On the steady non-Newtonian fluids in domains with noncompact boundaries

# On the steady non-Newtonian fluids in domains with noncompact boundaries

Yang Jiaqi   Yin Huicheng111Yang Jiaqi (yjqmath@163.com ) and Yin Huicheng (huichengnju.edu.cn, 05407njnu.edu.cn) are supported by the NSFC (No. 11571177) and the Priority Academic Program Development of Jiangsu Higher Education Institutions.
1. Department of Mathematics and IMS, Nanjing University, Nanjing 210093, China
2. School of Mathematical Sciences, Jiangsu Provincial Key Laboratory for Numerical
Simulation of Large Scale Complex Systems, Nanjing Normal University, Nanjing 210023, China
###### Abstract

In this paper, we study the steady non-Newtonian fluids in a class of unbounded domains with noncompact boundaries. With respect to the resulting mathematical problems, we establish the global existence of solutions with arbitrary large flux under some suitable conditions, and meanwhile, show the uniqueness of the solutions when the flux is sufficiently small. Our results are an extension or an improvement of those obtained in some previous references.

Keywords. Non-Newtonian fluid, steady, noncompact boundary, Leray problem, Ladyzhenskaya-Solonnikov problem, Korn-type inequality.

2010 Mathematical Subject Classification. 35Q30, 35B30, 76D05, 76D07.

## 1 Introduction

Although the steady Navier-Stokes equations have been investigated extensively (see [2]-[10], [12], [14]-[18], [21]-[29] and the references therein), the global well-posedness of a flow in a domain with noncompact boundaries is still an interesting question for arbitrary fluxes. A special case is that the domain is a distorted infinite cylinder or channel (see [10] and so on), namely, can be described as follows (see Figure 1 and Figure 2 below):

 Ω=2⋃i=0Ωi, (1.1)

where is a smooth bounded subset of , while and are disjoint regions which may be expressed in possibly different coordinate systems and by

 Ωi={(xi1,...,xid)∈Rd:xi1>0,(xi2,..,xid)∈Σi(xi1)},i=1,2, (1.2)

here represents the bounded cross section of for fixed .

Figure 1. Domain for

Figure 2. Domain for

Owing to the incompressibility of the fluids and the vanishing property of the current velocity on the boundary , we deduce that the flux of velocity ( stands for the unit outer normal direction of ) through is a constant independent of the variable , and s () satisfy

 α1+α2=0. (1.3)

When the cross section is independent of , which means that each outlet is a semi-infinite strip for or a semi-infinite straight cylinder for , respectively, will be simply denoted by . In this case, the classical Leray’s problem (see [19]) is to study the well-posedness of the following steady flows:

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩−μΔv+v⋅∇v+∇π=0in Ω,$div$ v=0in Ω,v=0,on ∂Ω,∫Σiv⋅n dS=αii=1,2,v→vi0as xi1→∞ in Ωi for i=1,2, (1.4)

where stands for the velocity of the Poiseuille flow corresponding to the given constant , which is determined by

 ⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩vi0=vi0(zi)e1,μd∑j=2∂2xijvi0(zi)=−Ciin Σi,vi0(zi)=0on ∂Σi (1.5)

with , and being a constant uniquely determined by .

Leray’s problem (1.4)-(1.5) has been extensively studied. In [14], under the smallness assumption of the flux , O.A.Ladyzhenskaya proved the existence of solution but the uniqueness argument was not given. In [2], C.J.Amick completed the proof of both existence and uniqueness when the flux is sufficient small. Alternately, O.A.Ladyzhenskaya and V.A.Solonnikov in [18] considered problem (1.4) together with (1.5) under the weaker assumption that the section is uniformly bounded with respect to the variable instead of the straight outlet in [2]. In this case, one cannot pose the condition of at infinity by Poiseuille flow since section changes for different . Consequently, the authors in [18] considered problem (1.4) in another way, which is called Ladyzhenskaya-Solonnikov Problem I (by prescribing a growth condition of with respect to the distance along the direction of each outlet instead of condition (1.5)), and they established the global existence of for arbitrary large flux by utilizing a variant of Saint-Venant’s principle. Furthermore, if the flux is sufficient small, they got the uniqueness of solution . In particular, if flux is small and both exits and are straight, then it has been shown that the solution to Ladyzhenskaya-Solonnikov Problem I tends to the corresponding Poiseuille solution of (1.5).

