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