Optimal results for the Brezzi-Pitkäranta approximation of viscous flow problems. (English) Zbl 1150.35508

The authors investigate a mixed boundary-value problem for the strongly elliptic system of second order partial differential equations in a bounded domain \(\Omega \subset \mathbb R^3\) with boundary \(\partial \Omega \) of class \(C^2\) \[ \begin{aligned} -\Delta v^{\varepsilon } + \nabla p^{\varepsilon }& = f' \quad \text{in } \Omega \\ -\varepsilon ^2\Delta p^{\varepsilon } + \text{div } v^{\varepsilon } &= f^4 \quad \text{in } \Omega \\ v^{\varepsilon } &= g' \quad \text{on } \partial \Omega \\ \partial _np^{\varepsilon }& = g_4 \quad \text{on } \partial \Omega. \end{aligned} \] They investigate the asymptotic behavior for solutions as \(\varepsilon \to 0 \). With \(g_4 =0 \), this system has to be considered as a singular perturbation of the Stokes system \[ \begin{aligned} -\Delta v^0 + \nabla p^0 &= f' \quad \text{in } \Omega \\ \text{div } v^0 &= f^4 \quad \text{in } \Omega \\ v^0 &= g' \quad \text{on } \partial \Omega. \end{aligned} \] Under additional regularity assumptions on the data, and energy estimates, the order of convergence with respect to \(\varepsilon \) and convergence in \(H^{s+1}\) and \(H^s\) norms are obtained for the velocity and pressure with \(s\in [0,3/2]\). They verify the asymptotic precision of the estimates by constructing the boundary layers.


35Q30 Navier-Stokes equations
76D07 Stokes and related (Oseen, etc.) flows
76N20 Boundary-layer theory for compressible fluids and gas dynamics