Fokker–Planck equation for dissipative 2D Euler equations with cylindrical noise
Authors:
Franco Flandoli, Francesco Grotto and Dejun Luo
Journal:
Theor. Probability and Math. Statist. 102 (2020), 117-143
MSC (2020):
Primary 60H15; Secondary 35Q31, 35Q84
DOI:
https://doi.org/10.1090/tpms/1130
Published electronically:
March 29, 2021
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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.
References
- S. Albeverio, M. Ribeiro de Faria, and R. Høegh-Krohn, Stationary measures for the periodic Euler flow in two dimensions, J. Statist. Phys. 20 (1979), no. 6, 585–595. MR 537263, DOI https://doi.org/10.1007/BF01009512
- Sergio Albeverio and Ana Bela Cruzeiro, Global flows with invariant (Gibbs) measures for Euler and Navier-Stokes two-dimensional fluids, Comm. Math. Phys. 129 (1990), no. 3, 431–444. MR 1051499
- Sergio Albeverio, Franco Flandoli, and Yakov G. Sinai, SPDE in hydrodynamic: recent progress and prospects, Lecture Notes in Mathematics, vol. 1942, Springer-Verlag, Berlin; Fondazione C.I.M.E., Florence, 2008. 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 Röckner. MR 2459087
- D. Barbato, F. Flandoli, and F. Morandin, Uniqueness for a stochastic inviscid dyadic model, Proc. Amer. Math. Soc. 138 (2010), no. 7, 2607–2617. MR 2607891, DOI https://doi.org/10.1090/S0002-9939-10-10318-9
- David Barbato, Hakima Bessaih, and Benedetta Ferrario, On a stochastic Leray-$\alpha $ model of Euler equations, Stochastic Process. Appl. 124 (2014), no. 1, 199–219. MR 3131291, DOI https://doi.org/10.1016/j.spa.2013.07.002
- David Barbato, Luigi Amedeo Bianchi, Franco Flandoli, and Francesco Morandin, A dyadic model on a tree, J. Math. Phys. 54 (2013), no. 2, 021507, 20. MR 3076367, DOI https://doi.org/10.1063/1.4792488
- Lisa Beck, Franco Flandoli, Massimiliano Gubinelli, and Mario Maurelli, Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness, Electron. J. Probab. 24 (2019), Paper No. 136, 72. MR 4040996, DOI https://doi.org/10.1214/19-ejp379
- Hakima Bessaih and Benedetta Ferrario, Invariant measures of Gaussian type for 2D turbulence, J. Stat. Phys. 149 (2012), no. 2, 259–283. MR 2988959, DOI https://doi.org/10.1007/s10955-012-0601-z
- Hakima Bessaih and Benedetta Ferrario, Inviscid limit of stochastic damped 2D Navier-Stokes equations, Nonlinearity 27 (2014), no. 1, 1–15. MR 3151089, DOI https://doi.org/10.1088/0951-7715/27/1/1
- Hakima Bessaih and Benedetta Ferrario, Statistical properties of stochastic 2D Navier-Stokes equations from linear models, Discrete Contin. Dyn. Syst. Ser. B 21 (2016), no. 9, 2927–2947. MR 3567794, DOI https://doi.org/10.3934/dcdsb.2016080
- Hakima Bessaih and Franco Flandoli, $2$-D Euler equation perturbed by noise, NoDEA Nonlinear Differential Equations Appl. 6 (1999), no. 1, 35–54. MR 1674779, DOI https://doi.org/10.1007/s000300050063
- Hakima Bessaih and Annie 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. MR 2982734, DOI https://doi.org/10.1137/110827235
- Luigi Amedeo Bianchi, Uniqueness for an inviscid stochastic dyadic model on a tree, Electron. Commun. Probab. 18 (2013), no. 8, 12. MR 3019671, DOI https://doi.org/10.1214/ECP.v18-2382
- Luigi Amedeo Bianchi and Francesco Morandin, Structure function and fractal dissipation for an intermittent inviscid dyadic model, Comm. Math. Phys. 356 (2017), no. 1, 231–260. MR 3694027, DOI https://doi.org/10.1007/s00220-017-2974-y
