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A Kato type theorem on zero viscosity limit of Navier-Stokes flows. (English) Zbl 0991.35059

The author presents a necessary and sufficient condition for the convergence of a solution \(u^{\nu}\) of the incompressible Navier-Stokes equations in a spatial domain \(\Omega\) to a solution \(u^0\) of the Euler equations at vanishing viscosity. He proves that \(u^{\nu} \rightarrow u^0\) as \(\nu\rightarrow 0\) in \(C([0,T], L^2(\Omega))\) iff there exist \(\delta (\nu)\) such that \(\lim_{\nu\rightarrow 0} \nu/\delta (\nu)=0\) and \(\lim_{\nu\rightarrow 0} \int_0^T \int_{\Gamma_{\delta}} |\nabla_t u^{\nu}_n|^2 dx dt=0\), where \(\Gamma_{\delta}\) is a \(\delta\)-neighborhood of the wall \(\partial \Omega\), \(\nabla_t\) denotes tangential derivatives, and \(u^{\nu}_n\) denotes the normal component of the velocity.

MSC:

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
35Q05 Euler-Poisson-Darboux equations
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