In [18], the authors also studied another problem for (1.4), i.e., Ladyzhenskaya -Solonnikov Problem II. At this time, the sections and of are not uniformly bounded and admit some certain rates of “growth”, i.e.,

 Ωi={x=(xi1,yi)∈Rd:xi1>0,|yi|≡√(xi2)2+...+(xid)2

where is a global Lipschitz function. Later, in a series of papers [24, 25, 26, 27], K.Pileckas shows that the Ladyzhenskaya -Solonnikov Problem II is uniquely solvable if flux is small. Simultaneously, it is shown in [24, 25, 26, 27] that the decay rate of solution at infinity is related to the inverse power of the functions and .

The Navier-Stokes model of incompressible fluids is based on the Stokes-hypothesis£¬ which simplifies the relation between the stress tensor and the velocity. However, a number of experiments show that many other incompressible fluids, including bloods, cannot be described by this model. In the late 1960s, see [15, 16], O.A.Ladyzhenskaya started a systematic investigation on the well-possedness of the boundary value problems associated to certain generalized Newtonian models. In contrast to Newtonian flows, for non-Newtonian flows, the viscosity coefficient is no longer constant, it depends on the magnitude of , i.e.,

 μ(D(v))=μ0+μ1|D(v)|p−2, (1.7)

where , , , and with . In this case, the corresponding Leray’s problem in the unbounded pipe domain is described as follows

 (1.8)

where is the Hagen-Poiseuille flow, which satisfies

 ⎧⎪⎨⎪⎩vPi=vPi(zi)e1,μ0Δ′ivPi+∇′i⋅(μ1|D(vPi)|p−2D(vPi))=−Ci,in ΣivPi(zi)=0,on ∂Σi (1.9)

here is a scalar function, , , and .

For the nonlinear equation systems in (1.8), O.A.Ladyzhenskaya [16] and J.L.Lions [20] proved the existence of the solution by the monotone operator theory in a bounded domain when . This result has been improved by some authors, in particular, in [5], the same result is established for . For noncompact boundaries, particularly, for piping-system, G.P.Galdi [8] proved that if , , and flux is small, then problem (1.8) together with (1.9) has a unique weak solution . If , by deriving some ¡°weighted¡± energy estimates, E. Marušić-Paloka in [21] established the existence and uniqueness of the weak solution to problem (1.8) with (1.9) when and the flux is small. Since the approach in [21] requires a detailed information about the dependence of on the cross-sectional coordinates, where an explicit background solution is known, it seems that the resulting proof in [21] is only suitable for the case of a circular cross section. For arbitrary large flux, motivated by Ladyzhenskaya and Solonnikov’s results in [18], the authors in [12] prove the existence and uniqueness of solution to the Ladyzhenskaya-Solonnikov Problem I for the non-Newtonian fluids when and . In this paper, we shall consider both Ladyzhenskaya-Solonnikov Problem I and Ladyzhenskaya-Solonnikov Problem II for the non-Newtonian fluids, and intend to establish some systematic results. Here we point out that the restriction of when is essentially required in the proof of [12] (one can see the statements of lines 8-9 from below on pages 3874 in [12]: “As far as we know, the Leray problem for (with small fluxes) is an open problem”), meanwhile only the corresponding Ladyzhenskaya-Solonnikov problem I is considered in [12]. We shall study problem (1.8) together with (1.9) for and (when , the condition will be needed). On the other hand, for the corresponding Ladyzhenskaya-Solonnikov Problem II of (1.8) (i.e., the outlets of may be permitted to be unbounded), we shall establish both the existence and uniqueness of the solution for and or and (hence ), especially, when the sections of are uniformly bounded, the resulting conclusions also hold for and (here we point out that this case has been solved in [12]).

