Zhang Qin

(Department of Mathematics and Statistic,Chongqing Jiaotong University,Chongqing 400074,China)

Abstract:The main object of this paper is to investigate the well-posedness theory of the incompressible inviscid jet flow with gravity in an semi-infinitely long symmetric nozzle. The main results read that given a mass flux in the inlet of the nozzle,we established the existence and the uniqueness of the incompressible jet flow problem with gravity in an semi-infinitely long symmetric nozzle,which contain a smooth free surface detaching at the boundary point of the lower nozzle wall.

Key words:existence and uniqueness,free streamline,inviscid,incompressible

1 Introduction and Main Results

In this paper,we will investigate the well-posedness of the incompressible,inviscid jet flow with gravity in a semi-infinitely long symmetric nozzle. Some existence and uniqueness results are established in this paper.

In the following,we would like to recall some known results about the mathematical results on the cavity and jet flows problem. In 1952,P. R. Garabedian,etc. in[1]investigated the axially symmetric finite cavity problem for Riabouchinsky model by using the variational approach. For general existence results on jet and cavity flows,we can refer to the references[2-4]. In 1981,H.W Alt,etc. developed a new variational approach to obtain the existence and the regularity for a minimum problem with free boundary in their breakthrough work[5]. Based on the work[5],some remarkable results on the existence and uniqueness of axially symmetric jet flow were established in[6],asymmetric jet flow in[7],jet flow with gravity in[8],axially symmetric infinite cavity in[9],and so on.

The motivation in this paper is to investigate the existence and uniqueness of the incompressible jet flow with gravity in a semi-infinitely long symmetric nozzle.

Denote the upper nozzle wall of the symmetric semi-infinitely long nozzle by

N1:y=H1>0,-∞

(1)

and the lower nozzle wall by

N2:x=g(y)∈C2,α(-∞,0),

(2)

with 0<α<1 and satisfying

(3)

Denote the symmetric axis of the symmetric nozzle asT:x=0,-∞

(4)

whereU(x,y)=(u(x,y),v(x,y))satisfying the irrotational condition that

×U=0,

(5)

P=P(x,y)denote the velocity field and the pressure,respectively,ande2=(0,1).

Furthermore,we assume that the nozzle wall and the symmetric axis are impermeable,then the flow satisfies the slip boundary condition

(u,v)·n=0,onN1∪N2∪T∪l,

(6)

wherenis the unit outward normal toN1∪N2∪T∪l.

There is an invariance along each streamline for the steady incompressible flow,namely,

(7)

1.1 Statement of the physical problem

Definition1(Jet flow with gravity problem) Suppose that the given semi-infinitely long nozzle wallN1,N2satisfy the conditions(1)-(3),given a mass fluxm0>0 of the incoming incompressible flow,and the atmospheric pressureP=Patm,does there exist a unique two-dimensional symmetric incompressible jet flow with gravity in the semi-infinitely long nozzle,which has a smooth free streamline leaving the vertexA=(-a,0)of the lower nozzle wall?

Definition2(A solution to the jet flow with gravity problem)Avector(u,v,ρ,Γ)is called a solution to the jet flow with gravity problem,provided that

(1)Γcan be expressed by a smooth functionx=f(y)∈C1(-∞,0),such that

(8)

Theorem1(Existence of the jet flow with gravity) Assume that the semi-infinitely long nozzle wallN1andN2satisfy the conditions(1)-(3),for any given mass fluxm0>0,then there exist a constantλ>0 and a solution(u,v,P,Γ)to the jet flow with gravity problem defined in Definition 2.

2 Mathematical Setting on Jet Flow with Gravity Problem

2.1 Stream function setting

In order to solve the jet flow with gravity problem,according to the first equation in(4),setu=ψy,v=-ψx,which combine with the irrotational condition(5)to obtain Δψ=0 in the flow fieldΩ0. Furthermore,we impose the Dirichlet boundary value conditions asψ=m0onN1∪T,andψ=0 onN2∪l∪Γ. Thus,the free boundary can be defined by

Γ=Ω∩∂{ψ>0},

(9)

whereΩis called as the possible flow field bounded byN1,N2,landT,which combining with equation(7)deduces that

(10)

wherevis the outer unit normal ofΓ. Therefore,we formulate the jet flow with gravity problem as the following boundary value problem for the stream function that

(11)

2.2 Variational approach and truncation

Denote the variational problem(Pλ)as

(12)

whereIAis the characteristic function of a setAande2=(0,1).

