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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
##### Keywords:
Navier-Stokes equation; Euler equation