## 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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