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The Stokes and Navier-Stokes equations with boundary conditions involving the pressure. (English) Zbl 0826.35093
The authors study the stationary Stokes and Navier-Stokes equations in a domain \(\Omega \subset \mathbb{R}^3\) where on some part of \(\partial \Omega\) a boundary condition for the pressure is prescribed. More precisely, the following problems are considered \[ \begin{aligned} - \nu \Delta u + \nabla p = f, \;\text{div} u = 0 \quad & \text{in } \Omega \\ u = u_0 \text{ on } \Gamma_1, \;u \times n = a \times n, p = p_0 \quad & \text{on } \Gamma_2 \\ u \cdot n = b \cdot n, \;\text{rot} u \times n = h \times n \quad & \text{on } \Gamma_3 \end{aligned} \] and \[ \begin{aligned} - \nu \Delta u + (u \cdot \nabla) u + \nabla p = f, \;\text{div} u = 0 \quad & \text{in } \Omega \\ u = u_0 \text{ on } \Gamma_1, \;u \times n = a \times n, p + {1 \over 2} |u |^2 = p_0 \quad & \text{on } \Gamma_2 \\ u \cdot n = b \cdot n, (\text{rot} u) \times n = h \times n \quad & \text{on } \Gamma_3. \end{aligned} \] Here \(\{\Gamma_1, \Gamma_2, \Gamma_3\}\) is a partition of \(\partial \Omega\). The above problems are then studied in a variational framework. For the Stokes problem, existence and uniqueness of a solution follows from the Lax-Milgram Lemma. In the case of the Navier-Stokes equations a further smallness condition has to be imposed. Finally, a variation of the boundary condition on \(\Gamma_2\) which allows one to prescribe the fluxes of the velocity on the connected components of \(\Gamma_2\) is also considered.

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids