A Journal "Theory of Probability and Mathematical Statistics"
2024
2023
2022
2021
2020
2019
2018
2017
2016
2015
2014
2013
2012
2011
2010
2009
2008
2007
2006
2005
2004
2003
2002
2001
2000
1999
1998
1997
1996
1995
1994
1993
1992
1991
1990
1989
1988
1987
1986
1985
1984
1983
1982
1981
1980
1979
1978
1977
1976
1975
1974
1973
1972
1971
1970


Archive

About   Editorial Board   Contacts   Template   Publication Ethics   Peer Review Process   Special Issues   History  

Theory of Probability and Mathematical Statistics



Fokker-Planck equations for dissipative 2D Euler equations with cylindrical noise

F. Flandoli, F. Grotto and D. Luo

Link

Abstract: After a short review of recent progresses in 2D Euler equations with random initial conditions and noise, some of the recent results are improved by exploiting a priori estimates on the associated infinite dimensional Fokker--Planck equation. The regularity class of solutions investigated here does not allow energy- or enstrophy-type estimates, but only bounds in probability with respect to suitable distributions of the initial conditions. This is a remarkable application of Fokker--Planck equations in infinite dimensions. Among the example of random initial conditions we consider Gibbsian measures based on renormalized kinetic energy.

Keywords: 2D Euler equations, Fokker--Planck equation, cylindrical noise, vorticity, energy-enstrophy measure

