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The evolution of a random vortex filament. (English) Zbl 1084.60030

The aim of this paper is to study an evolution problem for a model of a random vortex filament in three-dimensional incompressible fluid. The authors consider a generalization of the L. Rosenhead [Proc. R. Soc. London (A) 127, 590–612 (1930; JFM 56.1253.03)] model where the curve is not necessarily smooth, in order to take as initial condition a trajectory of a Brownian loop (or a fractional Brownian loop). The authors build a local solution for initial conditions which are Hölder continuous with exponent greater than 1/2. Then, they build a solution in a class of rough paths, in the sense of T. J. Lyons [Rev. Mat. Iberoam. 14, No. 2, 215–310 (1998; Zbl 0923.34056)], for initial conditions which are rough paths of Hölder regularity greater than 1/3. They also prove that the solution is Lipschitz continuous with respect to the initial data.

MSC:

60H05 Stochastic integrals
76B47 Vortex flows for incompressible inviscid fluids
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