- Patrick Billingsley, Convergence of probability measures, 2nd ed., Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1999. A Wiley-Interscience Publication. MR 1700749
- Guido Boffetta and Robert E. Ecke, Two-dimensional turbulence, Annual review of fluid mechanics. Volume 44, 2012, Annu. Rev. Fluid Mech., vol. 44, Annual Reviews, Palo Alto, CA, 2012, pp. 427–451. MR 2882604, DOI https://doi.org/10.1146/annurev-fluid-120710-101240
- Vladimir Bogachev, Giuseppe Da Prato, and Michael Röckner, Existence and uniqueness of solutions for Fokker-Planck equations on Hilbert spaces, J. Evol. Equ. 10 (2010), no. 3, 487–509. MR 2674056, DOI https://doi.org/10.1007/s00028-010-0058-y
- Vladimir Bogachev, Giuseppe Da Prato, and Michael Röckner, Uniqueness for solutions of Fokker-Planck equations on infinite dimensional spaces, Comm. Partial Differential Equations 36 (2011), no. 6, 925–939. MR 2765423, DOI https://doi.org/10.1080/03605302.2010.539657
- Vladimir I. Bogachev, Giuseppe Da Prato, Michael Röckner, and Stanislav 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. MR 3445207
- Vladimir I. Bogachev, Nicolai V. Krylov, Michael Röckner, and Stanislav V. Shaposhnikov, Fokker-Planck-Kolmogorov equations, Mathematical Surveys and Monographs, vol. 207, American Mathematical Society, Providence, RI, 2015. MR 3443169
- Dominic Breit, Eduard Feireisl, and Martina Hofmanová, Stochastically forced compressible fluid flows, De Gruyter Series in Applied and Numerical Mathematics, vol. 3, De Gruyter, Berlin, 2018. MR 3791804
- Zdzisław Brzeźniak, Franco Flandoli, and Mario Maurelli, Existence and uniqueness for stochastic 2D Euler flows with bounded vorticity, Arch. Ration. Mech. Anal. 221 (2016), no. 1, 107–142. MR 3483892, DOI https://doi.org/10.1007/s00205-015-0957-8
- Zdzisław Brzeźniak and Szymon Peszat, Stochastic two dimensional Euler equations, Ann. Probab. 29 (2001), no. 4, 1796–1832. MR 1880243, DOI https://doi.org/10.1214/aop/1015345773
- Fernanda Cipriano, The two-dimensional Euler equation: a statistical study, Comm. Math. Phys. 201 (1999), no. 1, 139–154. MR 1669421, DOI https://doi.org/10.1007/s002200050552
- Peter Constantin, Nathan Glatt-Holtz, and Vlad Vicol, Unique ergodicity for fractionally dissipated, stochastically forced 2D Euler equations, Comm. Math. Phys. 330 (2014), no. 2, 819–857. MR 3223489, DOI https://doi.org/10.1007/s00220-014-2003-3
- 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. MR 2610386, DOI https://doi.org/10.1016/j.jfa.2009.11.019
- G. Da Prato, F. Flandoli, E. Priola, and M. Röckner, Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift, Ann. Probab. 41 (2013), no. 5, 3306–3344. MR 3127884, DOI https://doi.org/10.1214/12-AOP763
- G. Da Prato, F. Flandoli, and M. Röckner, Fokker-Planck equations for SPDE with non-trace-class noise, Commun. Math. Stat. 1 (2013), no. 3, 281–304. MR 3197468, DOI https://doi.org/10.1007/s40304-013-0015-5
- G. Da Prato, F. Flandoli, M. Röckner, and A. Yu. Veretennikov, Strong uniqueness for SDEs in Hilbert spaces with nonregular drift, Ann. Probab. 44 (2016), no. 3, 1985–2023. MR 3502599, DOI https://doi.org/10.1214/15-AOP1016
- Giuseppe Da Prato and Arnaud Debussche, Ergodicity for the 3D stochastic Navier-Stokes equations, J. Math. Pures Appl. (9) 82 (2003), no. 8, 877–947 (English, with English and French summaries). MR 2005200, DOI https://doi.org/10.1016/S0021-7824%2803%2900025-4
- G. Da Prato, F. Flandoli, and M. Röckner, Continuity equation in LlogL for the 2D Euler equations under the enstrophy measure, arXiv e-prints arXiv:1711.07759, 2017.
