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,ω=（ω₁（x,t）,ω₂（x,t）,ω₃（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 μ is the kinematic viscosity,χ is the vortex viscosity,κ and γ 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 ω = 0 and χ = 0,the equations（0.0.3）reduce to the magnetohydrodynamic equations（MHD）.If both b = ω = 0 and χ = 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+α(0,T;（?）_∞,∞α).Our result reads:If u∈L2/1+α(0,T;（?）_∞,∞α)for 0<α<1,then the weak solution（u,ω,b）is regular on(0,T].For the case α = 0,using a different ap-proach we prove that if u∈L2（0,T;B∞,∞0）,then the weak solution（u,ω,b）is regular on(0,T].Assume u∈L2(0,7;（?）_∞,∞α),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≤τ≤t（||▽³u（τ,·）||₂² +||▽³ω（τ,·）||₂²+||▽³b（τ,·）||₂²）<∞.By the standard arguments of weak solution’s regularity,the weak solution（u,ω,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（?）dτ<∞ and the interpolation inequality in Besov space,we obtain ||▽_hu||₂ + ||▽_hω||₂-estimates.Step ₂:Using the estimates of ||▽_hu||₂ + ||▽_hω||2,the H¹ estimates of（u,ω）can be got further.Trans-forming the third direction derivative δ₃u,δ₃ω into horizontal gradient ▽_hu,▽_hω by the incompressibility condition and applying energy estimates again,we prove that if▽_hu,▽_hω ∈ L8/3（0,T;B∞,∞-1）,then the solution（u,ω）can be extended smoothly beyond T*>T.For the second blow-up criterion,we directly do the H1 estimates of（u,ω）.Trans-forming the third direction derivative δ₃u,δ₃ω into horizontal gradient ▽_hu,▽_hω 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,▽_hω in L¹（0,T;B∞,∞0）,we derive the H¹ estimates of（u,ω）,thus the solution（u,ω）can be extended smoothly beyond T^*>T.