Unique Ergodicity for Fractionally Dissipated, Stochastically Forced 2D Euler Equations

Communications in Mathematical Physics

Volumn 330, Issue 2, 2014, Pages 819-857

  Constantin, Peter ^a   Glatt Holtz, Nathan ^b   Vicol, Vlad ^a  

^a PRINCETON UNIVERSITY   (United States)

^b UNIVERSITY OF MINNESOTA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84903172604     PISSN: 00103616     EISSN: 14320916     Source Type: Journal    
DOI: 10.1007/s00220-014-2003-3     Document Type: Article

References (81)

1. Albeverio, S., Flandoli, F., Sinai, Y. G.: SPDE in Hydrodynamic: Recent Progress and Prospects, volume 1942 of Lecture Notes in Mathematics. Springer, Berlin, 2008. Lectures given at the C. I. M. E. Summer School held in Cetraro, August 29-September 3, Edited by Giuseppe Da Prato and Michael Röckner (2005).
2. Batchelor G. K.: Computation of the energy spectrum in homogeneous two-dimensional turbulence. Phys. Fluids 12(Suppl. II), 233-239 (1969).
3. Bernard D.: Influence of friction on the direct cascade of the 2d forced turbulence. EPL (Europhys. Lett.) 50(3), 333 (2000).
4. Bessaih H., Flandoli F.: Weak attractor for a dissipative Euler equation. J. Dyn. Differ. Equ. 12(4), 713-732 (2000).
5. Bessaih, H., Ferrario, B.: Inviscid limit of stochastic damped 2d Navier-Stokes equations. arXiv: 1212. 0509 (2012).
6. Bricmont, J., Kupiainen, A., Lefevere, R.: Ergodicity of the 2D Navier-Stokes equations with random forcing. Commun. Math. Phys. 224(1), 65-81 (2001) (Dedicated to Joel L. Lebowitz).
7. Baroud C. N., Plapp B. B., She Z.-S., Swinney H. L.: Anomalous self-similarity in a turbulent rapidly rotating fluid. Phys. Rev. Lett. 88(11), 114501 (2002).
8. Bouchet F., Simonnet E.: Random changes of flow topology in two-dimensional and geophysical turbulence. Phys. Rev. Lett. 102(9), 094504 (2009).
9. Bensoussan A., Temam R.: Équations aux dérivées partielles stochastiques non linéaires. I. Israel J. Math. 11, 95-129 (1972).
10. Córdoba A., Córdoba D.: A maximum principle applied to quasi-geostrophic equations. Commun. Math. Phys. 249(3), 511-528 (2004).
11. Castro A., Córdoba D., Gancedo F., Orive R.: Incompressible flow in porous media with fractional diffusion. Nonlinearity 22(8), 1791-1815 (2009).
12. Caffarelli L., Chan C. H., Vasseur A.: Regularity theory for parabolic nonlinear integral operators. J. Am. Math. Soc. 24(3), 849-869 (2011).
13. Constantin, P., Cordoba, D., Wu, J.: On the critical dissipative quasi-geostrophic equation. Indiana Univ. Math. J., 50(Special Issue), 97-107 (2001) (Dedicated to Professors Ciprian Foias and Roger Temam (Bloomington, IN, 2000)).
14. Constantin, P., Foias, C.: Navier-Stokes Equations. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1988).
15. Constantin P.: The littlewood-paley spectrum in two-dimensional turbulence. Theor. Comput. Fluid Dyn. 9(3-4), 183-189 (1997).
16. Castiglione P., Pumir A.: Evolution of triangles in a two-dimensional turbulent flow. Phys. Rev. E 64(5), 056303 (2001).
17. 2. Commun. Math. Phys. 275(2), 529-551 (2007).
18. Cruzeiro A. B.: Solutions et mesures invariantes pour des équations d'évolution stochastiques du type Navier-Stokes. Expo. Math. 7(1), 73-82 (1989).
19. Caffarelli L. A., Vasseur A.: Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann. of Math. (2) 171(3), 1903-1930 (2010).
20. Constantin P., Vicol V.: Nonlinear maximum principles for dissipative linear nonlocal operators and applications. Geom. Funct. Anal. 22(5), 1289-1321 (2012).
21. Chae D., Wu J.: The 2D Boussinesq equations with logarithmically supercritical velocities. Adv. Math. 230(4-6), 1618-1645 (2012).
22. Debussche, A.: Ergodicity Results for the Stochastic Navier-Stokes Equations: An Introduction. Topics in Mathematical Fluid Mechanics, Lect. Notes Math, pp. 23-108 (2013).
23. Da Prato G., Zabczyk J.: Stochastic Equations in Infinite Dimensions, volume 44 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1992).
24. Da Prato, G., Zabczyk, J.: Ergodicity for Infinite-Dimensional Systems, volume 229 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1996).
25. Eckmann J.-P., Hairer M.: Uniqueness of the invariant measure for a stochastic PDE driven by degenerate noise. Commun. Math. Phys. 219(3), 523-565 (2001).
