Singular vortex patches in a fluid of low viscosity. (Poches de tourbillon singulières dans un fluide faiblement visqueux.) (French) Zbl 1127.35037

The author studies the motion of a bidimensional, incompressible, small viscous fluid as the Navier-Stokes equations
\[ \partial_t v_\nu+ v_\nu\nabla v_\nu- \nu\Delta v_\nu= -\nabla p_\nu,\quad \operatorname{div} v_\nu= 0,\quad v_\nu|_{t= 0}= v^0,\tag{1} \]
near the motion of a fluid of null viscosity in the system of Euler equations
\[ \partial_t v+v\nabla v\neq -\nabla p,\quad\operatorname{div} v= 0,\quad v|_{t= 0}= v^0, \tag{2} \]
The used notations are evident. The motions are in the presence of singular vortex patches. The author considers only small viscosity, because he is interested in the comparison of the results for \(\nu\neq 0\) and \(\nu= 0\). He considers \(\lim_{\nu\to 0} v_\nu\) obtained from (1) and compares it with \(v\) obtained from (2). The singularities of the vortex patches concern the singularities which can appear on the boundary \(\partial\Omega\) of the domain \(\Omega\) occupied by the vortex patch: coin, cusp or other singularities.
The history of the presented problem is sketched in the introduction, followed by five chapters and an appendix, where there are analysed the following subjects: the velocity field, the singularities on \(\partial\Omega\), the dynamics of the singular patches and the comparison between the motion in (1) and (2). The paper is very rich in content and requires a high level of mathematical knowledge.


35Q30 Navier-Stokes equations
76D17 Viscous vortex flows
35A20 Analyticity in context of PDEs
42B25 Maximal functions, Littlewood-Paley theory
76B47 Vortex flows for incompressible inviscid fluids
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