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Remarks on the Euler and Navier-Stokes equations in \(R^ 2\). (English) Zbl 0598.35093
Nonlinear functional analysis and its applications, Proc. Summer Res. Inst., Berkeley/Calif. 1983, Proc. Symp. Pure Math. 45, Pt. 2, 1-7 (1986).
[For the entire collection see Zbl 0583.00018.]
The author considers the Navier-Stokes equation \[ (1)\quad \partial_ tu-\nu \Delta u+(u\cdot \partial)u+\partial p=0,\quad div u=0 \] for \(t\geq 0\), \(x\in R^ 2\) and the Euler equation \[ (2)\quad \partial_ tu+(u\cdot \partial)u+\partial p=0,\quad div u=0. \] The basic theorem runs as follows. Let \(s>2\), \(a^{\nu}\), \(a^ 0\in H^ s\) and let \(\lim_{\nu \to 0}\| a^{\nu}-a^ 0\|_{H^ s}=0\), then there are unique \(c([0,\infty);H^ s)\)-solutions of (1) and (2) with \(u^{\nu}(0)=a^{\nu}\), \(u^ 0(0)=a^ 0\). Moreover, \(u^{\nu}\to u^ 0\) in \(C([0,T];H^ s)\) for any \(T>0\).
Reviewer: T.Shaposhnikova

MSC:
35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids