Spatial Decay Estimates for the Solutions to Stokes Equations in Four Kinds of Semi-Infinite Cylinders

(Department of Apl lied Mathematics,Guangzhou Huashang Col lege,Guangzhou 511300,China)

Abstract:This paper investigates the spatial behavior of the solutions of the Stokes equations in a semi-infinite cylinder.We consider four kinds of semi-infinite cylinders with boundary conditions of Dirichlet type.For each type of cylinder we obtain the spatial decay estimates for the solutions.To make the attenuation meaningful,we derive the explicit bound for the total energy in terms of the initial boundary data.

Keywords:Spatial decay estimates;Stokes equations;Total energy

§1.Introduction

In 2003,Ansorge in the book[1]discussed a dimensionless problem arising from a Stokes flow with artificial time.The problem we study can be written as

whereui(x1,x2,x3,t)(i=1,2,3)andp(x1,x2,x3,t)are the dimensionless velocity fields and the dimensionless pressure,respectively.νandβ2are positive constants.The symbolΔis the Laplace operator,and a comma has been used to indicate partial differentiation with respect to the corresponding coordinate In addition,we have adopted the summation convention,in which a repeated Latin index in any term indicates a summation over that index from one to three and a repeated Greek index indicates a summation from one to two(e.g.,w,i w,i=

In a following paper,Song[31]considered the asymptotic behavior results of the equations(1.1)-(1.2).He defined a semi-infinite cylindrical pipeRwhose generator parallels to thex3-axis e.g.,

whereDis a bounded simply-connected region in(x1,x2)-plane with piecewise smooth boundary∂D.Supposing the velocity fields satisfy homogeneous Dirichlet conditions on the lateral surface of the cylinder,it was proved that the solutions of(1.1)-(1.2)either grew or decayed exponentially in the distance from the finite end of the cylinder.The effect of perturbing the equation parameters was also investigated.

In the present paper,we letΩadenote the interior of a semi-infinite cylindrical pipe of an arbitrary,smooth cross-section,whereais a positive constant.However the generator ofΩais no longer parallel to thex3-axis,i.e.,

whereDx3is a bounded simply-connected region which is parallel to(x1,x2)-plane and depends onx3,e.g.,

Clearly,Dadenotes the end(entrance)of the pipe in the planex3=aand∂Dadenotes the boundary ofDa.We consider the problems(1.1)and(1.2)inΩa×{t>0}with the following initial-boundary conditions

wherefi(i=1,2,3)is a vector function which is differentiable and vanishes on∂Dx3for allx3>a.

By considering four different types of cylinder regions,we show that various norms of solutions of(1.1)-(1.6)must decay exponentially(polynomial or logarithmic)with distance from the finite end of the cylinder.This kind of research can be regarded as a investigation of Saint-Venant’s principle.Saint-Venant’s principle is a famous mathematical and mechanical principle which was conjectured by B.de Saint-Venant in his paper[28].Early work on Saint-Venant’s principle primarily focused on the decay results for the initial-boundary value problems of elliptic equation(see,e,g.[4-6]).After the origin work of Boley[2],an extensive attention has been paid to the parabolic problems.We mention in particular the papers of Knowles[10,11],Quintanilla et al.[26,27],Knops and Quintanilla[8,9]for spatial decay results,Hameed et al.[3],Liu et al.[22,24],Scott et al.[29,30]for continuous dependence results and[7,12,15,16,18-21,23]for Phragm´en-Lindel¨of type alternative results.

This paper is partially inspired by Payne and Schaerer[25].They considered the biharmonic equation in three special types of domains in R2and obtained growth-decay estimates for some weighted and unweighted energy expressions in each type of domain.The main purpose of this paper is to generalize the results in[32]by using the ideas in[25],but our model is much more complex than[25],so our research is very meaningful.

§2.Basic inequality

We first refer to the methods in[14,17]to derive a differential inequality which plays a key role in this paper.

Lemma 2.1.([13])If w|∂Dz=0and Dz is a bounded simply-connected region with piecewise smooth boundary∂Dz,then

where r(z)=|Dz|is the area of Dz.

Next we will use Lemma 2.1 to derive a basic inequality which can lead to our main results.We define an”energy”function

wheredAis an element of cross-sectional area and 0<ηz0≥a>0.Using the divergence theorem in(2.1),we have

§3.Spatial decay results

In this section,we also consider four types of regions.

I.Ifr(z)=r0>0,this shows that the area of the cross-section of the cylinderΩaat anyx3=zis the same.

We summarize the above result in the following theorem.

Theorem 3.1.Let(ui,p)be a solution of problem(1.1)-(1.6)in the semi-infinite cylinderΩa defined in section 1.If r(z)=r0>0,then the function E(z,t)decays exponential ly as z→∞.To be precise,

where M2is a positive constant.

II.Assuming thatr(z)satisfies the following condition

Theorem 3.2.Let(ui,p)be a solution of problem(1.1)-(1.6)in the semi-infinite cylinderΩa defined in section 1.If the area of Dz satisfies(3.4),then the fol lowing inequality holds

where M3,M4are positive constants.

Remark 3.1.Whenγ=1,Theorem 3.2 does not hold.At this case,we recalculate(3.7)as

We summarize the above results in the following theorem.

Theorem 3.3.Let(ui,p)be a solution of problem(1.1)-(1.6)in the semi-infinite cylinderΩa defined in section 1.If the area of Dz satisfies(3.4)withγ=1,then the fol lowing inequality holds

Remark 3.2.Whenγ>1,we can’t obtain the spatial decay bound.From Theorem 3.2,we canget the energy bound for the solutions of(1.1)-(1.6).

III.We suppose thatr(z)satisfies

Combining(2.2),(2.3)and(3.14),we have the following theorem.

Theorem 3.4.Let(ui,p)be a solution of problem(1.1)-(1.6)in the semi-infinite cylinderΩadefined in section 1.If the area of Dz satisfies(3.12),then the fol lowing inequality holds

This shows that the solutions of(1.1)-(1.6)decay to zero in a polynomial way.

IV.Supposing thatr(z)satisfies

Combining(2.2),(2.3)and(3.18),we have the following theorem.

Theorem 3.5.Let(ui,p)be a solution of problem(1.1)-(1.6)in the semi-infinite cylinderΩa defined in section 1.If the area of Dz satisfies(3.15),then the fol lowing inequality holds

§4.Upper bound for the total energy

§5.Conclusion

In this paper,the equations(1.1)-(1.6)are reconsidered in a new semi-infinite cylinder.In a two-dimensional pipe,[25]obtained Phragm´en-Lindel¨of alternative results of biharmonic equation.As far as we know,there are few results in this type of three-dimensional cylinder region.On the other hand,there are many deeper problems to be studied in the future.First of all,we note that Leseduarte and Quintanilla[12]imposed dynamical nonlinear boundary conditions on the lateral side of the cylinder and proved a hragm´en-Lindel¨of alternative for the solutions.Yang and Zhou[34]studied a similar initial-boundary problem and obtained existence of the solution for heat equation.For more such papers one can see[33].Our idea is to impose nonlinear conditions on the side of the cylinder,so our problem will become more complex,but such research is more meaningful.In addition,we can continue to study the continuous dependence of coefficients in the equation as in[32].These are the issues we will continue to study in the future.