Global Well-posedness Of An Initial-boundary Value Problem Of The Two-dimensional Incompressible Navier-Stokes-Darcy System

The focus of this paper is to investigate on the well-posedness of the following incompressible coupling of time-dependent Navier-Stokes system with Darcy equations under certain initial-boundary value conditions in the coupled domain—domain ?2 which is bounded on the vertical axis in the two-dimensional space:(?)Where the coupled domain ? and two connected domains:?1 and ?2 satisfy(?),?1??2 ?(?)and(?)= ?.These boundaries are ?1 =(?)?1\? and ?2=(?)?2\?.Where the domain ?2 is filled by porous medium.In the first equation of the above system,the vector u = u(x,t)=(u1(x,t),u2(x,t))T denotes the velocity and the tensor S(p1,u)=-2?D(u)+p1I is the stress tensor of the fluid in ?1.Here and in the equations,the tensor D(u)=1/2[?u +(?u)T] denotes the deformation tensor,the scalar p1 = p1(x,t)represents the pressure of the fluid in?1,and I means the 2 x 2 identity tensor,and ? is the viscosity.Where p2 = p2(x,t)is the pressure of the fluid in the domain ?2 and G is a positive constant obtained form experimental data.In addition,the two-dimensional tensor K describes the permeability of porous medium in ?2 and we shall assume that K is uniformly bounded and strictly elliptic,i.e.,there are two constants ?>?>0 such that?|?|2??TK???|?|2(?)??R2.Based on the technique of the straightening the boundary,applying the properties of Dirichlet-Neumann operator,the a priori estimates of the above system can be closed.Then,under the energy estimates,the local well-posedness of the above coupled Navier-Stokes-Darcy system can be proved.Finally,with the local theorem in hand,the global well-posedness of the above coupled equations can be demonstrated via the continuation criterion without any smallness conditions imposed on the initial data in the Sobolev spaces.