Revista Matemática Iberoamericana


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Volume 17, Issue 2, 2001, pp. 331–373
DOI: 10.4171/RMI/297

Published online: 2001-08-31

Branching process associated with 2d-Navier Stokes equation

Saïd Benachour[1], Bernard Roynette[2] and Pierre Vallois[3]

(1) Université Henri Poincaré, Vandoeuvre lès Nancy, France
(2) Université Henri Poincaré, Vandoeuvre lès Nancy, France
(3) Université Henri Poincaré, Vandoeuvre lès Nancy, France

$\Omega$ being a bounded open set in $\mathbb R^2$, with regular boundary, we associate with Navier-Stokes equation in $\Omega$ where the velocity is null on ∂Ω, a non-linear branching process ($Y_t; t ≥ 0$). More precisely: $E_{ω0} (h,Y_t) = (ω, h)$, for any test function $h$, where ω = rot $u$, $u$ denotes the velocity solution of Navier-Stokes equation. The support of the random measure $Y_t$ increases or decreases of one unit when the underlying process hits ∂Ω; this stochastic phenomenon corresponds to the creation-annihilation of vortex localized at the boundary of Ω.

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Benachour Saïd, Roynette Bernard, Vallois Pierre: Branching process associated with 2d-Navier Stokes equation. Rev. Mat. Iberoamericana 17 (2001), 331-373. doi: 10.4171/RMI/297