- Giuseppe Da Prato and Jerzy Zabczyk, Stochastic equations in infinite dimensions, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 152, Cambridge University Press, Cambridge, 2014. MR 3236753
- Camillo De Lellis and László Székelyhidi Jr., The Euler equations as a differential inclusion, Ann. of Math. (2) 170 (2009), no. 3, 1417–1436. MR 2600877, DOI https://doi.org/10.4007/annals.2009.170.1417
- Charles L. Fefferman, Existence and smoothness of the Navier-Stokes equation, The millennium prize problems, Clay Math. Inst., Cambridge, MA, 2006, pp. 57–67. MR 2238274
- F. Flandoli, M. Gubinelli, and E. Priola, Well-posedness of the transport equation by stochastic perturbation, Invent. Math. 180 (2010), no. 1, 1–53. MR 2593276, DOI https://doi.org/10.1007/s00222-009-0224-4
- 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. MR 2802460, DOI https://doi.org/10.1016/j.spa.2011.03.004
- Franco Flandoli, Random perturbation of PDEs and fluid dynamic models, Lecture Notes in Mathematics, vol. 2015, Springer, Heidelberg, 2011. Lectures from the 40th Probability Summer School held in Saint-Flour, 2010; École d’Été de Probabilités de Saint-Flour. [Saint-Flour Probability Summer School]. MR 2796837
- Franco Flandoli, Weak vorticity formulation of 2D Euler equations with white noise initial condition, Comm. Partial Differential Equations 43 (2018), no. 7, 1102–1149. MR 3910197, DOI https://doi.org/10.1080/03605302.2018.1467448
- Franco Flandoli and Dejun Luo, $\rho $-white noise solution to 2D stochastic Euler equations, Probab. Theory Related Fields 175 (2019), no. 3-4, 783–832. MR 4026605, DOI https://doi.org/10.1007/s00440-019-00902-8
- Franco Flandoli and Dejun Luo, Convergence of transport noise to Ornstein-Uhlenbeck for 2D Euler equations under the enstrophy measure, Ann. Probab. 48 (2020), no. 1, 264–295. MR 4079436, DOI https://doi.org/10.1214/19-AOP1360
- Franco Flandoli and Dejun Luo, Kolmogorov equations associated to the stochastic two dimensional Euler equations, SIAM J. Math. Anal. 51 (2019), no. 3, 1761–1791. MR 3947288, DOI https://doi.org/10.1137/18M1175756
- Franco Flandoli and Dejun Luo, Energy conditional measures and 2D turbulence, J. Math. Phys. 61 (2020), no. 1, 013101, 22. MR 4047934, DOI https://doi.org/10.1063/1.5099359
- Franco Flandoli and Dejun Luo, Point vortex approximation for 2D Navier-Stokes equations driven by space-time white noise, J. Math. Anal. Appl. 493 (2021), no. 2, 124560, 21. MR 4144298, DOI https://doi.org/10.1016/j.jmaa.2020.124560
- Franco Flandoli, Mario Maurelli, and Mikhail 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. MR 3267550, DOI https://doi.org/10.1007/s00021-014-0187-0
- Franco Flandoli and Marco Romito, Markov selections for the 3D stochastic Navier-Stokes equations, Probab. Theory Related Fields 140 (2008), no. 3-4, 407–458. MR 2365480, DOI https://doi.org/10.1007/s00440-007-0069-y
- Paul Gassiat and Benjamin Gess, Regularization by noise for stochastic Hamilton-Jacobi equations, Probab. Theory Related Fields 173 (2019), no. 3-4, 1063–1098. MR 3936151, DOI https://doi.org/10.1007/s00440-018-0848-7
- Benjamin Gess and Mario Maurelli, Well-posedness by noise for scalar conservation laws, Comm. Partial Differential Equations 43 (2018), no. 12, 1702–1736. MR 3927370, DOI https://doi.org/10.1080/03605302.2018.1535604
- Nathan Glatt-Holtz, Vladimír Šverák, and Vlad Vicol, On inviscid limits for the stochastic Navier-Stokes equations and related models, Arch. Ration. Mech. Anal. 217 (2015), no. 2, 619–649. MR 3355006, DOI https://doi.org/10.1007/s00205-015-0841-6
- Nathan E. Glatt-Holtz and Vlad 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. MR 3161482, DOI https://doi.org/10.1214/12-AOP773
- Francesco Grotto, Stationary solutions of damped stochastic 2-dimensional Euler’s equation, Electron. J. Probab. 25 (2020), Paper No. 69, 24. MR 4119115, DOI https://doi.org/10.1214/20-ejp474
- Francesco Grotto and Marco Romito, A central limit theorem for Gibbsian invariant measures of 2D Euler equations, Comm. Math. Phys. 376 (2020), no. 3, 2197–2228. MR 4104546, DOI https://doi.org/10.1007/s00220-020-03724-1
- Svante Janson, Gaussian Hilbert spaces, Cambridge Tracts in Mathematics, vol. 129, Cambridge University Press, Cambridge, 1997. MR 1474726
- V. I. Judovič, Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat i Mat. Fiz. 3 (1963), 1032–1066 (Russian). MR 158189
- Robert H. Kraichnan, Remarks on turbulence theory, Advances in Math. 16 (1975), 305–331. MR 371254, DOI https://doi.org/10.1016/0001-8708%2875%2990116-4
- N. V. Krylov and M. Röckner, Strong solutions of stochastic equations with singular time dependent drift, Probab. Theory Related Fields 131 (2005), no. 2, 154–196. MR 2117951, DOI https://doi.org/10.1007/s00440-004-0361-z
- Sergei Kuksin and Armen Shirikyan, Mathematics of two-dimensional turbulence, Cambridge Tracts in Mathematics, vol. 194, Cambridge University Press, Cambridge, 2012. MR 3443633
- Sergei B. Kuksin, The Eulerian limit for 2D statistical hydrodynamics, J. Statist. Phys. 115 (2004), no. 1-2, 469–492. MR 2070104, DOI https://doi.org/10.1023/B%3AJOSS.0000019830.64243.a2
- Sergei B. Kuksin, Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2006. MR 2225710
- S. Kullback, A lower bound for discrimination in terms of variation, IEEE Trans. Inform. Theory 16 (1967), 126–127.
