The evolution of a random vortex filament
Hakima Bessaih, Massimiliano Gubinelli, and Francesco Russo
Source: Ann. Probab. Volume 33, Number 5
(2005), 1825-1855.
Abstract
We study an evolution problem in the space of continuous loops in a three-dimensional Euclidean space modeled upon the dynamics of vortex lines in 3d incompressible and inviscid fluids. We establish existence of a local solution starting from Hölder regular loops with index greater than 1/3. When the Hölder regularity of the initial condition X is smaller or equal to 1/2, we require X to be a rough path in the sense of Lyons [Rev. Mat. Iberoamericana 14 (1998) 215–310, System Control and Rough Paths (2002). Oxford Univ. Press]. The solution will then live in an appropriate space of rough paths. In particular, we can construct (local) solution starting from almost every Brownian loop.
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Permanent link to this document: http://projecteuclid.org/euclid.aop/1127395875
Digital Object Identifier: doi:10.1214/009117905000000323
Mathematical Reviews number (MathSciNet): MR2165581
Zentralblatt MATH identifier: 1084.60030
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