Rayleigh-Taylor instability for two_phase

OntheRayleigh-Taylorinstabilityforthetwo-phaseNavier-Stokesequations(JointworkwithJanPrüss)FreeBoundaryProblems,TheoryandApplicationsMSRI,March7-11,2011• Weconsiderthemotionoftwoimmiscible,incompressible,viscousfluidsoccupyingtheregionsΩ1(t)andΩ2(t),respectively.• Γ(t)(theinterfacebetweenthefluids,thefreeboundary)needstobedeterminedaspartoftheproblem.• ThemotionofthefluidsismodeledbytheNavier-Stokesequations:Γ(t)Γ(t)Ω2(t)Ω1(t)νONTHERAYLEIGH-TAYLORINSTABILITYFORTHETWO-PHASENAVIER-STOKESEQUATIONSJANPR¨USSANDGIERISIMONETTAbstract.Thetwo-phasefreeboundaryproblemwithsurfacetensionanddownforcegravityfortheNavier-Stokessystemisconsideredinasituationwheretheinitialinterfaceisclosetoequilibrium.Theboundarysymbolofthisproblemadmitszerosintheunstablehalfplaneincasetheheavyfluidisontopofthelightone,whichleadstothewell-knownRayleigh-Taylorinstability.InstabilityisprovedrigorouslyinanLp-settingbymeansofanabstractinstabilityresultduetoHenry[12].Ω2(t)={(x,y)∈Rn×R:yh(t,x),t≥0},Ω1(t)={(x,y)∈Rn×R:yh(t,x),t≥0},Γ(t)={(x,y)∈Rn×R:y=h(t,x),t≥0}.1.IntroductionOfconcernisthemotionoftwoimmiscible,viscous,incompressiblecapillaryfluids,fluid1andfluid2,thatoccupytheregionsΩi(t)={(x,y)∈Rn×R:(−1)i(y−h(t,x))0,t≥0},i=1,2.ThefluidsareseparatedbyasharpinterfaceΓ(t)={(x,y)∈Rn×R:y=h(t,x),t≥0}Date:August22,2009.2000MathematicsSubjectClassification.Primary:35R35;Secondary:35Q10,76D03,76D45,76T05.Keywordsandphrases.Navier-Stokesequations,freeboundaryproblem,sur-facetension,gravity,Rayleigh-Taylorinstability,well-posedness,analyticity.TheresearchofthesecondauthorwaspartiallysupportedbytheNSFGrantDMS-0600870.1• ρi=density• µi=viscosity• γa=accelerationofgravity• σ=surfacetension• κ=meancurvatureofΓ(t)• V=normalvelocityofΓ(t)2J.PR¨USSANDG.SIMONETTwithanunknownfunctionhthatneedstobedeterminedaspartoftheproblem.ThemotionofthefluidsisgovernedbytheincompressibleNavier-Stokesequationswithsurfacetensionanddownforcegravityandreadsasfollows,wherei=1,2;ρi%∂tu+(u·∇)u&−µiΔu+∇q=−ρiγaen+1inΩi(t)divu=0inΩi(t)−[[S(u,q)ν]]=σκνonΓ(t)[[u]]=0onΓ(t)V=u·νonΓ(t)u(0)=u0inΩi(0)Γ(0)=Γ0.(1)Theconstantsρi0andµi0denotethedensitiesandthevis-cositiesoftherespectivefluids,σstandsforthesurfacetensionandγaistheaccelerationofgravity.Moreover,S(u,q)isthestresstensordefinedbyS(u,q)=µi%∇u+(∇u)T&−qIinΩi(t),and[[v]]=(v|Ω2(t)−v|Ω1(t)&|Γ(t)denotesthejumpofthequantityv,definedontherespectivedomainsΩi(t),acrosstheinterfaceΓ(t).Fi-nally,κ=κ(t,·)isthemeancurvatureofthefreeboundaryΓ(t),ν=ν(t,