Let us comment on the proofs of our results. For the case of in (1.8), if one wants to directly deal with the nonlinear term for and apply the integration by parts for equation (1.8) multiplying the solution to obtain a priori estimate of , then the regularities of and for some positive number in bounded domains are required as pointed out in [12]. However this regularity is not expected for the weak solutions of (1.8) if as stated in [12] (see lines 15-16 of pages 3875). To overcome this kind of difficulty, the authors in [12] studied the following truncated modified problem

 ⎧⎪⎨⎪⎩−$div$ (1TD(vT)+μ1|D(vT)|p−2D(vT))+vT⋅∇vT+∇πT=0in Ω(T),$div$ vT=0in Ω(T),vT=0on ∂Ω(T), (1.10)

where . By deriving the uniform estimates of under the key assumption of and applying a local version of the Minty trick, the authors in [12] proved the existence and uniqueness of solution to the Ladyzhenskaya-Solonnikov Problem I of (1.8) when and . We now state our ingredients for treating problem (1.8) in this paper. At first, we consider the following truncated modified problem instead of (1.8)

 ⎧⎪⎨⎪⎩−$div$ (μ0D(vT)+μ1|D(vT)|p−2D(vT))+vT⋅∇vT+∇πT=0in Ω(T),$div$ vT=0in Ω(T),vT=0on ∂Ω(T). (1.11)

As in [12] and [18], we assume that the velocity of (1.11) has the form , where is the new unknown with zero flux, and is a specially constructed solenoidal field satisfying and admitting some other “good” properties. To obtain a priori estimates of , we have to control the nonlinear term . If one only assumes that is bounded as in [12], then it follows from Young inequality and Poincaré inequality that only the following estimate for is obtained

where and stands for a generic constant depending on . From (1), the authors in [12] obtained the crucial uniform estimate of for the solution to problem (1.10). To relax the restriction of power and get the uniform control for solution of problem (1.11), we need more properties of and other interesting observations. Note that, for , the leading term is in the energy estimate of (see (4.4) in ), if one can find a field such that

then instead of can be estimated; while, for and , the leading term is in the estimate of (see (4.4) in ), if one can construct a vector field such that in , then it follows from the Young inequality and Poincaré inequality that

which derives the uniform estimate of if . Thanks to [10] Lemma III.4.3 and [23] Lemma 2-Lemma 3, the aforementioned in (1.13) and (1.14) can be found. On the other hand, for Ladyzhenskaya-Solonnikov Problem I and II, the condition should be required (see problem (2.4) and problem (2.5) in ). Hence, by (1.14) we require such an inequality

 ∫t0|Σi(s)|1−p(d−2)(p−2)(d−1)ds⩽c∫t0|Σi(s)|1−dpd−1ds. (1.15)

In the case of , for Ladyzhenskaya-Solonnikov Problem I, (1.15) is automatically satisfied for any since is bounded, while for Ladyzhenskaya-Solonnikov Problem II, (1.15) is satisfied only for . Based on the uniform estimates of , inspired by [10] and [22], through choosing some suitable test functions and taking some delicate analysis on the resulting nonlinear terms, we can show a.e. in any compact subset of by establishing the uniform interior estimates of solution to (1.11). From this, together with some methods introduced in [18] for treating the Newtonian fluids and involved analysis on the resulting nonlinear terms in non-Newtonian fluids, we eventually complete the proofs on the existence and uniqueness of solution to the related Ladyzhenskaya-Solonnikov Problem I and Ladyzhenskaya-Solonnikov Problem II of (1.8) under some suitable conditions.

Our paper is organized as follows. In , the detailed descriptions on the resulting Ladyzhenskaya-Solonnikov Problems for the non-Newtonian flows are given. In , we present some preliminary conclusions which will be applied to prove our main results in subsequent sections. In , we establish the existence of the solutions to the bounded truncated problem corresponding to (1.8). In , we study the interior regularity of solutions obtained in . Based on and , we shall complete the proofs on Ladyzhenskaya-Solonnikov Problem I and Ladyzhenskaya-Solonnikov Problem II of (1.8) in and respectively.

## 2 Descriptions of Ladyzhenskaya-Solonnikov Problems for non-Newtonian fluids

We focus on the following non-Newtonian fluid problem in the domain with noncompact boundaries (see Figure 3 and Figure 4 below):

 (2.1)

where

 Ω=Ω0∪(N⋃i=1Ωi),

and

 Ωi={x∈Rn:xi1>0,yi=(xi2,..,xid)∈Σi(xi1)}.

Figure 3. Domain for .

