Abstract
In this paper, we consider the evolutionary Navier-Stokes equations subject to the nonslip boundary condition together with a Clarke subdifferential relation between the dynamic pressure and the normal component of the velocity. Under the Rauch condition, we use the Galerkin approximation method and a weak precompactness criterion to ensure the convergence to a desired solution. Moreover, a control problem associated with such system of equations is studied with the help of a stability result with respect to the external forces. At the end of this paper, a more general condition due to Z. Naniewicz, namely the directional growth condition, is considered and all the results are reexamined.
Highlights
In many engineering situations, one deals with fluid flow problems in tubes or channels, or for semipermeable walls and membranes
The model that usually describes this situation is repesented by the Navier-Stokes equations for incompressible viscous fluids with the nonslip boundary conditions together with a Clarke subdifferential relation between the dynamic pressure and the normal component of the velocity
Among the main applications of this theory, we mention the Newtonian and nonNewtonian Navier-Stokes equations and their variants with nonstandard boundary conditions ensuing from the multivalued nonmonotone friction law with leak, slip, or nonslip conditions
Summary
One deals with fluid flow problems in tubes or channels, or for semipermeable walls and membranes. In the case of the Navier-Stokes equations, the smallness condition links the growth condition constant, the coercivity constant, and the norm of the trace operator It is, not clear how it can be checked in a concrete situation. Among the disadvantages of the Rauch condition is that it ensures the existence of a solution, it does not allow the conclusion that the nonconvex functional is locally Lipschitz or even finite on the whole space. The present paper represents a continuation of our previous paper [32], where existence and optimal control questions involving the stationary Navier-Stokes problem with the multivalued nonmonotone boundary condition are studied.
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