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On regularity of stationary Stokes and

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MathematischeAnnalenmanuscriptNo.(willbeinsertedbytheeditor)

OnregularityofstationaryStokesandNavier-Stokes

equationsnearboundary

KYUNGKEUNKANG

Received:date/Revisedversion:date–cSpringer-Verlag2001

Abstract.Weobtainlocalestimatesofthesteady-stateStokessystem“withoutpressure”nearboundary.Asanapplicationofthelocalestimates,weprovethepartialregularityuptothebound-aryforthestationaryNavier-Stokesequationsinasmoothdomaininfivedimension.

MathematicsSubjectClassification(2000):35Q30,76D03

KYUNGKEUNKANGSchoolofMathematics,UniversityofMinnesota,Minneapolis,MN55455,USA(e-mail:kkang@math.umn.edu)thetracespaceof.

For,wedenotebythespaceoffunctionswhosederivatives

uptotheorderareH¨oldercontinuousin.

Forsimplicity,indicatestheaverageofagivenfunction

,namely,

denotethevolumeofunitballindimensionalspace.4

Proof.SeeTheorem3.1in[6,page77].

Wealsorecallawell-knownfactverifiedeasilybyiterations.

Lemma3.Letbeanonnegativeboundedfunctiondefinedinwhere

.Supposethatforwehave

whereandarenonnegativeconstantswith.Thenforall

wehave

whereisaconstantdependingonand.

Proof.SeeLemma3.1in[15,page161].

Infact,wewillneedaslightlymoregeneralversionofLemma3.

Lemma4.Letbeanonnegativeboundedfunctiondefinedinwhere

andbeapositiveinteger.Supposethatfor,wehave

whereandarenonnegativeconstantswith.Thenforall

wehave

whereisaconstantdependingonand.

Proof.ThiscanbeprovedbymodifyingtheproofofLemma3.1in[15,page

161].Themodificationisself-evident,andthereforeweomitthedetails.

Weconcludethissectionbyrecallingexistenceresultsfordivergenceequation

divinaboundeddomain(see[2]and[13]).,wheredependsonand,butnot.

Proof.SeeLemma3.1in[13,page121].

3.EstimatesfortheStokessystem

Inthissection,wewillprovetheestimate(1)fortheStokessystemmentionedin

theintroduction.LetbeaboundarypointofadomainwithLipschitz

boundary.Forconvenience,wedenotewith.Here

isthelargestpositiveradiussuchthatand6

.Suppose

thatandbestandardcutofffunctionsdefinedasfollows:

in

outsidein

outside

suchthatandaresupportedinand,respectivelyandforafixed

constanttheysatisfy

Wefirstconsiderthecase.Wenotefirstthatintheweakformulation

above,itiseasilyseenthatisinbythevariationalformulation(see

Lemma1.1in[13,page186]).Moreover,wemayuniquelytakeapressure

satisfying.Multiplyingto(5)andusingtheintegrationbyparts,

wehave

(8)

Usingtheenergyestimate(8),wecontrolthepressureintermsof.

Lemma6.LetbeaweaksolutionoftheStokessystem(5).Thenforevery

with,thefollowingestimateholds:,weobtain

Accordingtotheenergyestimate(8),wehave

Usingtheholefillingtechnique,weobtain

whichisindependentof.Sincearearbitrarynumbers,

duetotheLemma3,theassertion(9)iscompleted.

Sincepressureisestimatedintermsofvelocity,weeasilyhavethefollowing

Caccioppoliinequality.

Lemma7.LetbeaweaksolutionoftheStokessystem(5).Thenforevery

with,thefollowingestimateholds:8

where;

(12)

forany.Inwhatfollowswewillnotusethisestimatefor.

Thecaseandcanbeprovedbyanobviousmodificationofthe

proofabove,andthereforeweomitthedetails.

Sofar,weshowedthatandpressurecanbecontrolledbynearthebound-

ary.Nowwewillshow,furthermore,higherderivativesofandcanbealso

estimatedintermsofprovidedthatissufficientlysmooth.

Theorem1.Letbeadomainofclassandbeanintegerwith

.SupposeandbeaweaksolutionoftheStokes

system(5).Thenforeverywith,thefollowinglocalestimate

holds:

(13)

where.

Proof.If,theestimate(13)isduetoLemma6,Lemma7andRemark

1.Let(16)

foreach.However,as,thenapplying(16)as

,weseethatisconstant,andthereforebecauseonthe

.Thiscompletestheproof.

Similartheoremwasprovedin[8]byusingthereflectionprincipleforthe

Stokessystem(15).However,itdoesnotseemobviousthatthemethodin[8]

couldbeeasilyusedtoobtainlocalestimates(13).

Inremainingpartofthissection,usingresultsabove,wecanverifyesti-

mate(1)fortheStokessystem.Letbeasmoothdomainand

whereandisanyintegerwith.Forconvenience,let

usrecalltheStokessystem(5):

in

in

on10

iscontainedin.Thus,without

lossofgenerality,isassumedtobesmooth.Nowwearereadytoprove

estimatenearboundaryfortheStokessystemabove.

Theorem2.Letbeadomainofclassandbeanintegerwith

and.Supposeand

solvetheStokessystem(5)inaweaksense.Thenforanythe

followinglocalestimateholds:

(17)

where.

Proof.Wefirstinvestigateaprioriestimateforsmoothsolutions.Theideaisto

splitassumofand,whichsolvestheStokessystemwithnonzeroexternal

forceandzeroboundary,andwithzeroexternalforce,respectively.Firstwe

considerthefollowingStokessystem:

in

in

on

Itiswellknownthatthefollowingestimateholds(seeTheorem6.1in[13,page

231–232]).

(18)

whereisthenormalizedpressure,i.e..Hereweset

and.Thensolve

in

in

on

Usingestimate(13)oftheTheorem1,foranywehave

(19)

whereisthemidpointofand,i.e.

(20)

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2014年度全国职称英语等级考试卫生A真题及答案

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英语翻译

英语翻译

Thedevelopmentofmoldoscillation1.ForewordThemoldisintheheartofthecontinuouscastingmachineparts.

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Navier–Stokes equation

Navier–Stokes equation

Navier–Stokes equation [Wikipedia]

In physics, the Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term - hence describing viscous flow.

The equations are useful because they describe the physics of many things of academic and economic interest. They may be used to model the weather, ocean currents, water flow in a pipe and air flow around a wing. The Navier–Stokes equations in their full and simplified forms help with the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, and many other things. Coupled with Maxwell's equations they can be used to model and study magnetohydrodynamics.