are not perfect. But in my new paper, we have introduced the new kind of the weak solutions (Definition 1); also we can establish the stability of the weak solutions without any boundary value condition.


## 2. The Proof of Theorem2

For smallη>0, let (20)Sηs=∫0shητdτ,hηs=2η1-sη+.Obviously hη(s)∈C(R), and (21)hηs≥0,shηs≤1,Sηs≤1;limη→0⁡Sηs=sgn⁡s,limη→0⁡sSη′s=0.Proof of Theorem2.
Letu, v be two solutions of (1) with the initial values u0(x),v0(x). We can choose Sη(aβ(u-v)) as the test function. Then (22)∫ΩSηaβu-v∂u-v∂tdx+∫Ωaβ+1x∇up-2∇u-∇vp-2∇v·∇u-vSη′aβu-vdx+∫Ωax∇up-2∇u-∇vp-2∇v·∇aβu-vSη′aβu-vdx+∫ΩDibixu-vSηaβu-v+aβbixu-v·u-vxiSη′aβu-vdx+∫Ωbixu-v·axiβu-vSη′aβu-vdx+∫Ωcx,tu-vSηaβu-vdxdt=0.
Thus(23)limη→0∫ΩSηaβu-v∂u-v∂tdx=ddtu-v1,∫Ωaβ+1x∇up-2∇u-∇vp-2∇v·∇u-vSη′aβu-vdx≥0.
Since|∇a(x)|≤c in Ω, we have (24)∫Ωaβu-vSη′aβu-v∇up-2∇u-∇vp-2∇vdx=∫Ω:aβu-v<ηa-p-1/paβu-vSη′aβu-vap-1/p∇up-2∇u-∇vp-2∇vdx≤∫Ω:aβu-v<ηa-p-1/paβu-vSη′aβu-vpdx1/p∫Ω:aβu-v<ηax∇up+∇vpdxp-1/p.If {x∈Ω:u-v=0} has 0 measures, since (25)∫Ωa-p-1xdx<∞,consequently (26)∫Ω:aβu-v<ηa-p-1/paβu-vSη′aβu-vpdx≤c,limη→0∫Ω:aβu-v<ηax∇up+∇vpdxp-1/p=∫Ω:u-v=0ax∇up+∇vpdxp-1/p=0.If {x∈Ω:u-v=0} has a positive measure, obviously, (27)limη→0∫Ω:aβu-v<ηa-p-1/paβu-vSη′aβu-vpdx1/p=∫Ω:u-v=0a-p-1/paβu-vSη′aβu-vpdx1/p=0.By (21) and condition (10), using the Lebesgue dominated convergence theorem, in both cases, we have (28)limη→0∫Ωaβu-vSη′aβu-v∇up-2∇u-∇vp-2∇vdx=0.
While, by (9), |bi(x)|≤ca(x),(29)∫Ωbixu-v·axiβu-vSη′aβu-vdx≤∫Ωbixu-vSη′aβu-vaβu-vaxiβaβdx≤c∫ΩSη′aβu-vaβu-vbix∇aadx≤c∫ΩSη′aβu-vaβu-vdx.Then (30)limη→0∫Ωbixu-v·axiβu-vSη′aβu-vdx=0.
Moreover, by|bi(x)|≤ca(x), (31)∫Ωbixaβxu-vSη′aβu-vu-vxidx≤c∫Ωax∇up+∇vpdx1/p∫Ωaβxu-vSη′aβu-vp/p-1p-1/p.Therefore, we have (32)limη→0∫Ωbixu-vSη′aβu-vu-vxiaβxdx=0.limη→0∫ΩDibixu-vSηaβu-v+∫Ωcx,tu-vSηaβu-vdx≤cu-v1.
Now, letη→0 in (22). Then (33)ddtu-v1⩽cu-v1.It implies that (34)∫Ωux,t-vx,tdx⩽cT∫Ωu0-v0dx,∀t∈0,T.Theorem 2 is proved.Corollary 6.
Letu,v be two weak solutions of (1) with the initial values u0(x),v0(x), respectively. If (9) is true and it is supposed that (35)∫Ω∇up-1dx≤c,∫Ω∇vp-1dx≤c.then the stability (36)∫Ωux,t-vx,tdx≤c∫Ωu0x-v0xdx,a.e.t∈0,Tis true without any boundary value condition.Proof.
If (35) is true, then (30) is true by (21). Thus the corollary can be proved in a similar way as that of Theorem 2


