Thestudyofglobalclassicalsolutionstothe3DisentropiccompressibleNavier-Stokesequationswithgeneralinitialenergyisafundamentalprobleminmathematicalfluiddynamics.Theseequationsdescribethemotionofcompressibleviscousfluids,wherethepressuredependssolelyonthedensity(isentropicassumption).Theexistenceofglobalclassicalsolutionsisachallengingissueduetothenonlinearityandthepossibleformationofsingularities.Undergeneralinitialenergyconditions,theproblembecomesmorecomplexastheinitialdatamaylacksufficientregularityordecayproperties.Keydifficultiesincludecontrollingthedensityandvelocityfieldstopreventblow-up,especiallyinthepresenceofvacuumregions.Significantprogresshasbeenmadebyimposingsuitablesmallnessorstructuralconditionsontheinitialdata.However,thecaseoflargeinitialenergyremainslargelyopen,requiringadvancedanalyticaltechniquestohandletheinterplaybetweendissipation,nonlinearity,andpotentialsingularities.Thisresearchareahasimportantimplicationsforunderstandingthelong-timebehaviorofcompressibleflowsandhasconnectionstootherfieldssuchasastrophysicsandengineeringapplications.
