| Title:
|
Probabilistic models of vortex filaments (English) |
| Author:
|
Flandoli, Franco |
| Author:
|
Minelli, Ida |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
51 |
| Issue:
|
4 |
| Year:
|
2001 |
| Pages:
|
713-731 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A model of vortex filaments based on stochastic processes is presented. In contrast to previous models based on semimartingales, here processes with fractal properties between $1/2$ and $1$ are used, which include fractional Brownian motion and similar non-Gaussian examples. Stochastic integration for these processes is employed to give a meaning to the kinetic energy. (English) |
| Keyword:
|
stochastic integration |
| Keyword:
|
fractional Brownian motion |
| Keyword:
|
$p$-variation |
| Keyword:
|
vortex filaments |
| Keyword:
|
statistical fluid mechanics |
| MSC:
|
60H05 |
| MSC:
|
60H30 |
| MSC:
|
76F55 |
| MSC:
|
76M35 |
| idZBL:
|
Zbl 1001.60057 |
| idMR:
|
MR1864038 |
| . |
| Date available:
|
2009-09-24T10:46:44Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127682 |
| . |
| Reference:
|
[1] E. Alòs and D. Nualart: Stochastic calculus with respect to the fractional Brownian motion.Preprint. |
| Reference:
|
[2] E. Alòs, O. Mazet and D. Nualart: Stochastic calculus with respect to Gaussian processes.(to appear). MR 1849177 |
| Reference:
|
[3] J. Bell and D. Marcus: Vorticity intensification and the transition to turbulence in the three-dimensional Euler equation.Comm. Math. Phys. 147 (1992), 371–394. MR 1174419, 10.1007/BF02096593 |
| Reference:
|
[4] J. Bertoin: Sur une integrale pour les processus à $\alpha $-variatione bornée.Ann. Probab. 17 (1989), 1521–1535. MR 1048943, 10.1214/aop/1176991171 |
| Reference:
|
[5] H. Bessaih: Mean field theory for 3-D vortex filaments.In preparation. |
| Reference:
|
[6] E. Bolthausen: On the construction of the three dimensional polymer measure.Probab. Theory Related Fields 97 (1993), 81–101. Zbl 0794.60104, MR 1240717, 10.1007/BF01199313 |
| Reference:
|
[7] A. Chorin: Vorticity and Turbulence.Springer-Verlag, New York, 1994. Zbl 0795.76002, MR 1281384 |
| Reference:
|
[8] A. Colesanti and M. Romito: Some remarks on a probabilistic model of the vorticity field of a 3D fluid.Preprint. |
| Reference:
|
[9] F. Flandoli: On a probabilistic description of small scale structures in 3D fluids.(to appear). Zbl 1017.76074, MR 1899111 |
| Reference:
|
[10] F. Flandoli and M. Gubinelli: The Gibbs ensamble of a vortex filament.(to appear). MR 1892850 |
| Reference:
|
[11] U. Frisch: Turbulence.Cambridge University Press, Cambridge, 1995. Zbl 0832.76001, MR 1428905 |
| Reference:
|
[12] G. Gallavotti: Foundations of Fluid Mechanics.Quaderni FM2000–5, http://ipparco.roma1.infn.it/. |
| Reference:
|
[13] A. N. Kolmogorov: The local structure of turbulence in incompressible viscous fluid for very large Reynolds number.C. R. (Dokl.) Acad. Sci. USSR (N.S.) 30 (1941), 299–303. MR 1124922 |
| Reference:
|
[14] N. S. Landkof: Foundations of Modern Potential Theory.Springer-Verlag, New York, 1972. Zbl 0253.31001, MR 0350027 |
| Reference:
|
[15] P. L. Lions and A. Majda: Equilibrium statistical theory for nearly parallel vortex filaments.Comm. Pure Appl. Math. 53 (2000), 76–142. MR 1715529, 10.1002/(SICI)1097-0312(200001)53:1<76::AID-CPA2>3.0.CO;2-L |
| Reference:
|
[16] C. Marchioro, M. Pulvirenti: Mathematical Theory of Incompressible Nonviscous Fluids.Springer-Verlag, Berlin, 1994. MR 1245492 |
| Reference:
|
[17] I. Minelli: In preparation.. |
| Reference:
|
[18] D. Nualart, C. Rovira and S. Tindel: Probabilistic models for vortex filaments based on fractional Brownian motion.In preparation. |
| Reference:
|
[19] L. Onsager: Statistical hydrodynamics.Nuovo Cimento (9) 6 (1949), Supplemento no. 2, 279–287. MR 0036116, 10.1007/BF02780991 |
| Reference:
|
[20] Z. S. She, E. Jackson and S. A. Orszag: Structure and dynamics of homogeneous turbulence: models and simulations.Proc. Roy. Soc. Lond. Ser. A 434 (1991), 101–124. 10.1098/rspa.1991.0083 |
| Reference:
|
[21] B. Toth and W. Werner: The true self-repelling motion.Probab. Theory Relat. Fields 111 (1998), 375–452. MR 1640799, 10.1007/s004400050172 |
| Reference:
|
[22] A. Vincent and M. Meneguzzi: The spatial structure and statistical properties of homogeneous turbulence.J. Fluid Mech. 225 (1991), 1–25. 10.1017/S0022112091001957 |
| Reference:
|
[23] J. Westwater: On Edwards model for long polymer chains.Comm. Math. Phys. 72 (1980), 131–174. Zbl 0431.60100, MR 0573702, 10.1007/BF01197632 |
| Reference:
|
[24] L. C. Young: An inequality of Hölder type, connected with Stieltjes integration.Acta Math. 67 (1936), 251–282. MR 1555421, 10.1007/BF02401743 |
| Reference:
|
[25] M. Zähle: Integration with respect to fractal functions and stochastic calculus. I.Probab. Theory Related Fields 111 (1998), 333–374. MR 1640795, 10.1007/s004400050171 |
| . |