## 3. The Proofs of Theorems3 and 4

Proof of Theorem3.
By Definition1, for any φ1∈C01(Ω)¯, φ2∈L∞(0,T;Wloc1,p(Ω)), we have (37)∬QT∂u-v∂tφ1φ2+ax∇up-2∇u-∇vp-2∇v·∇φ1φ2+u-vbixixφ1φ2+bixφ1φ2xi-cx,tu-vφ1φ2dxdt=0.
For a small positive constantλ>0, Ωλ={x∈Ω:a(x)>λ} as before, let (38)ϕλx=1,ifx∈Ωλ,1λax,ifx∈Ω∖Ωλ.
Now, we can chooseφ1=ϕλ(x), φ2=χ[τ,s]Sη(u-v), and integrate them over QT; accordingly, (39)∫τs∫ΩϕλxSηu-v∂u-v∂tdxdt+∫τs∫Ωϕλxax∇up-2∇u-∇v2∇v·∇u-vSη′u-vdxdt+∫τs∫Ωax∇up-2∇u-∇vp-2∇v·∇ϕλxSηu-vdxdt+∫τs∫Ωu-vbixixϕλxSηu-v+bixϕλxSηu-vxidxdt-∫τs∫Ωcx,tu-vϕλxSηu-vdxdt=0.Clearly,(40)∫Ωϕλxax∇up-2∇u-∇vp-2∇v·∇u-vSη′u-vdx≥0,(41)∫Ωax∇up-2∇u-∇v2∇v·∇ϕλxSηu-vdx≤∫Ω∖Ωλax∇up-2∇u-∇v2∇v·∇ϕλxSηu-vdx≤∫Ω∖Ωλax∇up-2∇u-∇v2∇v∇ϕλxdx≤cλ∫Ω∖Ωλax∇up-1∇adx+∫Ω∖Ωλax∇vp-1∇adx.Since 1<p≤2, |∇a|≤c, (42)∫Ω∖Ωλ∇apdx≤cλ≤cλp-1,then (43)cλ∫Ω∖Ωλax∇apdx1/p≤cλλ∫Ω∖Ωλ∇apdx1/p≤c.By (41)-(43), using the Ho¨lder inequality, (44)∫Ωax∇up-2∇u-∇v2∇v·∇ϕλxSηu-vdx≤cλ∫Ω∖Ωλax∇up-1∇adx+∫Ω∖Ωλax∇vp-1∇adx≤cλ∫Ω∖Ωλa∇apdx1/p∫Ω∖Ωλax∇updxp-1/p+cλ∫Ω∖Ωλax∇apdx1/p∫Ω∖Ωλax∇vpdxp-1/p≤c∫Ω∖Ωλax∇updxp-1/p+c∫Ω∖Ωλax∇vpdxp-1/p.Thus (45)limλ→0∫Ωax∇up-2∇u-∇vp-2∇v·∇ϕλxSηu-vdx=0.There is one more point that I should touch on is that, by that ∫Ω‍a-1/p-1(x)dx<c, using (21) and the Lebesgue dominated convergence theorem,(46)limη→0∫Ωϕλbixu-vSη′u-vu-vxidx≤limη→0∫Ωa-1/pSη′u-vu-vp/p-1dxp-1/p∫Ωax∇up+∇vpdx1/p=0,limλ→0∫Ωϕλxibixu-vSηu-vdx≤limλ→0cλ∫Ω∖Ωλ∇adx≤limλ→0cλ∫Ω∖Ωλax∇apdx1/p∫Ω∖Ωλa-1/p-1xdxp-1/p=0,by (43) and ∫Ω‍a-1/p-1(x)dx<c, while (47)limη→0∫Ωu-vbixixϕλxSηu-vdx≤∫Ωu-vdxis obviously true.
By (46)-(47), (48)limη→0∫Ωu-vbixixϕλxSηu-v+bixϕλxSηu-vxidx≤∫Ωu-vdx.
At last,(49)limη→0⁡limλ→0∫τs∫ΩϕλxSηu-v∂u-v∂tdxdt=∫τsddtu-vL1Ωdt.
Now, after lettingλ→0, let η→0 in (37). Then, using (40), (45)–(49), and by the Gronwall inequality, we have (50)∫Ωux,t-vx,tdx⩽c∫Ωu0-v0dx.Proof of Theorem4.
In the first place, similar to the proof of Theorem3, we have (39)–(41). There is one more point that we should touch on that since u(x) and v(x) satisfy (16), using the Ho¨lder inequality, we have (51)∫Ωax∇up-2∇u-∇v2∇v·∇ϕλxSηu-vdx≤cλ∫Ω∖Ωλa∇apdx1/p∫Ω∖Ωλax∇updxp-1/p+cλ∫Ω∖Ωλax∇apdx1/p∫Ω∖Ωλax∇vpdxp-1/p≤c∫Ω∖Ωλa∇apdx1/p+c∫Ω∖Ωλax∇apdx1/p,which goes to zero as λ→0 since that a(x)∈C1(Ω). Thus we have (45) too. Last but not the least, since ∫Ω‍a-1/p-1(x)dx<∞, similar to the proof of Theorem 3, we have (46)–(49). So, as the proof of Theorem 3, we know that stability (11) is true.


