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On solvability of the Stokes problem in Sobolev power weight spaces. (English) Zbl 0595.35085
The author is concerned with the boundary value problem in power weighted Sobolev spaces with weight $$d^{\epsilon}(x)=(dist(x,\partial \Omega))^{\epsilon}$$, ($$\epsilon\in R)$$ for the Stokes equations $$-\nu \Delta u+\text{grad} p=f$$, div u$$=g$$ in $$\Omega$$, $$u=\phi$$ on $$\partial \Omega$$, in a bounded domain $$\Omega$$ in $$R^ N$$ with Lipschitz boundary $$\partial \Omega$$, where $$\nu >0$$ and $$\int_{\Omega}gdx=\int_{\partial \Omega}\phi,n dS$$. Let $$L^ 2(\Omega;d^{\epsilon})$$ be the Lebesgue space with weight $$d^{\epsilon}$$, $W^{1,2}(\Omega,d^{\epsilon})=\{u\in L^ 2(\Omega;d^{\epsilon})| \quad D^{\alpha}u\in L^ 2(\Omega;d^{\epsilon}),\quad | \alpha | \leq 1\}$ and $L^ 2_ 0(\Omega;d^{\epsilon})=\{\phi \in L^ 2(\Omega;d^{\epsilon})| \quad \int_{\Omega}\phi dx=0\}.$ The main result is the following theorem: There exists an open interval J $$(0\in J)$$ such that for every $$\epsilon \in J$$ the Stokes problem has a unique weak solution $(u,p)\in [W^{1,2}(\Omega;d^{\epsilon})]^ N\times L^ 2_ 0(\Omega;d^{\epsilon})$ whenever $$f\in [W_ 0^{1,2}(\Omega;d^{-\epsilon})]^ N,$$ $$g\in L^ 2(\Omega;d^{\epsilon})$$ and $$\phi \in [W^{1,2}(\Omega;d^{\epsilon})]^ N.$$
Reviewer: R.Iino
##### MSC:
 35Q30 Navier-Stokes equations 35J55 Systems of elliptic equations, boundary value problems (MSC2000) 35A05 General existence and uniqueness theorems (PDE) (MSC2000)
##### Keywords:
boundary value problem; power weighted Sobolev spaces
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