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Les équations de Stokes et de Navier-Stokes avec des conditions aux limites sur la pression. (On Stokes and Navier-Stokes equations with boundary condition on the pressure). (French) Zbl 0687.35069
Nonlinear partial differential equations and their applications, Lect. Coll. de France Semin., Vol. IX, Paris/Fr. 1985-86, Pitman Res. Notes Math. Ser. 181, 179-264 (1988).
[For the entire collection see Zbl 0653.00012.]
The authors study the stationary Stokes and Navier-Stokes equations with non standard boundary conditions, in which the pressure is given on some part of the boundary. Examples of flows leading to these equations are the flow in a network of pipes and the flow around an obstable. In this paper, the variational formulations of these problems are given and are proved to be equivalent to the boundary value problems. Moreover, existence and uniqueness results are proved for the variational problems. Depending on the geometry of the problem, the bilinear form in the variational formulations is considered to satisfy two types of coercivity conditions. Thus, the existence and uniqueness results are proved by a variation of the classical methods such as the Lax-Milgram lemma and the Galerkin’s approximation method. Furthermore, the authors present a determination of the normal derivative of the pressure and of the fluxes of the velocity on parts of the boundary. Finally, some numerical results are given in the last section of the paper.
Reviewer: M.A.Boudourides

##### MSC:
 35Q30 Navier-Stokes equations 35J65 Nonlinear boundary value problems for linear elliptic equations 35J20 Variational methods for second-order elliptic equations 65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems 65N99 Numerical methods for partial differential equations, boundary value problems