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Poche de tourbillon pour Euler 2D incompressible dans un ouvert à bord

Nicolas Depauw (1998)

Journées équations aux dérivées partielles

Nous considérons l'équation d'Euler pour un fluide incompressible dans un domaine borné régulier du plan. Pour une donnée initiale avec un tourbillon de type poche, i.e valant 1 sur un ouvert lisse à bord höldérien et 0 en dehors, nous prouvons l'existence d'une solution de même type, pour tout temps si la poche initiale est décollée du bord du domaine et seulement localement en temps si la poche initiale est tangente au bord. Nous contrôlons l'influence du bord grâce à la théorie des problèmes...

Poches de tourbillon singulières dans un fluide faiblement visqueux.

Taoufik Hmidi (2006)

Revista Matemática Iberoamericana

In this paper, we study the singular vortex patches in the two-dimensional incompressible Navier-Stokes equations. We show, in particular, that if the initial vortex patch is C1+s outside a singular set Σ, so the velocity is, for all time, lipschitzian outside the image of Σ through the viscous flow. In addition, the correponding lipschitzian norm is independent of the viscosity. This allows us to prove some results related to the inviscid limit for the geometric structures of the vortex patch.

Scattering for 1D cubic NLS and singular vortex dynamics

Valeria Banica, Luis Vega (2012)

Journal of the European Mathematical Society

We study the stability of self-similar solutions of the binormal flow, which is a model for the dynamics of vortex filaments in fluids and super-fluids. These particular solutions χ a ( t , x ) form a family of evolving regular curves in ℝ 3 that develop a singularity in finite time, indexed by a parameter a > 0 . We consider curves that are small regular perturbations of χ a ( t 0 , x ) for a fixed time t 0 . In particular, their curvature is not vanishing at infinity, so we are not in the context of known results of local existence...

Steady vortex rings with swirl in an ideal fluid: asymptotics for some solutions in exterior domains

Tadie (1999)

Applications of Mathematics

In this paper, the axisymmetric flow in an ideal fluid outside the infinite cylinder ( r ≤ d ) where ( r , θ , z ) denotes the cylindrical co-ordinates in ℝ 3 is considered. The motion is with swirl (i.e. the θ -component of the velocity of the flow is non constant). The (non-dimensional) equation governing the phenomenon is (Pd) displayed below. It is known from e.g. that for the problem without swirl ( f q = 0 in (f)) in the whole space, as the flux constant k tends to ∞ , 1) dist ( 0 z , ∂ A ) = O ( k 1 / 2 ) ; diam A = O ( exp ( - c 0 k 3 / 2 ) ) ; 2) ( k 1 / 2 Ψ ) k ∈ ℕ converges to a vortex cylinder U m (see...

Système d'Euler incompressible et régularité microlocale analytique

Pascal Gamblin (1994)

Annales de l'institut Fourier

Dans cet article on étudie la régularité analytique (ou Gevrey) des courbes intégrales de champs de vecteurs solutions non nécessairement lipschitziennes du système d’Euler incompressible. On en déduit que le front d’onde analytique (ou Gevrey) de ces solutions est localisé dans la variété caractéristique de l’opérateur linéarisé.

The vortex method for 2D ideal flows in the exterior of a disk

Diogo Arsénio, Emmanuel Dormy, Christophe Lacave (2014)

Journées Équations aux dérivées partielles

The vortex method is a common numerical and theoretical approach used to implement the motion of an ideal flow, in which the vorticity is approximated by a sum of point vortices, so that the Euler equations read as a system of ordinary differential equations. Such a method is well justified in the full plane, thanks to the explicit representation formulas of Biot and Savart. In an exterior domain, we also replace the impermeable boundary by a collection of point vortices generating the circulation...

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