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Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary. (English) Zbl 0559.35067

Seminar on nonlinear partial differential equations, Publ., Math. Sci. Res. Inst. 2, 85-98 (1984).
[For the entire collection see Zbl 0542.00009.]
The paper is concerned with the question of convergence of the nonstationary incompressible Navier-Stokes flow \(u=u_{\nu}\) to the Euler flow \(\bar u\) as the viscosity \(\nu\) tends to zero. Here u and \(\bar u\) are weak solutions of the Navier-Stokes equation \(\partial_ tu-\nu \Delta u+(u \text{grad})u+\text{grad} p=f,\quad div u=0,\quad u|_{\partial \Omega}=0\) and the Euler equation \(\partial_ t\bar u+(\bar u \text{grad})\bar u+\text{grad} p=\bar f,\quad div \bar u=0,\quad \bar u_ n|_{\partial \Omega}=0.\) The main result of the paper is the equivalence between the relations: (i) u(t)\(\to \bar u(t)\) in \(L^ 2(\Omega)\) uniformly in \(t\in [0,T]\), (ii) u(t)\(\to \bar u(t)\) weakly in \(L^ 2(\Omega)\) for each \(t\in [0,T]\), \((iii)\quad \nu \int^{T}_{0}\| \text{grad} u\|^ 2dt\to 0,\) (iii’) \(\nu\int^{T}_{0}\| \text{grad} u\|^ 2_{\Gamma_{c\nu}}dt\to 0\) if \(\nu\) \(\to 0\) and u(0)\(\to \bar u(0)\) in \(L^ 2(\Omega)\), \(\int^{T_ 0}_{0}\| f-\bar f\| dt\to 0\) as \(\nu\) \(\to 0\), where \(\Gamma_{c\nu}\subset \Omega\) is the boundary strip of width \(c\nu\) with \(c>0\) fixed but arbitrary.
Reviewer: I.Bock

MSC:

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs

Citations:

Zbl 0542.00009