## 4. The Local Stability

Proof of Theorem5.
Letu, v be two solutions of (1) with the initial values u0(x),v0(x), respectively. From the definition of the weak solution, if g(s)=s, for any φ1∈C01(Ω)¯, φ2∈L∞(0,T;Wloc1,p(Ω)), (52)∬QTφ1φ2∂u-v∂tdxdt=-∬QTax∇up-2∇u-∇vp-2∇v∇φ1φ2dxdt-∬QTu-vbixiφ1φ2+bixφ1φ2xidxdt+∬QTu-vcx,tφ1φ2dxdt.In particular, we choose (53)φ1=aβ,φ2=χτ,su-v,where χ[τ,s] is the characteristic function on [τ,s] and the constant β≥1. Denoting Qτs=Ω×[τ,s], then (54)∬Qτsax∇up-2∇u-∇vp-2∇v∇u-vaβdxdt=∬Qτsa1+β∇up-2∇u-∇vp-2∇v∇u-vdxdt+∬Qτsax∇up-2∇u-∇vp-2∇vu-v∇aβdxdt.Clearly, (55)∬Qτsa1+β∇up-2∇u-∇vp-2∇v∇u-vdxdt≥0.For the second term on the right-hand side of (54), since |∇a|≤c, (56)∬Qτsu-vax∇up-2∇u-∇vp-2∇v∇aβdxdt≤∬Qτsu-vax∇up-1+∇vp-1∇aβdxdt≤c∫τs∫Ωax∇up+∇vpdxdtp-1/p·∫τs∫Ωax∇aβpu-vpdxdt1/p≤c∫τs∫Ωax∇up+∇vpdxdtp-1/p·∫τs∫Ωa1+pβ-1u-vpdxdt1/p≤c∫τs∫Ωa1+pβ-1u-vpdxdt1/p.Now, since β≥1 implies 1+p(β-1)≥β, we have (57)∬Qτsu-va∇up-2∇u-∇vp-2∇v∇aβdxdt≤c∫τs∫Ωaβu-vpdxdt1/p.If p≥2, (58)∫τs∫Ωaβu-vpdxdt1/p≤c∫τs∫Ωaβu-v2dxdt1/p.If 1<p<2, by Ho¨lder inequality, (59)∫τs∫Ωaβu-vpdxdt1/p≤c∫τs∫Ωaβu-v2dxdt1/2.Moreover, by β≥1, u,v∈L∞(QT), we easily see that (60)∬Qτsbixu-vu-vaβxidxdt=∬Qτsbixu-v2axiβdxdt+∬Qsbixu-vu-vxiaβdxdt≤c∫τs∫Ωλu-vaβ-1dx+∫τs∫Ωaβ-1/pp′u-vp′dxdt1/p′∫τs∫Ωax∇up+∇vpdxdt1/p≤c∫τs∫Ωaβu-v2dxdt1/2+c∫τs∫Ωaβu-v2dxdt1/p′.At last, it is easily to deduce that (61)∬Qτsbixixu-v2+cx,tu-v2aβdxdt≤c∫τs∫Ωaβu-v2dxdt1/2,∬Qτsu-vaβ∂u-v∂tdxdt=∫Ωaβux,s-vx,s2dx-∫Ωaβux,τ-vx,τ2dx.
By (54)–(61), we let λ→0 in (52). Then (62)∫Ωρβux,s-vx,s2dx-∫Ωaβux,τ-vx,τ2dx≤c∫0s∫Ωρβux,t-vx,t2dxdtq,where q<1. By (62), we easily show that (63)∫Ωaβux,s-vx,s2dx≤∫Ωaβux,τ-vx,τ2dx.Thus, by the arbitrariness of τ, we have (64)∫Ωaβux,s-vx,s2dx≤∫Ωaβu0-v02dx.The proof is complete.



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