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Existence of vortex sheets in dimension two. (Existence de nappes de tourbillon en dimension deux.) (French) Zbl 0780.35073
Dans le §1 l’auteur étudie l’existence de nappes de tourbillon en dimension 2. Soient \(x=(x_ 1,x_ 2)\) les coordonnées en \(\mathbb{R}^ 2\), \(\varphi\) une distribution sur \(\mathbb{R}^ 2\) à valeurs dans \(\mathbb{R}\), \[ \partial_ j\varphi= {{\partial\varphi} \over {\partial x_ j}}, \qquad \nabla\varphi={}^ t(\partial_ 1 \varphi,\partial_ 2 \varphi), \qquad \nabla^ \perp \varphi={}^ t(-\partial_ 2 \varphi, \partial_ 1 \varphi). \] Théorème. Si \(\omega_ 0\) est une mesure de Radon positive à support compact sur \(\mathbb{R}^ 2\) qui est dans l’espace de Sobolev \(H^{-1}(\mathbb{R}^ 2)\) et \(v_ 0=\nabla^ \perp \Delta^{-1}\omega_ 0\), alors il existe deux fonctions \(v\), \(p\), avec \(v\in L^ \infty_{\text{loc}}(\mathbb{R}\); \(L^ 2_{\text{loc}}(\mathbb{R}^ 2; \mathbb{R}^ 2))\) et \(p\in L^ \infty_{\text{loc}} (\mathbb{R};{\mathcal S}'(\mathbb{R}^ 2))\), solutions du système d’Euler \[ \partial v/\partial t+\text{div}(v \otimes v)=-\nabla p, \qquad \text{div} v=0,\;v|_{t=0}=v_ 0, \] où \(v\otimes v\) est la matrice \((v_ i v_ j)_{1\leq i,j\leq 2}\). Dans le §2 l’auteur prouve le théorème analogue sur une surface riemannienne composite, connexe, orientée.
Reviewer: S.Cinquini (Pavia)

MSC:
35Q30 Navier-Stokes equations
35L60 First-order nonlinear hyperbolic equations
76B47 Vortex flows for incompressible inviscid fluids
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