26. Khanin K., Mazel A., Sinai Y.: Invariant measures for Burgers equation with stochastic forcing. Ann. of Math. (2) 151(3), 877-960 (2000).
27. Weinan, E., Mattingly, J. C., Sinai, Y.: Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation. Commun. Math. Phys. 224(1), 83-106, (2001) (Dedicated to Joel L. Lebowitz).
28. Eyink G. L.: Exact results on stationary turbulence in 2d: consequences of vorticity conservation. Phys. D Nonlinear Phenom. 91(1), 97-142 (1996).
29. Ferrario B.: Stochastic Navier-Stokes equations: analysis of the noise to have a unique invariant measure. Ann. Mat. Pura Appl. (4) 177, 331-347 (1999).
30. Foias C., Jolly M. S., Manley O. P., Rosa R.: Statistical estimates for the Navier-Stokes equations and the Kraichnan theory of 2-D fully developed turbulence. J. Stat. Phys. 108(3-4), 591-645 (2002).
31. Falkovich G., Lebedev V.: Nonlocal vorticity cascade in two dimensions. Phys. Rev. E (3) 49(3), R1800-R1803 (1994).
32. Flandoli F., Maslowski B.: Ergodicity of the 2-D Navier-Stokes equation under random perturbations. Commun. Math. Phys. 172(1), 119-141 (1995).
33. Foias C., Prodi G.: Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension 2. Rend. Sem. Mat. Univ. Padova 39, 1-34 (1967).
34. Frisch, U.: Turbulence. Cambridge University Press, Cambridge, The legacy of A. N. Kolmogorov (1995).
35. Frisch U., Sulem P.-L.: Numerical simulation of the inverse cascade in two-dimensional turbulence. Phys. Fluids 27, 1921 (1984).
36. Foias C., Temam R.: Gevrey class regularity for the solutions of the Navier-Stokes equations. J. Funct. Anal. 87(2), 359-369 (1989).
37. Glatt-Holtz, N., Kukavica, I., Vicol, V., Ziane, M.: Existence and regularity of invariant measures for the three dimensional stochastic primitive equations (2013) (submitted).
38. Glatt-Holtz, N., Sverak, V., Vicol, V.: On inviscid limits for the stochastic Navier-Stokes equations and related models. arXiv: 1302. 0542 (2013).
39. Glatt-Holtz N., Vicol V.: Local and global existence of smooth solutions for the stochastic euler equations on a bounded domain. Ann. Probab. 42(1), 80-145 (2014).
40. Hmidi T., Keraani S., Rousset F.: Global well-posedness for Euler-Boussinesq system with critical dissipation. Comm. Partial Differ. Equ. 36(3), 420-445 (2011).
41. Hairer M., Mattingly J. C.: Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing. Ann. Math. (2) 164(3), 993-1032 (2006).
42. Hairer M., Mattingly J. C.: Spectral gaps in Wasserstein distances and the 2D stochastic Navier-Stokes equations. Ann. Probab. 36(6), 2050-2091 (2008).
43. Hairer M., Mattingly J. C.: A theory of hypoellipticity and unique ergodicity for semilinear stochastic pdes. Electron. J. Probab. 16(23), 658-738 (2011).
44. Held I. M., Pierrehumbert R. T., Garner S. T., Swanson K. L.: Surface quasi-geostrophic dynamics. J. Fluid Mech. 282, 1-20 (1995).
45. Kiselev A.: Regularity and blow up for active scalars. Math. Model. Nat. Phenom. 5(4), 225-255 (2010).
46. Kiselev A., Nazarov F., Volberg A.: Global well-posedness for the critical 2D dissipative quasi-geostrophic equation. Invent. Math. 167(3), 445-453 (2007).
47. Kraichnan R. H.: Inertial ranges in two-domensional turbulence. Phys. Fluids 10, 1417-1423 (1967).
48. p-valued processes. Probab. Theory Relat. Fields 147(3-4), 583-605 (2010).
49. Kuksin S., Shirikyan A.: A coupling approach to randomly forced nonlinear PDE's. I. Commun. Math. Phys. 221(2), 351-366 (2001).
50. Kuksin S., Shirikyan A.: Coupling approach to white-forced nonlinear PDEs. J. Math. Pures Appl. (9) 81(6), 567-602 (2002).
51. Kuksin, S., Shirikyan, A.: Mathematics of Two-Dimensional Turbulence. Number 194 in Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (2012).
52. Kuksin S. B.: The Eulerian limit for 2D statistical hydrodynamics. J. Stat. Phys. 115(1-2), 469-492 (2004).
53. Kupiainen, A.: Ergodicity of two dimensional turbulence. arXiv: 1005. 0587v1 [math-ph] 05 (2010).
54. Malliavin P.: Stochastic analysis, volume 313 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1997).
55. Masmoudi N.: Remarks about the inviscid limit of the Navier-Stokes system. Commun. Math. Phys. 270(3), 777-788 (2007).
