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The Stokes-system in exterior domains: \(L^ p\)-estimates for small values of a resolvent parameter. (English) Zbl 0738.35061

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^ 3\) with \(C^ 2\)-boundary \(\partial\Omega\). The author considers the resolvent problem for the Stokes system in an exterior domain: \[ -\nu\Delta u+\lambda\cdot u+\nabla\pi=f, \text{ div}u=0, \text{ in } \Omega_ 1=\mathbb{R}^ 3\backslash\overline\Omega, \;\;u\mid_{\partial\Omega}=0. (*) \] The problem (\(*\)) may be solved in the following sense: for \(\lambda\in\mathbb{C}\backslash]-\infty,0[\), \(\nu\in]0,\infty[\), \(p\in]1,\infty[\), \(f\in L^ p(\Omega_ 1)^ 3\), the set \[ S\{\nu,\lambda,p,f\}\equiv\{(u,\pi)\in W^{2,p}(\Omega_ 1)^ 3\times\tilde H^{1,p}(\Omega_ 1): u\in W_ 0^{1,p}(\Omega_ 1)^ 3, -\nu\Delta u+\lambda \cdot u+\nabla\pi=f, \text{div} u=0\} \] contains a pair \((u,\pi)\), with \(u\) uniquely determined, and with \(\pi\) unique up to an additive constant. Here \(\tilde H^{1,p}(\Omega_ 1)=\{g\in W^{1,p}_{loc}(\Omega_ 1):\nabla g\in L^ p(\Omega_ 1)^ 3\}\). Using the method of integral equation, the author estimates the solution \((u,\pi)\) in \(L^ p\)-norms for small values of \(|\lambda|\).

MSC:

35Q30 Navier-Stokes equations
76D07 Stokes and related (Oseen, etc.) flows
47A25 Spectral sets of linear operators
45E05 Integral equations with kernels of Cauchy type
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