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

ABSTRACT

We study the motion of a viscous incompressible fluid in a bounded region of ${\mathbb{R}}^3$ 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.

COMMUNICATED BY:

• G. P. Panasenko

KEYWORDS:

• Navier–Stokes
• slip boundary conditions
• well-posedness
• variational formulation
• suspensions

AMS SUBJECT CLASSIFICATIONS:

• 76D03
• 76D05
• 76T20

Disclosure statement

No potential conflict of interest was reported by the authors.