2The Navier-Stokes and Euler Equations(7)

math

2TheNavier–StokesandEulerEquations–FluidandGasDynamics

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whereDg

denotesthematerialderivativeofthescalarfunctiong:

Dg=gt+u.gradg.Dt

Thus,fortwo-dimensional ows,thevorticitygetsconvectedbythevelocity eld,isdiffusedwithdiffusioncoef cientνandexternallyproduced/destroyedbythecurloftheexternalforce.Forthreedimensional owsanadditionaltermappearsinthevorticityformulationoftheNavier–Stokesequations,whichcorrespondstovorticitydistortion.

TheNavier–StokesandEulerequationshadtremendousimpactonappliedmathematicsinthe20thcentury,e.g.theyhavegivenrisetoPrandtl’s9boundarylayertheorywhichisattheoriginofmodernsingularperturbationtheory.NeverthelesstheanalyticalunderstandingoftheNavier–Stokesequationsisstillsomewhatlimited:Inthreespacedimensions,withsmooth,decaying(inthefar eld)initialdatumandforce eld,aglobal-in-timeweaksolutionisknowntoexist(Leraysolution10),howeveritisnotknownwhetherthisweaksolutionisuniqueandtheexistence/uniquenessofglobal-in-timesmoothsolutionsisalsounknownforthree-dimensional owswitharbitrarilylargesmoothinitialdataandforcing elds,decayinginthefar eld.Infact,thisispreciselythecontentofaClayInstituteMillenniumProblem11withanawardofUSD1000000!!Averydeeptheorem(see[2])provesthatpossiblesingularitysetsofweaksolutionsofthethree-dimensionalNavier–Stokesequationsare‘small’(e.g.theycannotcontainaspace-timecurve)butithasnotbeenshownthattheyareempty…

Weremarkthatthetheoryoftwodimensionalincompressible owsismuchsimpler,infactsmoothglobal2 dsolutionsexistforarbitrarilylargesmoothdataintheviscidandinviscidcase(see[6]).

Whyisitsoimportanttoknowwhethertime-globalsmoothsolutionsoftheincompressibleNavier–Stokessystemexistforallsmoothdata?Ifsmoothnessbreaksdownin nitetimethen–closetobreak-downtime–thevelocity elduofthe uidbecomesunbounded.Obviously,weconceive owsofviscousreal uidsassmoothwithalocally nitevelocity eld,sobreakdownofsmoothnessin nitetimewouldbehighlycounterintuitive.Hereournaturalconceptionoftheworldsurroundingusisatstake!

ThetheoryofmathematicalhydrologyisadirectimportantconsequenceoftheNavier–Stokesor,resp.,Eulerequations.The owofriversingeneral–andinparticularinwaterfallslikethefamousonesoftheRioIguassuontheArgentinian-Brazilianborder,oftheOranjeriverintheSouthAfricanAugra-biesNationalParkandothersshownintheFigs.2.1–2.6,areoftenmodeledbythesocalledSaint–Venantsystem,namedaftertheFrenchcivilengineer9

10

11http://www. /msc/prandtl.htmhttp://www-groups.dcs.st-and.ac.uk/～history/Mathematicians/Leray.html/millennium/Navier–Stokes_Equations/