Global Well-posedness Of The Incompressible Magneto-micropolar Equations

In this thesis,we investigate the 3D incompressible Magneto-micropolar system:where u =?u₁?x,t?,u₂?x,t?,u₃?x,t??denotes the velocity field,?=??₁?x,t?,?₂?x,t?,?₃?x,t??the micro-rotational velocity,b =?b1?x,t?,b2?x,t?,b3?x,t??the the magnetic field,and p is the hydrostatic pressure,and ? is the kinematic viscosity,? is the vortex viscosity,? and ? are spin viscosities,1/v is the magnetic Reynold.If b = 0,the equations?0.0.3?reduce to the micropolar fluid equations:If both ? = 0 and ? = 0,the equations?0.0.3?reduce to the magnetohydrodynamic equations?MHD?.If both b = ? = 0 and ? = 0,the equations?0.0.3?reduce to the famous Navier-Stokes equations.For the three dimensional system?0.0.3?and?0.0.4?,the regularity of weak solu-tions is still an open problem.It is necessary to study them as an important topic in the research of global well-posedness.Firstly,in the third chapter,we research the regularity criteria of weak solutions to the incompressible magneto-micropolar fluid equations?0.0.3?in three dimensions.By the Littlewood-Paley decomposition,the velocity field u can be decomposed into high frequency and low frequency,which are estimated by the energy estimate of H1,respectively.We derive the H1 estimates by u?L2/1+?(0,T;???_?,??).Our result reads:If u?L2/1+?(0,T;???_?,??)for 0<?<1,then the weak solution?u,?,b?is regular on(0,T].For the case ? = 0,using a different ap-proach we prove that if u?L2?0,T;B?,?0?,then the weak solution?u,?,b?is regular on(0,T].Assume u?L2(0,7;???_?,??),if T0 is close to T enough,then???sufficiently small.Applying a logarithmic Sobolev inequality,we obtain estimation of H3 on the small interval,i.e.supTo???t?||?³u??,·?||₂² +||?³???,·?||₂²+||?³b??,·?||₂²?<?.By the standard arguments of weak solution's regularity,the weak solution?u,?,b?is regular on(0,T].In the fourth chapter,we study the blow-up criterion of the smooth solutions to the 3D incompressible micropolar system?0.0.4?.For the first blow-up criterion,the proof can be accomplished in two steps.Step 1:The L² estimates of horizontal deriva-tive.By the condition???d?<? and the interpolation inequality in Besov space,we obtain ||?_hu||₂ + ||?_h?||₂-estimates.Step ₂:Using the estimates of ||?_hu||₂ + ||?_h?||2,the H¹ estimates of?u,??can be got further.Trans-forming the third direction derivative ?₃u,?₃? into horizontal gradient ?_hu,?_h? by the incompressibility condition and applying energy estimates again,we prove that if?_hu,?_h? ? L8/3?0,T;B?,?-1?,then the solution?u,??can be extended smoothly beyond T*>T.For the second blow-up criterion,we directly do the H1 estimates of?u,??.Trans-forming the third direction derivative ?₃u,?₃? into horizontal gradient ?_hu,?_h? by the incompressibility condition,the horizontal gradient can be decomposed into low frequency>intermediate frequency and high frequency parts,which can be estimated respectively.By the boundedness of ?_hu,?_h? in L¹?0,T;B?,?0?,we derive the H¹ estimates of?u,??,thus the solution?u,??can be extended smoothly beyond T^*>T.