·)istheunitnormalfieldonΓ(t),andV=V(t,·)isthenormalvelocityofΓ(t).Hereweusetheconventionthatν(t,·)pointsfromΩ1(t)intoΩ2(t),andthatκ(x,t)isnegativewhenΩ1(t)isconvexinaneighborhoodofx∈Γ(t).System(1)comprisesthetwo-phaseNavier-Stokesequationswithsurfacetensionsubjecttogravity.Inordertoeconomizeournotation,wesetρ=ρ1χΩ1(t)+ρ2χΩ2(t),µ=µ1χΩ1(t)+µ2χΩ2(t),whereχdenotestheindicatorfunction.Itisconvenienttointroducethemodifiedpressure˜q:=q+ργay.Withthisconventionsystem(1)2J.PR¨USSANDG.SIMONETTwithanunknownfunctionhthatneedstobedeterminedaspartoftheproblem.ThemotionofthefluidsisgovernedbytheincompressibleNavier-Stokesequationswithsurfacetensionanddownforcegravityandreadsasfollows,wherei=1,2;ρi%∂tu+(u·∇)u&−µiΔu+∇q=−ρiγaen+1inΩi(t)divu=0inΩi(t)−[[S(u,q)ν]]=σκνonΓ(t)[[u]]=0onΓ(t)V=u·νonΓ(t)u(0)=u0inΩi(0)Γ(0)=Γ0.(1)Theconstantsρi0andµi0denotethedensitiesandthevis-cositiesoftherespectivefluids,σstandsforthesurfacetensionandγaistheaccelerationofgravity.Moreover,S(u,q)isthestresstensordefinedbyS(u,q)=µi%∇u+(∇u)T&−qIinΩi(t),and[[v]]=(v|Ω2(t)−v|Ω1(t)&|Γ(t)denotesthejumpofthequantityv,definedontherespectivedomainsΩi(t),acrosstheinterfaceΓ(t).Fi-nally,κ=κ(t,·)isthemeancurvatureofthefreeboundaryΓ(t),ν=ν(t,·)istheunitnormalfieldonΓ(t),andV=V(t,·)isthenormalvelocityofΓ(t).Hereweusetheconventionthatν(t,·)pointsfromΩ1(t)intoΩ2(t),andthatκ(x,t)isnegativewhenΩ1(t)isconvexinaneighborhoodofx∈Γ(t).System(1)comprisesthetwo-phaseNavier-Stokesequationswithsurfacetensionsubjecttogravity.Inordertoeconomizeournotation,wesetρ=ρ1χΩ1(t)+ρ2χΩ2(t),µ=µ1χΩ1(t)+µ2χΩ2(t),whereχdenotestheindicatorfunction.Itisconvenienttointroducethemodifiedpressure˜q:=q+ργay.Withthisconventionsystem(1)2J.PR¨USSANDG.SIMONETTfollows,wherei=1,2;ρi%∂tu+(u·∇)u&−µiΔu+∇q=−ρiγaen+1inΩi(t)divu=0inΩi(t)−[[S(u,q)ν]]=σκνonΓ(t)[[u]]=0onΓ(t)V=u·νonΓ(t)u(0)=u0inΩi(0)Γ(0)=Γ0.(1)Theconstantsρi0andµi0denotethedensitiesandtheviscosi-tiesoftherespectivefluids,σstandsforthesurfacetensionandγaistheaccelerationofgravity.Moreover,S(u,q)isthestresstensordefinedbyS(u,q)=µi%∇u+(∇u)T&−qIand[[v]]=(v|Ω2(t)−v|Ω1(t)&|Γ(t)denotesthejumpofthequantityv,definedontherespectivedomainsΩi(t),acrosstheinterfaceΓ(t).Finally,κ=κ(t,·)isthemeancurvatureofthefreeboundaryΓ(t),ν=ν(t,·)istheunitnormalfieldonΓ(t),andV=V(t,·)isthenormalvelocityofΓ(t).Hereweusetheconventionthatν(t,·)pointsfromΩ1(t)intoΩ2(t),andthatκ(x,t)isnegativewhenΩ1(t)isconvexinaneighborhoodofx∈Γ(t).System(1)comprisesthetwo-phaseNavier-Stokesequationswithsurfacetensionsubjecttogravity.Inorde