## 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

Zbl 0542.00009