Bibliography:
1. S. Albeverio, M. Ribeiro de Faria, and R. Hoegh-Krohn, Stationary measures for the periodic Euler ow in two dimensions, J. Statist. Phys. 20 (1979), no. 6, 585-595.
2. S. Albeverio and A. B. Cruzeiro, Global ows with invariant (Gibbs) measures for Euler and Navier-Stokes two-dimensional uids, Comm. Math. Phys. 129 (1990), no. 3, 431-444.
3. S. Albeverio, F. Flandoli, and Y. G. Sinai, SPDE in hydrodynamic: recent progress and prospects, Lecture Notes in Mathematics vol. 1942, Lectures given at the C.I.M.E. Summer School held in Cetraro, August 29 - September 3, 2005, Edited by Giuseppe Da Prato and Michael Rockner, Springer-Verlag, Berlin; Fondazione C.I.M.E., Florence, 2008.
4. D. Barbato, F. Flandoli, and F. Morandin, Uniqueness for a stochastic inviscid dyadic model, Proc. Amer. Math. Soc. 138 (2010), no. 7, 2607-2617.
5. D. Barbato, H. Bessaih, and B. Ferrario, On a stochastic Leray-$\alpha$ model of Euler equations, Stochastic Process. Appl. 124 (2014), no. 1, 199-219.
6. D. Barbato, L. A. Bianchi, F. Flandoli, and F. Morandin, A dyadic model on a tree, J. Math. Phys. 54 (2013), no. 2, 021507, 20.
7. L. Beck, F. Flandoli, M. Gubinelli, and M. Maurelli, Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness, Electron. J. Probab. 24 (2019), paper no. 136, 72.
8. H. Bessaih and B. Ferrario, Invariant measures of Gaussian type for 2D turbulence, J. Stat. Phys. 149 (2012), no. 2, 259-283.
9. H. Bessaih and B. Ferrario, Inviscid limit of stochastic damped 2D Navier-Stokes equations, Nonlinearity 27 (2014), no. 1, 1-15.
10. H. Bessaih and B. Ferrario, Statistical properties of stochastic 2D Navier-Stokes equations from linear models, Discrete Contin. Dyn. Syst. Ser. B 21 (2016), no. 9, 2927-2947.
11. H. Bessaih and F. Flandoli, 2-D Euler equation perturbed by noise, NoDEA Nonlinear Di erential Equations Appl. 6 (1999), no. 1, 35-54.
12. H. Bessaih and A. Millet, Large deviations and the zero viscosity limit for 2D stochastic Navier-Stokes equations with free boundary, SIAM J. Math. Anal. 44 (2012), no. 3, 1861-1893.
13. L. A. Bianchi, Uniqueness for an inviscid stochastic dyadic model on a tree, Electron. Commun. Probab. 18 (2013), paper no. 8, 12.
14. L. A. Bianchi and F. Morandin, Structure function and fractal dissipation for an intermittent inviscid dyadic model, Comm. Math. Phys. 356 (2017), no. 1, 231-260.
15. P. Billingsley, Convergence of probability measures, Wiley Series in Probability and Statistics: Probability and Statistics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, second edition, 1999.
16. G. Boffetta and R. E. Ecke, Two-dimensional turbulence, Annu. Rev. Fluid Mech. vol. 44, Annual Reviews, Palo Alto, CA, 2012, pp. 427-451.
17. V. I. Bogachev, G. Da Prato, M. Rockner, Existence and uniqueness of solutions for Fokker-Planck equations on Hilbert spaces, J. Evol. Equ. 10 (2010), no. 3, 487-509.
18. V. I. Bogachev, G. Da Prato, M. Rockner, Uniqueness for solutions of Fokker-Planck equations on infinite dimensional spaces, Comm. Partial Differential Equations 36 (2011), no. 6, 925-939.
19. V. I. Bogachev, G. Da Prato, M. Rockner, S. V. Shaposhnikov, An analytic approach to infinite-dimensional continuity and Fokker-Planck-Kolmogorov equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 14 (2015), no. 3, 983-1023.
20. V. I. Bogachev, N. V. Krylov, M. Rockner, S. V. Shaposhnikov, Fokker-Planck-Kolmogorov equations, Mathematical Surveys and Monographs vol. 207, Amer. Math. Soc., Providence, RI, 2015.
21. D. Breit, E. Feireisl, and M. Hofmanova, Stochastically forced compressible uid ows, De Gruyter Series in Applied and Numerical Mathematics vol. 3, De Gruyter, Berlin, 2018.
22. Z. Brzezniak, F. Flandoli, and M. Maurelli, Existence and uniqueness for stochastic 2D Euler ows with bounded vorticity, Arch. Ration. Mech. Anal. 221 (2016), no. 1, 107-142.
23. Z. Brzezniak and S. Peszat, Stochastic two dimensional Euler equations, Ann. Probab. 29 (2001), no. 4, 1796-1832.
24. F. Cipriano, The two-dimensional Euler equation: a statistical study, Comm. Math. Phys. 201 (1999), no. 1, 139-154.
25. P. Constantin, N. Glatt-Holtz, and V. Vicol, Unique ergodicity for fractionally dissipated, stochastically forced 2D Euler equations, Comm. Math. Phys. 330 (2014), no. 2, 819-857.
26. G. Da Prato and F. Flandoli, Pathwise uniqueness for a class of SDE in Hilbert spaces and applications, J. Funct. Anal. 259 (2010), no. 1, 243-267.
27. G. Da Prato, F. Flandoli, E. Priola, and M. Rockner, Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift, Ann. Probab. 41 (2013), no. 5, 3306-3344.
28. G. Da Prato, F. Flandoli, M. Rockner, Fokker-Planck equations for SPDE with non-trace-class noise, Commun. Math. Stat. 1 (2013), no. 3, 281-304.
29. G. Da Prato, F. Flandoli, M. Rockner, and A. Yu. Veretennikov, Strong uniqueness for SDEs in Hilbert spaces with nonregular drift, Ann. Probab. 44 (2016), no. 3, 1985-2023.
30. G. Da Prato and A. Debussche, Ergodicity for the 3D stochastic Navier-Stokes equations, J. Math. Pures Appl. (9) 82 (2003), no. 8, 877-947.
31. G. Da Prato, F. Flandoli, and M. Rockner, Continuity equation in LlogL for the 2D Euler equations under the enstrophy measure, arXiv e-prints arXiv:1711.07759, 2017.