Suppose that is a solenoidal field and holds on . Then

 |αi|p=|∫Σi(t)v⋅n dS|p⩽|Σi(t)|p−1∫Σi(t)|v|p dS⩽c|Σi(t)|dpd−1−1∫Σi(t)|∇′v|p dS,

where stands for a generic constant. This means

 |αi|p∫t0|Σi(s)|1−dpd−1ds⩽c∫Ωi(t)|∇v|pdx,

where for . Hence, if and

 Ii(t)≡∫t0|Σi(s)|1−dpd−1ds→+∞ as t→+∞, (2.2)

then

 Qi(t)≡∫Ωi(t)|∇v|pdx→+∞ % as t→+∞. (2.3)

Figure 4. Domain for .

From (2.2) and (2.3), it is natural to consider the following two problems

Suppose that there are two positive constant and such that  for  . We look for a pair vector field to fulfill

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩−$div$ (μ0D(v)+μ1|D(v)|p−2D(v))+v⋅∇v+∇π=0in Ω,$div$ v=0in Ω,v=0on ∂Ω,∫Σi(xi1)v⋅n dS=αi% with N∑i=1αi=0,supt>0t−1Qi(t)<∞for 1≤i≤N, (2.4)

where is defined in (2.3) for , and is defined as for .

Suppose that for , while for , we look for a pair vector field such that

 (2.5)

where is defined in (2.2) for , and is defined as for .

In subsequent sections, we shall focus on the studies on these two problems above. The obtained results will be stated in Theorem 6.1-Theorem 6.2 and Theorem 7.3-Theorem 7.4 respectively. In addition, for notational convenience, we introduce some function spaces as follows:

 D1,p(Ω)={u∈L1loc(Ω):∇u∈Lp(Ω)}, D1,p0(Ω)={\ completion\ of\ C∞0(Ω)\ in\ the\ semi-norm\ ∥∇u∥p,Ω≡(∫Ω|∇u|pdx)1p}, D−1,p′(Ω)=(D1,p(Ω))′,D−1,p′0(Ω)=(D1,p0(Ω))′, D(Ω)={u∈C∞0(Ω):∇⋅u=0}, D1,p0(Ω)={completion\ of\ D(Ω) \ in\ the\ semi-norm\ ∥∇u∥p,Ω}.

## 3 Preliminary results

In this part, some preliminary results will be listed so that we can apply them to study the described problems in . It follows from the proof of Appendix of [18] that we have

Lemma 3.1. Let and . Then

 ∥u∥r,ω⩽c(d,r,q)max{1,(t2−t1)−1dmaxx1∈[t1,t2]|Σ(x1)|1d(d−1)}|ω|1d+1r−1q∥∇u∥q,ω,

where , if ; , if ; , if .

The following result will play a crucial role in estimating pressure in problems (2.4)-(2.5).

Lemma 3.2. (Theorem III.3.3 of [10]) For a bounded Lipschitzian domain , suppose that

 f∈Lp(ω) and ∫ωfdx=0, (3.1)

where . Then one can find a vector field such that

 ⎧⎪⎨⎪⎩∇⋅w=0,w∈W1,p0(ω),∥w∥1,p⩽M(ω)∥f∥p, (3.2)

where is a constant depending only on the Lebesgue’s measure of domain .

Remark 3.1. Assume that is a star-shaped domain with respect to a ball B with the radius . Then it follows from Theorem III.3.1 of [10] that the positive constant in (3.2) satisfies . This property will be useful in order to solve problem (2.5).

Remark 3.2. If with , then one can find a vector field (see Remark III.3.12 of [10]) such that

 ∥w∥1,p⩽M(ω)∥f∥p  and  ∥w∥1,r⩽M(ω)∥f∥r.

Next, we list some results, whose proofs can be found in Lemma 2.3 of [18] or Lemma 3.1 of [12].

Lemma 3.3. Let be a fixed constant with and . In addition, we suppose that is a monotonically increasing function, equal to zero for and equal to infinity for .

