Well-posedness of steady state Navier-Stokes equations with slip boundary conditions

摘要

We study the motion of a viscous incompressible fluid in a bounded region of containing finitely many rigid solid particles of small size. On the boundary of the particles, we prescribe the slip boundary condition of the form developed in Fujita H. [A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions. Mathematical fluid mechanics and modeling (Kyoto, 1994). Srikaisekikenkysho Kkyroku No. 888; 1994. p. 199-216] and Le Roux C, Tani A. [Steady solutions of the Navier-Stokes equations with threshold slip boundary conditions. Math Methods Appl Sci. 2007;30(5):595-624]. We derive a weak formulation of the problem and obtain an equivalent variational inequality formulation. Sufficient conditions for existence, uniqueness, and continuous dependence on data are obtained.