On a probabilistic description of small scale structures in 3D fluids. (English) Zbl 1017.76074

Flandoli, Franco

Ann. Inst. Henri Poincaré, Probab. Stat. 38, No. 2, 207-228 (2002).

Summary: We introduce a vortex structure based on three-dimensional Brownian motion. The interaction energy \(H_{xy}\) between different vortex filaments \((x+W_t)\) and \((y+W_t)\) is rigorously defined and proved to be finite. The divergence of \(H_{xy}\) as \(|x-y|\to 0\) is analyzed and used to prove that the total energy of vortex structure is finite, under suitable assumptions. We also establish a relation with the intersection local time.

MSC:

76M35	Stochastic analysis applied to problems in fluid mechanics
76F55	Statistical turbulence modeling
60K40	Other physical applications of random processes

Keywords:

small scale structures; vortex structure; three-dimensional Brownian motion; interaction energy; total energy; intersection local time