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September 2005 The evolution of a random vortex filament
Hakima Bessaih, Massimiliano Gubinelli, Francesco Russo
Ann. Probab. 33(5): 1825-1855 (September 2005). DOI: 10.1214/009117905000000323


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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Hakima Bessaih. Massimiliano Gubinelli. Francesco Russo. "The evolution of a random vortex filament." Ann. Probab. 33 (5) 1825 - 1855, September 2005.


Published: September 2005
First available in Project Euclid: 22 September 2005

zbMATH: 1084.60030
MathSciNet: MR2165581
Digital Object Identifier: 10.1214/009117905000000323

Primary: 60H05 , 76B47

Keywords: path-wise stochastic integration , rough path theory , Vortex filaments

Rights: Copyright © 2005 Institute of Mathematical Statistics


Vol.33 • No. 5 • September 2005
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