Since the functionalJλ(ψ)is unbounded for anyψ∈K,thus we need to truncate the domainΩ,namely,Ωμ=Ω∩{x≥-μ}. Therefore,the truncated functional is that

(13)

Denote the truncated variational problem(Pλ,μ)as finding aψλ,μsuch that

where the corresponding admissible set is that

in whichN1,μ=N1∩{x≥-μ},N2,μ=N2∩{x≥-μ} andI1={(-μ,y)|H2,-μ≤y≤H1},whereH2,-μ=max{y|-μ=g(y)}.

3 Existence of the Jet Flow with Gravity

3.1 Existence of minimizer to the truncated variational problem

Lemma1ProblemPλ,μhas a solution.

ProofHere,we just need to obtain the boundedness of the functionalJλ,μ,namely,there exists a functionψ0∈Kμ,such that the functionalJλ,μ<+∞,thus,the variational problemPλ,μhas a minimizer. Takey0small,and define

then we can extendψ0into the domainΩμy≤y0}so that it belongs to the admissible setKμ. Then,

For simplicity,we setψ=ψλ,μin the following.

Lemma2Ifψis a minimum,then

(14)

for any vectorη=(η1,η2)∈(C1(E))2,vεis the unit outward normal toΩμ∩∂{ψ≥ε}.

Particularly,

(15)

for any vectorη=(η1,η2)∈(C1(E))2,η=0 onΩμI0,η·vε≤0 onI0,I0={ψ=0}andvis the unit outward normal toI0.

ProofFor any realε,|ε|small,letτε(x,y)=(x+εη1(x,y),y+εη2(x,y))and defineψε(τε(x,y))=ψ(x,y). Thenψε∈Kμand

(Dτε(r,y))-1=(I+ε·ηI-εDη)(detDτε)-1and detDτε=1+ε·η+o(ε),

whereIis the identity matrix.

Hence we have

(16)

in whichER=E∩{(x,y)|x2+y2≤R},R>0 sufficiently large.

Then,by a series of calculations,we have the following estimates,takingR→+∞,the linear term ofεin inequality(16)vanishes,thus

(17)

owing to the facts thatvεis parallel toψon the streamlines and the first equation in(11),thus,the proof of(14)is completed.

which directly implies the inequality(15).

Γλ,μ=Ωμ∩{y<0}∩∂{ψ>0}.

(18)

ProofSupposeψ1andψ2are two minimizer and set

(19)

Then,takingε→0 showsψ1(x,y)≥ψ2(x,y)inΩμ. Similarly,we can obtain thatψ1(x,y)≤ψ2(x,y)inΩμ. Hence,ψ1=ψ2inΩμ.

Next,setψ1=ψ2,combining with inequality(19),to yield thatψ(x+ε,y)≥ψ(x,y)inΩμ. Therefore,we finish the proof of Lemma 3.

3.2 Fundamental properties of the free-boundary

Owing to the condition that the monotonicity of the minimizerψ(x,y)with respect tox,we setΓλ,μ:x=fλ,μ(y),then

Ωμ∩{y<0}∩{ψ>0}={(x,y)|fλ,μ(y)

Therefore,by using the non-oscillation Lemma 9 in Appendix,we can prove that the free boundaryx=fλ,μ(y)is continuous in(-∞,0],which is similar to the arguments to Lemma 5.4 in[6],thus we omit a part of the proof here.

Proposition1fλ,μ(y)is a continuous function in(-∞,0). Furthermore,fλ,μ(y)is analysis.

ProofIn view of Lemma 5.4 in[6],then it suffices to show thatfλ,μ(y+0)=fλ,μ(y-0)=fλ,μ(y)for anyy<0. Firstly,suppose that there exists a pointy0∈(-∞,0)such thatfλ,μ(y+0)≠fλ,μ(y0),then without loss of generality,we assume thatfλ,μ(y+0)>fλ,μ(y0). Then there exist some constantsε1,ε2,such that there is a strip that

Eε1,ε2={(x,y)|fλ,μ(y0)-ε1

By virtue of the monotonicity ofψwith respect tox,we deduce that {(x,y0)|fλ,μ(y0)-ε1

3.3 The continuous fit and the smooth fit of the free boundary

Next,we will consider the continuous fit and the smooth fit conditions of the free boundary. Before that,we need to give some basic and important lemmas. The lemmas are as follows:

ProofThis proof of this Lemma has been given in section 4.7 in[5],thus,we omit the processes here.

Lemma5For anym0>0,ifλ>0 is sufficiently small,thenfλ,μ(0)>-a.

which implies that it is impossible for sufficiently smallλ. Hence,we complete the proof of this Lemma.

Lemma6For anym0>0,ifλ>0 is sufficiently large,thenfλ,μ(0)<-a.

