The evolution of a random vortex filament

The evolution of a random vortex filament Article

Bessaih, H, Gubinelli, M, Russo, F. (2005). The evolution of a random vortex filament . 33(5), 1825-1855. 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. © Institute of Mathematical Statistics, 2005.