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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)
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