Noise prevents collapse of vlasov-poisson point charges

Communications on Pure and Applied Mathematics

Volumn 67, Issue 10, 2014, Pages 1700-1736

  Delarue, François ^a   Flandoli, Franco ^b   Vincenzi, Dario ^a  

^a UNIVERSITÉ DE NICE SOPHIA ANTIPOLIS   (France)

^b UNIVERSITY OF PISA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84905754793     PISSN: 00103640     EISSN: 10970312     Source Type: Journal    
DOI: 10.1002/cpa.21476     Document Type: Article

References (35)

1. Ambrosio, L. Transport equation and Cauchy problem for BV vector fields. Invent. Math. 158 (2004), no. 2, 227-260. doi: 10.1007/s00222-004-0367-2
2. Attanasio, S.; Flandoli, F. Zero-noise solutions of linear transport equations without uniqueness: an example. C. R. Math. Acad. Sci. Paris 347 (2009), no. 13-14, 753-756. doi: 10.1016/j.crma.2009.04.027
3. Bafico, R.; Baldi, P. Small random perturbations of Peano phenomena. Stochastics 6 (1981/82), no. 3-4, 279-292. doi: 10.1080/17442508208833208
4. Bec, J.; Cencini, M.; Hillerbrand, R. Heavy particles in incompressible flows: the large Stokes number asymptotics. Phys. D 226 (2007), no. 1, 11-22. doi: 10.1016/j.physd.2006.10.007
5. Carmack, L.; Majda, G. Concentrations in the one-dimensional Vlasov-Poisson equations: additional regularizations. Phys. D 121 (1998), no. 1-2, 127-162. doi: 10.1016/S0167-2789(98)00029-3
6. Celani, A.; Rubenthaler, S.; Vincenzi, D. Dispersion and collapse in stochastic velocity fields on a cylinder. J. Stat. Phys. 138 (2010), no. 4-5, 579-597. doi: 10.1007/s10955-009-9875-1
7. Delort, J.-M. Existence de nappes de tourbillon en dimension deux. J. Amer. Math. Soc. 4 (1991), no. 3, 553-586. doi: 10.2307/2939269
8. DiPerna, R. J.; Lions, P.-L. Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989), no. 3, 511-547. doi: 10.1007/BF01393835
9. Dirr, N.; Luckhaus, S.; Novaga, M. A stochastic selection principle in case of fattening for curvature flow. Calc. Var. Partial Differential Equations 13 (2001), no. 4, 405-425. doi: 10.1007/s005260100080
10. E, W.; Vanden-Eijnden, E. A note on generalized flows. Phys. D 183 (2003), no. 3-4, 159-174. doi: 10.1016/S0167-2789(03)00183-0
11. 2-solutions at zero-diffusivity in the Kraichnan model of a passive scalar. Preprint, 1996. arxiv:chao-dyn/9605008
12. Falkovich, G.; Gawȩdzki, K.; Vergassola, M. Particles and fields in turbulence. Rev. Mod. Phys. 73 (2001), no. 4, 913-975. doi: 10.1103/RevModPhys.73.913
13. Flandoli, F. Random perturbation of PDEs and fluid dynamic models. Lectures from the 40th Probability Summer School Held in Saint-Flour, 2010. Lecture Notes in Mathematics, 2015. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18231-0
14. Flandoli, F.; Gubinelli, M.; Priola, E. Well-posedness of the transport equation by stochastic perturbation. Invent. Math. 180 (2010), no. 1, 1-53. doi: 10.1007/s00222-009-0224-4
15. Flandoli, F.; Gubinelli, M.; Priola, E. Full well-posedness of point vortex dynamics corresponding to stochastic 2D Euler equations. Stochastic Process. Appl. 121 (2011), no. 7, 1445-1463. doi: 10.1016/j.spa.2011.03.004
16. Gradinaru, M.; Herrmann, S.; Roynette, B. A singular large deviations phenomenon. Ann. Inst. H. Poincaré Probab. Statist. 37 (2001), no. 5, 555-580. doi: 10.1016/S0246-0203(01)01075-5
17. Hakulinen, V. Passive advection and the degenerate elliptic operators Mn. Comm. Math. Phys. 235 (2003), no. 1, 1-45. doi: 10.1007/s00220-002-0778-0
