为交流近年来在偏微分方程及其应用领域所取得的最新研究成果，研讨相关的前沿课题，同时促进偏微分方程相关领域的专家，特别是青年学者之间的交流与合作，华南理工大学数学学院和广东工业大学数学与统计学院将于2023 年 5 月 18 日举办《2023 年偏微分方程青年研讨会》。会议详细信息如下：
一、学术委员会
陈智奇李用声卫雪梅朱长江(主席)
二、组织委员会
洪光益彭红云谢启林翟小平
三、主办单位：
广东工业大学数学与统计学院
广东工业大学数学与统计学院(代章)
2023 年 5 月 17 日
会议日程(5月18日星期四)
ABSTRACT
GlobalexistenceforaclassoflargesolutiontocompressibleNavier-Stokesequationswithvacuum
洪广益
Inthistalk,weareconcernedwiththeCauchyproblemofthethree-dimensionalisentropic compressibleNavier-Stokesequations.Weprovetheglobalexistenceanduniquenessof classicalsolutionswithlargeinitialenergyandvacuum,undertheassumptionsthatthefluid isnearlyisothermal(i.e.,theadiabaticexponent$\gamma$issufficientlycloseto$1$),and thatthefar-fielddensity$\tilde{\rho}$iseithervacuumorclosetovacuum.Tothebestofour knowledge,weestablishthefirstresultontheglobalexistenceoflarge-energysolutionswith vacuumtothethree-dimensionalcompressibleNavier-Stokesequationsforthecasesof vacuumandnonvacuumfar-fieldconstantstates,whichgeneralizestheresultin[X.D.Huang,J.Li,Z.P.Xin,Comm.PureAppl.Math., 2012]onclassicalsolutionswithvacuumandsmall energy(largeoscillations).ThisisjointworkwithXiaofengHou,HongyunPengandProf.ChangjiangZhu.
Asymptoticstabilityofsolutionstoahyperbolic-ellipticcoupledsystemoftheradiatinggaswithboundaryeffect
季善明
Thistalkisconcernedwiththeasymptoticstabilityofthesolutiontoaninitial-boundary valueproblemwithboundaryeffectforahyperbolic-ellipticcoupledsystemoftheradiating gascorrespondingtotherarefactionwavecaseandshockwavecaserespectively.Forthe scalarviscousconservationlawandrarefactionwavecase,itisknownbyLiu,Matsumura,andNishihara(SIAMJ.Math.undefined. 1998)thatthesolutiontendstowardrarefactionwaveor stationarysolutionorsuperpositionofthesetwokindofwaves.Motivatedbytheirwork,we provethestabilityoftheabovethreetypesofwavepatternsforthehyperbolic-elliptic coupledsystemoftheradiatinggaswithsmallperturbation.Asingularphaseplaneanalysis methodisintroducedtoshowtheexistenceandthepreciseasymptoticbehaviorofthe stationarysolution.Thestabilityofrarefactionwave,stationarysolution,andtheirsuperposition,isprovedbyapplyingthestandardL^2-energymethod.Theasymptotic stabilityofshockwavecaseisalsoconsidered.ThisisajointworkwithMinyiZhangand ChangjiangZhu.
ExistenceofMulti-dimensionalContactDiscontinuitiesfortheidealCompressibleElastodynamics
解斌强
Contactdiscontinuitiesaremosttypicalinterfacialwavesforsystemsofhyperbolic conservations.Weprovetheexistenceofthemulti-dimensionalcompressibleelastodynamic contactdiscontinuitiesinboth2Dand3DinSobolevspaceswithoutanyadditional constraintswhichremovethestabilityconditionin{GuiqiangChen,PaoloSecchiandTao Wang,StabilityofMultidimensionalThermoelasticContactDiscontinuities,Arch.Rational Mech.undefined.237 (2020) 1271– 1323}.ThekeyingredientsofouranalysisareLagrangian coordinates,thetransversalityofthedeformationtensor,andanviscousapproximation.This talkisbasedonajointworkwithJiaxuLi(CUHK).
Cauchyproblemofdegeneratediffusionequationswithtimedelay
徐田媛
Thistalkisaboutpropagationspeedofdegeneratediffusionequationswithtimedelay.Weshowtheasymptoticspreadingspeedanditscoincidencewiththecriticalwavespeedof sharpwave,andprovethattheinitialperturbationortheboundaryofthecompactsupportof thesolutionpropagatesatthecriticalwavespeedforthetime-delayeddegeneratediffusion equations.Differentfromtheexistingstudiesrelatedtospreadingspeeds,thetimedelay makesthecriticalspeedoftravelingwavesslowdowninamorecomplicatedfashionsuch thatthecriticalspeedcannotbedeterminedbythecharacteristicequation,andthedegenerate diffusioncausesthelossofregularityforthesolutions.Byaphasetransformtechniquewe constructupperandlowersolutionswithsemi-compactsupportsandthenwedeterminethe asymptoticspreadingspeed.