Fix the aboveλasλμ,combining with Lemma 4,Lemma 5 and Lemma 6,we can give the continuous fit conditionfλμ,μ(0)=-a. Finally,we will check that the smooth fit conditionf′λμ,μ=g′(0)indeed holds,and the fact can be obtained along the similar argument in[5]and[6],hence we omit it here.

Proposition2fλμ,μ(0)=-aholds,and thenN2∩Γλμ,μcontinuously differentiable in a neighborhood ofA. Furthermore,ψλμ,μis continuously differentiable in {ψλμ,μ>0}∩Bδ(A),for someδ>0.

4 The Existence and Uniqueness of the Jet Flow with Gravity Problem

4.1 The existence of the jet flow with gravity problem

To establish the existence of the solution to jet flow with gravity problem,we will take limitμ→∞ to the solutionψλμ,μto the truncated variational problem(Pλμ,μ)and show the limitψλis indeed a solution to the variational problemPλ(12). By virtue of the uniform gradient estimate |ψλμn,μn|≤Cin any compact subset ofΩwhich can be referred on Lemma 10 in Appendix,it follows from the similar arguments in Section 12 in Chapter 3 in[3]that there exists a subsequence{ψλμn,μn}and{λμn},such thatλμn→λandψλ,μn→ψλweakly inuniformly in any compact subset ofΩ,asn→∞. In the following,we will verify thatψλis in fact a minimizer to the variational problem(Pλ),and solves the boundary value problem(11). In particular,the continuous fit and smooth fit conditions are fulfilled.

Sinceψλμn,μnis a minimizer to the functionJλμn,μn,thenJλμn,μn(ψλμn,μn)≤Jλμn,μn(ψμn)for anyψμn∈Kμn. For any bounded domainD⊂Ω,we can choose a sufficiently largeμn,such thatD⊂Ωμn. Choosingψμn=ψλμn,μnon ∂D,and extendingψμnwithψλμn,μnoutsideD,hence we can conclude

JD(ψλμn,μn)≤JD(ψμn).

(20)

Therefore,similar to the proof in Lemma 5.4 in[5]and takingn→∞,one hasJD(ψλ)≤JD(ψ),for anyψ∈Kwithψ=ψλon ∂D,namely,ψλis a minimizer to the variational problem(Pλ).

Moreover,ψλ(x,y)is monotonic increasing with respect tox. In fact,for any(x1,y),(x2,y)∈Ωwithx1>x2,there exists a compact subsetGofΩ,such that(x1,y),(x2,y)∈G. For sufficiently largen,we haveG⊂Ωμn,and it follows from Lemma 3 that

ψλμn,μn(x1,y)≥ψλμn,μn(x2,y).

(21)

Sinceψλμn,μnconverges toψλuniformly in any compact subset ofΩ,then there exists a subsequenceψλμn,μnand lettingn→+∞ in(21),one hasψλ(x1,y)≥ψλ(x2,y),for any(x1,y),(x2,y)∈Ωwithx1>x2. The monotonicity ofψλ(x,y)with respect toximplies that the free boundary isy-graph. Then it follows that there exists a continuous functionfλ(y),such that the free boundaryΓλofψλcan be described as

Γλ:x=fλ(y)fory∈(-∞,0].

4.2 Uniqueness of the jet flow with gravity problem

This part shows the uniqueness of the two-dimensional symmetric jet flow with gravity problem.

Theorem2For anym0>0,then there exists a uniqueλsuch that the solution(u,v,ρ,Γ)established in Theorem 1 is unique.

thus,

(22)

Thus there exists a smallestσ0such that the above condition holds. By the maximum principle,we have

(23)

5 Appendix

In this part,we will mention some Lemmas that has been proved in references[6],which is important for the proof of the Theorem 1.

Lemma7There exists a enough large positive constantCindependent ofμandλ,such that ifψis a minimum,then for any ballBR(X0)∈ΩμwithX0=(x0,y0),

implies thatψλ,μ>0 inBR(X0).

Lemma8For any small 00 such that for all ballsBR(X0)with centerX0∈Ωμandrsmall,the following holds

impliesψλ,μ=0 inBκr(X0).

Lemma9LetGbe a domain inΩμbounded by two disjointed arcsγ1,γ2of the free boundary,y=β1,y=β2.Suppose that the acrsγi(i=1,2)lie in{β10,then

|β2-β1|≤Cmax{|α1-α2|,|ζ1-ζ2|},

whereCis a constant depending only onλ,dandm0.

Lemma10LetX0be a free boundary point inGandG∈Ωμ,then there exists a constantC>0 depending only onλ,Gsuch that for any minimizerψλ,μ,