| The micropolar fluid equations can describe the motion of viscous fluid with suspended particles.When the velocity field and the micro-rotation field of the 2D incompress-ible micropolar fluid equations have full dissipation,the global existence and uniqueness of solutions could be obtained easily.However,the global regularity for the inviscid equations is currently out of reach.In this thesis,we study the global well-posedness and regularity of 2D and 21/2D incompressible micropolar fluid equations in the whole space by employing energy method,commutator estimates and anisotropic Sobolev inequality.In Chapter three,the global regularity of solutions for the 2D incompressible micropolar fluid equations with partial velocity dissipation is considered.By using energy estimates to prove the global bound of‖u,w‖H~2,and combining with the logarithmic interpolation inequal-ity,we obtain the higher-order energy inequalities of the equations.Moreover,we overcome the difficulties caused by nonlinear terms by using commutator estimates.Then,we establish the global regularity of solutions for the equations.In Chapter four,the global well-posedness of the 21/2D incompressible micropolar fluid equations is studied.We make use of the anisotropic Sobolev inequality,energy estimates,and combine the special structure of the equations to overcome difficulties brought by the nonlinear terms.Then,the existence and uniqueness of solutions for the equations in the 21/2D space is obtained.In Chapter five,we study the global regularity of solutions for the 21/2D incompressible mi-cropolar fluid equations with partial velocity dissipation.By utilizing the anisotropic Sobolev inequality,Gronwall’s inequality,and combining with the energy estimates,we overcome the difficulties caused by the partial velocity dissipation and the lack of divergence free condition of angular velocity.At last,we establish the global regularity of solutions for the micropolar fluid equations with only horizontal velocity dissipation or vertical velocity dissipation. |
