## 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.

### MSC:

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