Solutions globales avec nappe de tourbillon pour les equations d’Euler dans le plan. (Global solutions with vorticity for the plane Euler equations).(French)Zbl 0644.76020

Sémin. Équations Dériv. Partielles 1985-1986, Exposé No. 7, 11 p. (1986).
In this elegant note the authors discuss the evolution of a two- dimensional vortex of density $$\Omega (t,x_ 1)$$ concentrated on the curve $$\Gamma_ t:x_ 2=y(t,x_ 1)$$ via the equations derived by Birkhoff. Let $$B_ 0$$ be the Banach space of functions u which are the Fourier transforms of a bounded Radon measure given the norm $$\| u\| =\int_{R}d| \hat u|$$. Further let $$B_{\alpha}$$ be the Banach space of functions u(t), continuous in t, which are the Fourier transforms of a bounded Radon measure for each t and such that $$| e^{\alpha t}\hat u(t)|$$ is dominated by a positive bounded Radon measure $$\mu$$ and let $$| u|_{\alpha}=\inf \mu$$. Setting $$\Omega =2(1+\omega)$$ with $$\Lambda$$ given by $${\hat \Lambda}$$u$$=2\pi | \xi | \hat u(\xi)$$, (x,$$\omega)$$ are solutions of $\frac{d}{dt}\left( \begin{matrix} y_ x\\ \omega \end{matrix} \right)-\left( \begin{matrix} 0\\ \Lambda \end{matrix} \begin{matrix} \Lambda \\ 0\end{matrix} \right)\left( \begin{matrix} y_ x\\ \omega \end{matrix} \right)-\left( \begin{matrix} F(y_ x,\omega)_ x\\ G(y_ x,\omega)_ x\end{matrix} \right)=0$ together with the initial condition $$y_ x(0,x)=y_{ox}.$$
The operators F and G are each decomposed by the sums of a Hilbert transform and transforms with regular kernels and the above problem is manipulated into the form of a fixed point of an associated operator T on the Banach space $$B_{\alpha}\times B_{\alpha}$$. The principal result states that for some $$\epsilon >0$$ and $$y_{ox}\in B_ o$$ of mean zero and norm $$\leq \epsilon$$, there exists $$d>0$$, $$d\ni (y_ x,\omega)\in B_{\alpha}\times B_{\alpha}$$ is the distributional solution of f whose mean is zero and $$(y_ x,\omega)\to 0$$ uniformly as $$t\to \infty$$.
Reviewer: M.Thompson

MSC:

 76B47 Vortex flows for incompressible inviscid fluids 44A15 Special integral transforms (Legendre, Hilbert, etc.) 35Q99 Partial differential equations of mathematical physics and other areas of application
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