32. G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications vol. 152, Cambridge University Press, Cambridge, second edition, 2014.
33. C. De Lellis and L. Szekelyhidi Jr., The Euler equations as a differential inclusion, Ann. of Math. (2) 170 (2009), no. 3, 1417-1436.
34. C. L. Fefferman, Existence and smoothness of the Navier-Stokes equation, The millennium prize problems, Clay Math. Inst., Cambridge, MA, 2006, pp. 57-67.
35. F. Flandoli, M. Gubinelli, and E. Priola, Well-posedness of the transport equation by stochastic perturbation, Invent. Math. 180 (2010), no. 1, 1-53.
36. F. Flandoli, M. Gubinelli, and E. Priola, Full well-posedness of point vortex dynamics corresponding to stochastic 2D Euler equations, Stochastic Process. Appl. 121 (2011), no. 7, 1445-1463.
37. F. Flandoli, Random perturbation of PDEs and uid dynamic models, Lecture Notes in Mathematics vol. 2015, Lectures from the 40th Probability Summer School held in Saint-Flour, 2010, Ecole d'Ete de Probabilites de Saint-Flour, Saint-Flour Probability Summer School. Springer, Heidelberg, 2011.
38. F. Flandoli, Weak vorticity formulation of 2D Euler equations with white noise initial condition, Comm. Partial Differential Equations 43 (2018), no. 7, 1102-1149.
39. F. Flandoli and D. Luo, $\rho$-white noise solution to 2D stochastic Euler equations, Probab. Theory Relat. Fields 175 (2019), no. 3-4, 783-832.
40. F. Flandoli and D. Luo, Convergence of transport noise to Ornstein-Uhlenbeck for 2D Euler equations under the enstrophy measure, to appear in Ann. Probab., see arXiv:1806.09332, 2018.
41. F. Flandoli and D. Luo, Kolmogorov equations associated to the stochastic two dimensional Euler equations, SIAM J. Math. Anal. 51 (2019), no. 3, 1761-1791.
42. F. Flandoli and D. Luo, Energy conditional measures and 2D turbulence, J. Math. Phys. 61 (2020), no. 1, 013101, 22.
43. F. Flandoli and D. Luo, Point vortex approximation for 2D Navier-Stokes equations driven by space-time white noise, arXiv e-prints arXiv:1902.09338, 2019.
44. F. Flandoli, M. Maurelli, and M. Neklyudov, Noise prevents infinite stretching of the passive field in a stochastic vector advection equation, J. Math. Fluid Mech. 16 (2014), no. 4, 805-822.
45. F. Flandoli and M. Romito, Markov selections for the 3D stochastic Navier-Stokes equations, Probab. Theory Related Fields 140 (2008), no. 3-4, 407-458.
46. P. Gassiat and B. Gess, Regularization by noise for stochastic Hamilton-Jacobi equations, Probab. Theory Related Fields 173 (2019), no. 3-4, 1063-1098.
47. B. Gess and M. Maurelli, Well-posedness by noise for scalar conservation laws, Comm. Partial Differential Equations 43 (2018), no. 12, 1702-1736.
48. N. Glatt-Holtz, V. Sverak, and V. Vicol, On inviscid limits for the stochastic Navier-Stokes equations and related models, Arch. Ration. Mech. Anal. 217 (2015), no. 2, 619-649.
49. N. E. Glatt-Holtz and V. C. Vicol, Local and global existence of smooth solutions for the stochastic Euler equations with multiplicative noise, Ann. Probab. 42 (2014), no. 1, 80-145.
50. F. Grotto, Stationary Solutions of Damped Stochastic 2-dimensional Euler's Equation, arXiv e-prints arXiv:1901.06744, 2019.
51. F. Grotto and M. Romito, A Central Limit Theorem for Gibbsian Invariant Measures of 2D Euler Equation, Commun. Math. Phys. (2020).
52. S. Janson, Gaussian Hilbert spaces, volume 129 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1997.
53. V. I. Judovic, Non-stationary ows of an ideal incompressible uid, Z. Vycisl. Mat. i Mat. Fiz. 3 (1963), 1032-1066.
54. R. H. Kraichnan, Remarks on turbulence theory, Advances in Math. 16 (1975), 305-331.
55. N. V. Krylov and M. Rockner, Strong solutions of stochastic equations with singular time dependent drift, Probab. Theory Related Fields 131 (2005), no. 2, 154-196.
56. S. Kuksin and A. Shirikyan, Mathematics of two-dimensional turbulence, Cambridge Tracts in Mathematics vol. 194, Cambridge University Press, Cambridge, 2012.
57. S. B. Kuksin, The Eulerian limit for 2D statistical hydrodynamics, J. Statist. Phys. 115 (2004), no. 1-2, 469-492.
58. S. B. Kuksin, Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zurich, 2006.
59. S. Kullback, A lower bound for discrimination in terms of variation, IEEE Trans. Inform. Theory 16 (1967), 126-127.
60. P.-L. Lions, Mathematical topics in uid mechanics. Vol. 2, Oxford Lecture Series in Mathematics and its Applications vol. 10, Compressible models, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1998.
61. A. J. Majda and A. L. Bertozzi, Vorticity and incompressible ow, Cambridge Texts in Applied Mathematics vol. 27, Cambridge University Press, Cambridge, 2002.
62. M. Maurelli, Wiener chaos and uniqueness for stochastic transport equation, C. R. Math. Acad. Sci. Paris 349 (2011), no. 11-12, 669-672.
63. S. Schochet, The weak vorticity formulation of the 2-D Euler equations and concentration-cancellation, Comm. Partial Differential Equations 20 (1995), no. 5-6, 1077-1104.
64. J. Simon, Compact sets in the space Lp(0; T;B), Ann. Mat. Pura Appl. (4) 146 (1987), 65-96.
65. A. Ju. Veretennikov, Strong solutions and explicit formulas for solutions of stochastic integralequations, Mat. Sb. (N.S.) 111(153) (1980), no. 3, 434-452, 480.