简介：FranklandFürediin[1]conjecturedthatther-graphwithmedgesformedbytakingthefirstmsetsinthecolexorderingofN（r）hasthelargestLagrangianofallr-graphswithmedges.Denotethisr-graphbyCr,mandtheLagrangianofahypergraphbyλ（G）.Inthispaper,wefirstshowthatif(3t-1)≤m<（3t）,Gisaleft-compressed3-graphwithmedgesandonvertexset[t],thetriplewithminimumcolexorderinginGcis（t—2—i）（t—2）t,thenλ（G）≤λ(C3,m).Asanimplication,theconjectureofFranklandFiirediistruefor（3t）-6≤m≤（3t）.

简介：Fornon-negativeintegersi,jandk,wedenotethegeneralizednetasN_(i,j,k),whichisatrianglewithdisjointpathsoflengthi,jandk,attachedtodistinctverticesofthetriangle.Inthispaper,weprovethatevery3-connected{K_(1,3),N_(8-i,i,1}-freegraphishamiltonian,where1≤i≤4.

简介：Inthispaper,wefirstshowtheglobalexistence,uniquenessandregularityofweaksolutionsforthehyperbolicmagnetohydrodynamics(MHD)equationsinR~3.ThenweestablishthatthesolutionswithinitialdatabelongingtoH~m(R~3)∩L~1(R~3)havethefollowingtimedecayrate:║▽~mu(x,t)║~2+║▽~mb(x,t)║~2+║▽~(m+1)u(x,t)║~2+║▽~(m+1)b(x,t)║~2≤c(1+t)~(-3/2-m)forlarget,wherem=0,1.

简介：Inthispaper,westudytheCauchyproblemforthe3DgeneralizedNavier-Stokes-Boussinesqequationswithfractionaldiffusion:{ut+（u·▽）u+v∧2αu=-▽p+θe（3）,e3=（0,0,1）T,θt+（u·▽）θ=0,Dicu=0.Withthehelpofthesmoothingeffectofthefractionaldiffusionoperatorandalogarithmicestimate,weprovetheglobalwell-posednessforthissystemwithα≥5/4.Moreover,theuniquenessandcontinuityofthesolutionwithweakerinitialdataisbasedonFourierlocalizationtechnique.Ourresultsextendonesonthe3DNavier-Stokesequationswithfractionaldiffusion.

简介：Inthispaper,weinvestigatetheglobalexistenceandlongtimebehaviorofstrongsolutionsforcompressiblenematicliquidcrystalflowsinthreedimensionalwholespace.TheglobalexistenceofstrongsolutionsisobtainedbythestandardenergymethodundertheconditionthattheinitialdataareclosetotheconstantequilibriumstateinH~2-framework.IftheinitialdatasinL~1-normarefiniteadditionally,theoptimaltimedecayratesofstrongsolutionsareestablished.WiththehelpofFouriersplittingmethod,onealsoestablishesoptimaltimedecayratesforthehigherorderspatialderivativesofdirector.