## Vanishing viscosity plane parallel channel flow and related singular perturbation problems.(English)Zbl 1160.35329

Summary: We study a special class of solutions to the three-dimensional Navier-Stokes equations $$\partial_t u^\nu + \nabla_{u^\nu} u^\nu + \nabla p^\nu=\nu\Delta_u^\nu$$, with no-slip boundary condition, on a domain of the form $$\Omega = {(x,y,z) : 0 \leq z \leq 1}$$, dealing with velocity fields of the form $$u^\nu (t,x,y,z) = (v^\nu (t,z),w^\nu (t,x,z),0)$$, describing plane-parallel channel flows. We establish results on convergence $$u^\nu \rightarrow u^0$$ as $$\nu \rightarrow 0$$, where $$u^0$$ solves the associated Euler equations. These results go well beyond previously established $$L^2$$-norm convergence, and provide a much more detailed picture of the nature of this convergence. Carrying out this analysis also leads naturally to consideration of related singular perturbation problems on bounded domains.

### MSC:

 35B25 Singular perturbations in context of PDEs 35K20 Initial-boundary value problems for second-order parabolic equations 35Q30 Navier-Stokes equations
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