Steady Stokes flows with threshold slip boundary conditions. (English) Zbl 1129.35424
Summary: We prove the existence, uniqueness and continuous dependence on the data of weak solutions to boundary-value problems that model steady flows of incompressible Newtonian fluids with wall slip in bounded domains. The flows satisfy the Stokes equations and a nonlinear slip boundary condition: for slip to occur, the magnitude of the tangential traction must exceed a prescribed threshold, which is independent of the normal stress, and where slip occurs the tangential traction is equal to a prescribed, possibly nonlinear, function of the slip velocity. In addition, a Dirichlet condition is imposed on a component of the boundary if the domain is rotationally symmetric. The method of proof is based on a variational inequality formulation of the problem and fixed point arguments which utilize wellposedness results for the Stokes problem with a slip condition of the ‘friction type’.
MSC:
| 35J85 | Unilateral problems; variational inequalities (elliptic type) (MSC2000) |
| 35Q35 | PDEs in connection with fluid mechanics |
| 35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |
| 76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
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