| Navier-Stokes equation as one of the fundamental equations in fluid dynamics,has a long history.It describes the motion of fluid with viscosity and has elementary applications in engineering.The gently water flow,flashy stream,turbulence near fight wings,tornadoes near the sea level and shock waves in an explosion,all these phenomenons can seek explanations from Navier-Stokes equations.In this paper,we will consider the existence of solutions of Navier-Stokes equations satisfying homogeneous boundary conditions,in multi-connected bounded domains with symmetry.The Clay Mathematics Institute has called the existence and smoothness of solutions of Navier-Stokes equations as one of the seven most important open problems in mathematics.Navier-Stokes equations has drawn many attentions and numerous results have been achieved.Mathematicians once believe that the study on5-dimensional stationary problems might lead to new ideas to 3-dimensional evolutionary problems.On the other hand,Korobkov,Pileckas and Russo proved the existence of solutions to the non-homogeneous boundary value problems on the general multi-connected bounded domains.Their results was published in Annals of Math,2015.Therefore,the existence of solutions on multi-connected higher dimensional domain is indeed an interesting problem.The boundary function can be extended as a curl if its flux on every connected components of boundary is zero.However,the extension can not be achieved if only the compatibility condition is satisfied.We will construct the virtual drain function,such that it concentrates all the fluxes of the boundary function,and has support in a neighborhood of the symmetric plane.Subtracting the virtual drain from the boundary function,the remaining part,with zero flux,then can be extended as a curl.A uniform bound for all possible solutions can be proved by Leray inequality.The existence of weak solutions can then be proved by Leray-Schauder degree theory. |
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