Abstract
A time-dependent Stokes fluid flow problem is studied with nonlinear boundary conditions described by the Clarke subdifferential. We present equivalent weak formulations of the problem, one of them in the form of a hemivariational inequality. The existence of a solution is shown through a limiting procedure based on temporally semi-discrete approximations. Uniqueness of the solution and its continuous dependence on data are also established. Finally, we present a result on the existence of a solution to an optimal control problem for the hemivariational inequality.
Highlights
Let Ω be a bounded connected domain in Rd (d = 2 or 3) with a C2 boundary Γ
The following surjectivity result for pseudomonotone and coercive operators will be applied later in the paper
Denote by BV (I; X) the space of functions of bounded total variation on I defined as follows
Summary
Let Ω be a bounded connected domain in Rd (d = 2 or 3) with a C2 boundary Γ. Let T0 > 0 and define Q = Ω × (0, T0). We consider hemivariational inequalities for the nonstationary Stokes system ut − ν∆u + ∇h = f in Q,. The system (1)–(2) is to be supplemented by initial and boundary conditions. The initial condition is u(0) = u0 in Ω.
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