Martin FUCHSJan MüLLER

Fachbereich 6.1 Mathematik,Universit?t des Saarlandes,Postfach 15 11 50,D–66041 Saarbrücken,Germany

E-mail:fuchs@math.uni-sb.de;jmueller@math.uni-sb.de

Abstract We consider entire solutions u of the equations describing the stationary fl ow of a generalized Newtonian fl uid in 2D concentrating on the question,if a Liouville-type result holds in the sense that the boundedness of u implies its constancy.A positive answer is true for p- fl uids in the case p>1(including the classical Navier-Stokes system for the choice p=2),and recently we established this Liouville property for the Prandtl-Eyring fl uid model,for which the dissipative potential has nearly linear growth.Here we fi nally discuss the case of perfectly plastic fl uids whose fl ow is governed by a von Mises-type stress-strain relation formally corresponding to the case p=1.It turns out that,for dissipative potentials of linear growth,the condition ofμ-ellipticity with exponentμ <2 is sufficient for proving the Liouville theorem.

Key words generalized Newtonian fl uids;perfectly plastic fl uids;von Mises fl ow;Liouville theorem

1 Introduction

In this note we look at entire solutions u:R2→R2of the homogeneous equation

together with the incompressibility condition

here u denotes the velocity fi eld of a fl uid and π :R2→R is the a priori unknown pressure function.In equation(1.1)and in what follows we adopt the convention of summation with respect to indices repeated twice.By ε(u)we denote the symmetric gradient of the fi eld u and TDrepresents the deviatoric part of the stress tensor being characteristic for the fl uid under consideration.For further mathematical and also physical explanations we refer to the monographs of Ladyzhenskaya[14],Galdi[8,9]and of M′alek,Necˇas,Rokyta,R?uˇziˇcka[15]as well as to the book[6].A case of particular interest arises,when TDis of the type

for a potential H such that

holds with a given density h:[0,∞)→ [0,∞)of class C2.Combining(1.3)and(1.4),we see that

holds,and equation(1.5)includes as particular cases

(i)power law models

(ii)Prandtl-Eyring fl uids h(t)=tln(1+t),t≥0.

As a matter of fact we recover the Navier-Stokes equation(NSE)just by letting h(t)=t2,and in their fundamental paper[13],Koch,Nadirashvili,Seregin and Sverˇak obtained the following Liouville-type result as a byproduct of their investigations on the regularityof solutions to the instationary variant of(NSE).

Theorem 1.1Suppose that u:R2→R2is a solution of(1.1)and(1.2).Let in addition(1.3)and(1.4)hold for the choice h(t)= νt2with some positive constant ν.Then,if

u must be a constant vector.

For(NSE),di ff erent types of Liouville theorems were studied.For example,Gilbarg and Weinberger showed in their paper[10]the constancy of fi nite energy solutions to(NSE)in the plane,more precisely,the conclusion of Theorem 1 remains valid if(1.6)is replaced by Z

We wish to remark that the proofs of the above results use the linearity of the leading part of(NSE)in an essential way.However,referring to the results obtained in[2,3,7,21]and[22],we could show by applying appropriate arguments.

Theorem 1.2Suppose that u∈C1(R2,R2)is a(weak)solution of equations(1.1)and(1.2)on the whole plane with TDgiven by(1.3)and(1.4)for a function h being subject to(i)or(ii).Suppose that either(1.6)holds or that(1.7)is replaced by

Then u is a constant vector.

Remark 1.3In the case of(NSE)weak solutions of some local Sobolev class are automatically smooth.This is in general not clear for generalized Newtonian fl uids and we therefore assume at least u∈C1.A slightly weaker formulation can be found in[12].

Remark 1.4A comprehensive survey of Theorem 2 including further related results is given in the paper[4].

Let us now turn to the von Mises fl ow(see e.g.,[11,16,20])for which we formally have

which means(recall(1.4))H(ε(u))=|ε(u)|.Since this density is neither di ff erentiable nor strictly convex,equation(1.3)and thereby identity(1.1)can only be interpreted in a very weak sense,and we have no idea,if in this setting a Liouville-type result can be expected.For this reason we follow the ideas of[1]and replace the density from(1.9)through a family hμ,μ >1,of more regular densities being still of linear growth and such that hμ(|ε(u)|)→ |ε(u)|as μ → ∞.For example we may take(μ>1)

where we have abbreviated

Note that actually

and if we formally letμ=1 in the fi rst line of(1.11),then-up to negligible terms-we recover the Prandtl-Eyring model(ii).In the same spirit,the choiceμ<1 leads to p- fl uids with value p:=2?μ.Of course our considerations are not limited to the particular density hμ.More general,we can choose any function h:[0,∞)→ [0,∞)of class C2such that

for any t≥0 and with exponent

c1,c2denoting positive constants.It is immediate that the functions hμde fi ned in(1.10)satisfy(1.12)–(1.14).Moreover,if we de fi ne H according to(1.4),then conditions(1.12)–(1.14)imply theμ-ellipticity of H,i.e.,

being valid for(2 × 2)-matrices ε,σ.In addition,the potential H is of linear growth.More precisely we have(see Lemma 2.7 in[1])

with c1and c2from(1.13)and(1.14),respectively.Now we can state our main result.

Theorem 1.5Let u∈C1(R2,R2)denote a(weak)solution of(1.1)and(1.2)on the whole plane with deviatoric stress tensor de fi ned according to(1.3)and(1.4),where h satis fi es conditions(1.12)–(1.15).In addition we assume that

If then the velocity fi eld u is bounded,it must be a constant vector.