(i) Assume that the nondecreasing, nonnegative smooth functions z(t) and , not identically equal to zero, satisfy the following inequalities for all ,

 z(t)⩽Ψ(z′(t))+(1−δ)φ(t), (3.3)

and

 φ(t)⩾δ−1Ψ(φ′(t)). (3.4)

If

 z(T)⩽φ(T), (3.5)

then for all ,

 z(t)⩽φ(t). (3.6)

(ii) Assume that inequalities (3.3) and (3.4) are fulfilled for all . Then (3.6) holds for if

 liminft→∞z(t)φ(t)<1, (3.7)

or if z(t) has an order of growth for , less than the order of growth of the positive solutions to the equation

 ~z(t)=δ−1Ψ(~z′(t)). (3.8)

(iii) Assume that the nonidentical zero nonnegative functions z(t), satisfying the homogenous inequality

 z(t)⩽δ−1Ψ(z′(t)) for t⩾t0, (3.9)

increases unboundedly for . If holds for and , then

 liminft→∞t−mm−1z(t)>0; (3.10)

if, however, holds for , then

 liminft→∞z(t)exp(−tc0)>0. (3.11)

The following Korn-type inequality can be referred in Theorem 3.2 of [13] or Theorem 1 of [9].

Lemma 3.4. Let K be a cone in and . If , then there is a skew-symmetric matrix A with constant coefficients such that

 ∫K|∇(u(x)−Ax)|pdx⩽C∫K|D(u)(x)|pdx, (3.12)

where the positive constant C does not depend on the function itself.

Remark 3.3. If , then holds in (3.12).

Finally, we state a conclusion as follows, whose proof can be found in [10] Lemma III.4.3, and [23] Lemma 2-Lemma 3.

Lemma 3.5. Assume that the domain and the numbers are defined in problem (2.4) or problem (2.5). Let . Then for any fixed , there exists a smooth divergence-free vector field which vanish in a neighborhood of (), and which satisfies

 (i) |a|⩽c(ε)α|Σi(t)|−1 and |∇a|⩽c(ε)α|Σi(t)|−dd−1for x∈Ωi(t) and 1≤i≤N. (ii) ∫Σi(t)a⋅n dS=αifor 1≤i≤N. (iii) ∫Ω0a2w2dx⩽εα2∫Ω0|∇w|2dxfor % any w∈D(Ω). (iv) ∫Ωi(t2)∖Ωi(t1)a2w2dx⩽εα2∫Ωi(t2)∖Ωi(t1)|∇w|2dxfor any w∈D(Ω), t2>t1>0, and 1≤i≤N.

## 4 Existence of solutions to problems (2.4) and (2.5) in bounded truncated domains

In this part, for the following problem in the bounded domain

 ⎧⎪⎨⎪⎩−$div$ (μ0D(vT)+μ1|D(vT)|p−2D(vT))+vT⋅∇vT+∇πT=0in Ω(T),$div$ vT=0in Ω(T),vT=0on ∂Ω(T), (4.1)

we intend to find a weak solution such that

 μ0(D(uT)+D(a),D(ψ))+μ1(|D(uT)+D(a)|p−2(D(uT)+D(a)),D(ψ))=(uT⋅∇ψ,uT)+(uT⋅∇ψ,a)+(a⋅∇ψ,uT)+(a⋅∇ψ,a), ∀ ψ∈D(Ω(T)), (4.2)

where the vector value function is given in Lemma 3.5.

Theorem 4.1. Let and or and . Then there is a vector field such that (4.2) holds, and , if ; , if .

Proof. Although the proof of Theorem 4.1 is standard as in [8]-[9] and [12], where the authors treated problem (4.2) for different vector value function , we still give out the detailed proof for the sake of completeness.

. ,

Let { be a basis in . We look for a series such that satisfies

 (4.3)

Multiplying both sides of (4.3) by and summing over yield

 (4.4)

Using Schwarz inequality we get

 μ0(D(a),D(uTm))⩾−μ02∥D(uTm)∥22−μ02∥D(a)∥22. (4.5)

 ∥D(uTm)+D(a)∥pp⩾12p−1∥D(uTm)∥pp−∥D(a)∥pp. (4.6)

We also notice that, by Hölder inequality and Young inequality,

 ∣∣μ1(|D(uTm)+D(a)|p−2(D(uTm)+D(a)),D(a))∣∣⩽μ12p∥D(uTm)∥pp+c∥D(a)∥pp, (4.7)

where and below denotes by a generic positive constant. By Lemma 3.5 (iii) and (iv) we have that for any fixed ,