Research On The Well-Posedness Of Solutions For A Class Of Anisotropic Non-Newtonian Micropolar Fluid Equations

In this paper,we mainly study the initial boundary value problem of the following anisotropic non-Newtonian micropolar fluid equations:（?）Where Ω（?）R³ is a bounded region with sufficiently smooth boundary,QT=Ω×[0,T],Γ_T=（?）Ω×[0,T].In this model,the unknown vector function u represents the fluid velocity;w represents the angular velocity;Diu=（（?）_iu₁,（?）_iu₂,（?）_iu₃）;P is the pressure term;f and g are given external force terms;The exponent of q_i is a given constant,satisfying 1<q_i<∞（i=1,2,3）;ν₁,μ_r,ν₂ represent the kinematic viscosity coefficient,vortex viscosity coefficient and spin viscosity coefficient,and they are positive constants.For the above model,we construct the approximate solution of the equation,estimate the approximate solution,take the limit of the approximate solution,use Galerkin method and monotone operator theory to overcome the difficulties of the strong nonlinearity and strong coupling of the equations,prove the existence of the weak solution of the equations.The results were then compared with the isotropic case.Finally,the energy inequality of the equation is given and the solution is proved to be exponentially degenerate in terms of time.