56. Mattingly J. C.: Ergodicity of 2D Navier-Stokes equations with random forcing and large viscosity. Commun. Math. Phys. 206(2), 273-288 (1999).
57. Mattingly, J. C.: The dissipative scale of the stochastics Navier-Stokes equation: regularization and analyticity. J. Stat. Phys. 108(5-6), 1157-1179 (2002) (Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays).
58. Mattingly J. C.: Exponential convergence for the stochastically forced Navier-Stokes equations and other partially dissipative dynamics. Commun. Math. Phys. 230(3), 421-462 (2002).
59. Mattingly, J. C.: On recent progress for the stochastic Navier Stokes equations. In: Journées "Équations aux Dérivées Partielles", pages Exp. No. XI, 52. Univ. Nantes, Nantes (2003).
60. Majda A. J., Bertozzi A. L.: Vorticity and Incompressible Flow, volume 27 of Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002).
61. Mattingly J. C., Pardoux É.: Malliavin calculus for the stochastic 2D Navier-Stokes equation. Commun. Pure Appl. Math. 59(12), 1742-1790 (2006).
62. Muscalu C., Schlag W.: Classical and Multilinear Harmonic Analysis, vol. 137. Cambridge University Press, Cambridge (2013).
63. Masmoudi N., Young L.-S.: Ergodic theory of infinite dimensional systems with applications to dissipative parabolic PDEs. Commun. Math. Phys. 227(3), 461-481 (2002).
64. Novikov E. A.: Functionals and the random-force method in turbulence theory. Soviet Phys. JETP 20, 1290-1294 (1965).
65. Nualart, D.: The Malliavin Calculus and Related Topics. Probability and its Applications (New York), 2nd edn. Springer, Berlin (2006).
66. Nualart, D.: Malliavin calculus and its applications, volume 110 of CBMS Regional Conference Series in Mathematics. In: Published for the Conference Board of the Mathematical Sciences, Washington, DC (2009).
67. Pauluis O., Balaji V., Held I. M.: Frictional dissipation in a precipitating atmosphere. J. Atmos. Sci. 57(7), 989-994 (2000).
68. Pauluis O., Held I. M.: Entropy budget of an atmosphere in radiative-convective equilibrium. Part I: Maximum work and frictional dissipation. J. Atmos. Sci. 59(2), 125-139 (2002).
69. Prévôt, C., Röckner, M.: A Concise Course on Stochastic Partial Differential Equations, Volume 1905 of Lecture Notes in Mathematics. Springer, Berlin (2007).
70. Pumir A., Shraiman B. I, Chertkov M.: Geometry of lagrangian dispersion in turbulence. Phys. Rev. Lett. 85(25), 5324-5327 (2000).
71. Paret J., Tabeling P.: Experimental observation of the two-dimensional inverse energy cascade. Phys. Rev. Lett. 79(21), 4162-4165 (1997).
72. Paret J., Tabeling P.: Intermittency in the two-dimensional inverse cascade of energy: Experimental observations. Phys. Fluids 10, 3126 (1998).
73. Rozovskiǐ, B. L.: Stochastic Evolution Systems, volume 35 of Mathematics and its Applications (Soviet Series). (Linear theory and applications to nonlinear filtering, Translated from the Russian by A. Yarkho), Kluwer, Dordrecht (1990).
74. Robert R., Sommeria J.: Relaxation towards a statistical equilibrium state in two-dimensional perfect fluid dynamics. Phys. Rev. Lett. 69(19), 2776-2779 (1992).
75. Roncal, L., Stinga, P. R.: Fractional laplacian on the torus. arXiv preprint arXiv: 1209. 6104 (2012).
76. Smith, K. S., Boccaletti, G., Henning, C. C., Marinov, I., Tam, C. Y., Held, I. M., Vallis, G. K.: Turbulent diffusion in the geostrophic inverse cascade. J. Fluid Mech. 469 13-48 (2002).
77. Tabeling P.: Two-dimensional turbulence: a physicist approach. Phys. Rep. 362(1), 1-62 (2002).
78. Taylor M. E.: Pseudodifferential Operators and Nonlinear PDE, volume 100 of Progress in Mathematics. Birkhäuser Boston Inc., Boston (1991).
79. Vishik, M. I., Komech, A. I., Fursikov, A. V.: Some mathematical problems of statistical hydromechanics. Uspekhi Mat. Nauk, 34(5(209)), 135-210 (1979) (256).
80. Voth, G. A., La Porta, A., Crawford, A. M., Alexander, J., Bodenschatz, E.: Measurement of particle accelerations in fully developed turbulence. J. Fluid Mech. 469(9), 121-160 (2002).
81. Wu J.: The quasi-geostrophic equation and its two regularizations. Comm. Partial Differ. Equ. 27(5-6), 1161-1181 (2002).