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A new approach to the \(L^ 2\)-regularity theorems for linear stationary nonhomogeneous Stokes systems. (English) Zbl 0885.35088

For a bounded domain \(\Omega\subset \mathbb{R}^n\), \(n\geq 2\), the author studies the system \(-\mu\Delta u+\nabla p= f\), \(\lambda p+\nabla u=g\) in \(\Omega\), \(u|_{\partial\Omega}= \varphi\), where \(f\), \(g\), and \(\varphi\) are given functions, and \(\lambda\geq 0\) is a real parameter (for \(\lambda=0\) we have the usual nonhomogeneous Stokes system). The main result is an elementary proof of the following theorem: let \(k\) be a nonnegative integer, \(\partial\Omega\in C^{k,1}\), \(f\in H^{k- 1}\), \(g\in H^k\), \(\varphi\in H^{k+ 1/2}(\partial\Omega)\), and the compatibility condition \(\int_\Omega gdx= \int_{\partial\Omega} \varphi\cdot vds\) holds. Then there is a unique solution \((u_\lambda, p_\lambda)\in H^{k+ 1}\times H^k\) of the problem. Moreover, \[ \mu|u_\lambda|_{k+ 1}+(1+ \lambda\mu)|p_\lambda|_k\leq \text{const}(|f|_{k- 1}+\mu|g|_k+ \mu|\varphi|_{k+ 1/2,\partial\Omega}), \] where \(|\cdot|_l\) denotes the canonical norm in \(H^l\). The proof uses the technique of linearization and flattening of the boundary.
Reviewer: O.Titow (Berlin)

MSC:

35Q30 Navier-Stokes equations
76D07 Stokes and related (Oseen, etc.) flows
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