## 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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