18. Hauray, M.; Jabin, P.-E. N-particles approximation of the Vlasov equations with singular potential. Arch. Ration. Mech. Anal. 183 (2007), no. 3, 489-524. doi: 10.1007/s00205-006-0021-9
19. Kotelenez, P. Stochastic ordinary and stochastic partial differential equations: transition from microscopic to macroscopic equations. Stochastic modelling and applied probability, 58. Springer, New York, 2008.
20. Krylov, N. V.; Röckner, M. Strong solutions of stochastic equations with singular time dependent drift. Probab. Theory Related Fields 131 (2005), no. 2, 154-196. doi: 10.1007/s00440-004-0361-z
21. Kunita, H. Stochastic differential equations and stochastic flows of diffeomorphisms. École d'été de probabilités de Saint-Flour, XII-1982, 143-303. Lecture Notes in Mathematics, 1097. Springer, Berlin, 1984. doi: 10.1007/BFb0099433
22. Le Jan, Y. On isotropic Brownian motions. Z. Wahrsch. Verw. Gebiete 70 (1985), no. 4, 609-620. doi: 10.1007/BF00531870
23. Le Jan, Y.; Raimond, O. Integration of Brownian vector fields. Ann. Probab. 30 (2002), no. 2, 826-873. doi: 10.1214/aop/1023481009
24. Lions, P.-L.; Perthame, B. Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system. Invent. Math. 105 (1991), no. 2, 415-430. doi: 10.1007/BF01232273
25. Lototskii, S. V.; Rozovskii, B. L. The passive scalar equation in a turbulent incompressible Gaussian velocity field. Uspekhi Mat. Nauk 59 (2004), no. 2(356), 105-120; translation in Russian Math. Surveys 59 (2004), no. 2, 297-312. doi: 10.1070/RM2004v059n02ABEH000719
26. Majda, A. J.; Bertozzi, A. L. Vorticity and incompressible flow. Cambridge Texts in Applied Mathematics, 27. Cambridge University Press, Cambridge, 2002.
27. Majda, A. J.; Majda, G.; Zheng, Y. X. Concentrations in the one-dimensional Vlasov-Poisson equations. I. Temporal development and non-unique weak solutions in the single component case. Phys. D 74 (1994), no. 3-4, 268-300. doi: 10.1016/0167-2789(94)90198-8
28. Marchioro, C.; Pulvirenti, M. Mathematical theory of incompressible nonviscous fluids. Applied Mathematical Sciences, 96. Springer, New York, 1994. doi: 10.1007/978-1-4612-4284-0
29. Monin, A. S.; Yaglom, A. M. Statistical fluid mechanics. Vol. II. MIT Press, Cambridge, Mass., 1975.
30. Nguyen, M. D.; Nualart, D.; Sanz, M. Application of Malliavin calculus to a class of stochastic differential equations. Probab. Theory Related Fields 84 (1990), no. 4, 549-571. doi: 10.1007/BF01198319
31. Nualart, D. The Malliavin calculus and related topics. Probability and Its Applications (New York). Springer, New York, 1995.
32. Poupaud, F. Diagonal defect measures, adhesion dynamics and Euler equation. Methods Appl. Anal. 9 (2002), no. 4, 533-561.
33. Souganidis, P. E.; Yip, N. K. Uniqueness of motion by mean curvature perturbed by stochastic noise. Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004), no. 1, 1-23. doi: 10.1016/S0294-1449(03)00029-5
34. Yaglom, A. M. Certain types of random fields in n-dimensional space similar to stationary stochastic processes. Teor. Veroyatnost. i Primenen 2 (1957), 292-338.
35. Zheng, Y. X.; Majda, A. J. Existence of global weak solutions to one-component Vlasov-Poisson and Fokker-Planck-Poisson systems in one space dimension with measures as initial data. Comm. Pure Appl. Math. 47 (1994), no. 10, 1365-1401. doi: 10.1002/cpa.3160471004