Remark 1.6Limitation(1.18)enters for two reasons.First,as it will become evident from the proof of Theorem 1.5,it plays the role of a technical restriction making it possible to manage certain quantities.Second,the results obtained in[1]suggest that there is some hope for the existence of regular weak solutions in caseμ<2 motivating our assumption u∈C1,whereas counterexamples taken from a slightly di ff erent setting(see[5])indicate that forμ>2 equations of the form(1.1)may fail to have solutions even on bounded domains,which can be found in some Sobolev space.Due to the linear growth of H stated in(1.17)and with respect to the ellipticity condition(1.16),the space of functions with bounded deformation(see e.g.[17–19])seems to be the appropriate class for discussing(1.1)but it is not evident how to give a reasonable“very weak” formulation of equation(1.1)and to investigate the Liouville property in this setting.

Remark 1.7We emphasize that our arguments for the proof of Theorem 1.5 fail in dimension n≥3.However,for the three dimensional Stokes-type problem,

we have a variant of Theorem 1.3 from[7],i.e.under the assumptions of Theorem 1.5 on the function H and the parameterμ,every entire solution u∈ C1(R3,R3)of the above system,for which|x|?α|u(x)|is bounded with some α ∈ [0,1/2),must be constant.The proof follows along the lines of[7].In the case n=2,this result holds true even for the optimal parameter range α∈[0,1),cf.Theorem 1.1 in[7].

2 The Proof of the Main Result

Proof of Theorem 1.5We follow the arguments outlined in Section 3 of[7]keeping the notation introduced in Theorem 1.5 and assuming that all the hypothesis of Theorem 1.5 are valid.We start with

Lemma 2.1There exists a constant c=c<∞such that

holds for any square QR(x0)? R2,QR(x0):=?x∈R2:|xi?x0i|0,x0∈R2.

Proof of Lemma 2.1From equations(1.1)and(1.2)we infer(recalling also(1.3))after an integration by parts

for a constant c>0 which is independent of R and x0.Now choosing ? = η2u?w in(2.2),we arrive at the identity

Let us look at the quantities on the l.h.s.of(2.3):for any tensor ε it holds(recall(1.4)and(1.5))

and the convexity of h(see(1.13))together with(1.12)implies

thus th′(t)≥ h(t)and therefore

Using the boundedness of DH(compare(1.14)),we see

on account of|?η|≤ c/R by the choice of η and due to the boundedness of u.Again by the boundedness of DH it follows from the properties of w

Let us note that during our calculations c is a generic constant independent of R and x0.Combining the above estimates with(2.4)and returning to(2.3),we get

For the quantities occurring on the r.h.s.of(2.3)we can quote(3.6)and(3.7)in[7],hence

and by inserting(2.6)into(2.5)the claim of Lemma 2.1 follows. ?

Up to now neither the condition ofμ-ellipticity(see(1.13)and(1.16))nor the bound(1.18)on the parameterμhave entered our discussion,which means that estimate(2.1)is valid under much weaker hypotheses as required in Theorem 1.5.The full strength of our assumptions is needed in the next step.We return to equation(2.2)and replace ? by ?α? for ?∈,div?=0.After an integration by parts we obtain

At this stage we recall our assumption u∈C1(R2,R2),which enables us with the help of the di ff erence-quotient technique to deduce u∈from equation(2.2)and to justify(2.7).Note also that(2.7)corresponds exactly to formula(3.10)in[7]and with the choice of ? as in this reference,we deduce identity(3.12).Without any changes we can pass to inequality(3.18)from this reference,i.e.,setting ω :=D2H(ε(u))(?αε(u),?αε(u)),we have

for any δ>0 and all squares Q2R(x0)with arbitrary ? ∈(Q32R(x0)).Specifying ? as 0≤ ? ≤ 1,? =1 on QR(x0)and|??|≤ c/R,we my furthermore pass to inequality(3.19)from[7]which reads as

where we use H?lder’s inequality and condition(1.16).Let us choose τ:= δc(δ)?1R2in(2.10)with c(δ)from(2.9).Then it follows with a new constant?c(δ),

and if we select τ:= δc(δ)?1R in(2.10),we fi nd

In(2.11)and(2.12)we have to get rid of the quantityRQ2R(x0)?2(1+|ε(u)|)μdx by absorbing it into the left-hand sides.It holds for any λ>0,

on account of Young’s inequality and due to our assumption μ <2.By choosing λ proportional to R2,we infer from(2.11),

whereas λ～R in combination with(2.12)yields

We insert(2.14)and(2.15)into(2.9),replacing δ by δ/3 and get for any δ>0 and all squares Q2R(x0)?R2,

To inequality(2.16)we can apply Lemma 3.1 from[7]to get

for arbitrary squares QR(x0).Clearly,Θ(R)is bounded and hence

Our goal is to show Θ(R)→ 0 as R → ∞ since,together with inequality(2.17),this implies ε(?u)≡ 0,hence?2u≡ 0 which means that u is an affine function.The assumption on the boundedness of u then gives the assertion of Theorem 1.5.

We start with the term R?2RQ2R(x0)|ε(u)|dx by noting that due to h′(0)=h(0)=0,we have(for β>0 arbitrarily small)

which means that

Consequently,it holds

by(2.18),we may absorb the term αT2R(x0)?2|ε(u)|2dx in the middle-term of estimate(2.22)for α sufficiently small,thus

and the boundedness of u together with(2.23)and(2.20)therefore implies

It remains to discuss the quantity

We have already established that the second factor goes to 0 for R→∞and it therefore suffices to show that the fi rst factor is bounded.Arguing as above,we see that