半空间上一类广义～Navier-Stokes方程解的大时间行为

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1、ABSTRACTThemathematicalmodelsoffluiddynamicshavebeenattractedmoreandmoreattentioninthepasthalfcenturyduetothelargeapplicationstothefieldssuchasme-teorologyscience，marinescience，aeronauticalscienceandbiologymedicine．ItiswellknownthattheclassicNavier—St

2、okesequationsareacceptedasarightsystemmodellingtheabovefluidflows．However，Thenonlineardampingeffectrequiredinsomecomplexfluidssuchastheporousmediaflowwithfrictioneffects．Inthisthesis，weconsideroneclassgeneralizedNavier-Stokesequationswhichexhibittheno

3、nlineardampingV6+V丌+16尸一1b=，(1)Hereband7raxeunknownvelocityfieldsandpressurefields，16P1bisthenonlineardampingterm．ThepurposeofthisthesisistoinvestigatethelargetimebehaviorofsolutionsoftheabovegeneralizedNavier-Stokesequations．Moreprecisely,withtheaido

4、fthespectraldecompositiontechniqueandLp—Lqestimatesofthelinearheatequations，wewillderivesomedecayresultsofweaksolutionstothegeneralizedNavier-Stokesequations(1)．Thethesisisorganizedasfollows．Inthechapter1，beforeintroducingsomefundamentalmainpay’sicalb

5、ackgroundofNavier-Stokesequations，wereviewtherelatedpreviousworkoftheequations(1)．Thenwegavesomepreliminarywhichisnecessaryinourthesis．Inthechapter2，firstlyweapplythespectraldecompositionapproachtoderiveanestimationofthenonlinearoperator：Theorem0．1Sup

6、posethatbisaweaksolutiono

7、theequations(1)inthehalfspaces．ThenE(A)B(b)II≤cIIbll2入{+c俐8^，3．Soweobtainthetimedecayrateresultsoftheweaksolutionstothe3DgeneralizedNavier-StokesequationsinR王byborrowingthespectraldecompositionapproachduetoBorchersandMiyakawa

8、[6]andthemodifiedLp—L2estimatesofBaeandChoe[5]．6．+，k60=舶划卜一6—6ZaV酞，l_l●l●--，、I__II【Theoremo．2Supposethatbisaweaksolutiontotheequations《i)．Then?lJehave：(1)．1imIIblI=0，b0∈L；t--*oo(2)llb(t)ll≤Ct一(募一i)，t≥1，bo∈L；NL”(1≤7．<2)．0．3Supposethatbisaweaksolution∥俐

9、，bo∈L；AL7forl≤r<2．Then如7’t≥1，wehave：6(洲≤优一-s(·l_1)11，forfR。+Iz360

10、r如<∞·Inthechapter3，webegintoconsidertheasymptoticstabilityforthenontrivialsolutionoftheequations(1)withnon-zeroforce．Moreprecisely，thefollowingresultwillbestudied．0．4Supposebo∈H