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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
Full Text: Numdam EuDML