- Pierre-Louis Lions, Mathematical topics in fluid mechanics. Vol. 2, Oxford Lecture Series in Mathematics and its Applications, vol. 10, The Clarendon Press, Oxford University Press, New York, 1998. Compressible models; Oxford Science Publications. MR 1637634
- Andrew J. Majda and Andrea L. Bertozzi, Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics, vol. 27, Cambridge University Press, Cambridge, 2002. MR 1867882
- Mario Maurelli, Wiener chaos and uniqueness for stochastic transport equation, C. R. Math. Acad. Sci. Paris 349 (2011), no. 11-12, 669–672 (English, with English and French summaries). MR 2817388, DOI https://doi.org/10.1016/j.crma.2011.05.006
- Steven 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. MR 1326916, DOI https://doi.org/10.1080/03605309508821124
- Jacques Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4) 146 (1987), 65–96. MR 916688, DOI https://doi.org/10.1007/BF01762360
- A. Ju. Veretennikov, Strong solutions and explicit formulas for solutions of stochastic integral equations, Mat. Sb. (N.S.) 111(153) (1980), no. 3, 434–452, 480 (Russian). MR 568986
References
- S. Albeverio, M. Ribeiro de Faria, and R. Höegh-Krohn, Stationary measures for the periodic Euler flow in two dimensions, J. Statist. Phys. 20 (1979), no. 6, 585–595. MR 537263
- S. Albeverio and A. B. Cruzeiro, Global flows with invariant (Gibbs) measures for Euler and Navier-Stokes two-dimensional fluids, Comm. Math. Phys. 129 (1990), no. 3, 431–444. MR 1051499
- 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 Röckner, Springer-Verlag, Berlin; Fondazione C.I.M.E., Florence, 2008. MR 2459087
- D. Barbato, F. Flandoli, and F. Morandin, Uniqueness for a stochastic inviscid dyadic model, Proc. Amer. Math. Soc. 138 (2010), no. 7, 2607–2617. MR 2607891
- 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. MR 3131291
- 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. MR 3076367
- 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. MR 4040996
- H. Bessaih and B. Ferrario, Invariant measures of Gaussian type for 2D turbulence, J. Stat. Phys. 149 (2012), no. 2, 259–283. MR 2988959
- H. Bessaih and B. Ferrario, Inviscid limit of stochastic damped 2D Navier-Stokes equations, Nonlinearity 27 (2014), no. 1, 1–15. MR 3151089
- 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. MR 3567794
- H. Bessaih and F. Flandoli, $2$-D Euler equation perturbed by noise, NoDEA Nonlinear Differential Equations Appl. 6 (1999), no. 1, 35–54. MR 1674779
- 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. MR 2982734
- L. A. Bianchi, Uniqueness for an inviscid stochastic dyadic model on a tree, Electron. Commun. Probab. 18 (2013), paper no. 8, 12. MR 3019671
- 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. MR 3694027
- 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. MR 1700749
- G. Boffetta and R. E. Ecke, Two-dimensional turbulence, Annu. Rev. Fluid Mech. vol. 44, Annual Reviews, Palo Alto, CA, 2012, pp. 427–451. MR 2882604
- V. I. Bogachev, G. Da Prato, M. Röckner, Existence and uniqueness of solutions for Fokker–Planck equations on Hilbert spaces, J. Evol. Equ. 10 (2010), no. 3, 487–509. MR 2674056
- V. I. Bogachev, G. Da Prato, M. Röckner, Uniqueness for solutions of Fokker–Planck equations on infinite dimensional spaces, Comm. Partial Differential Equations 36 (2011), no. 6, 925–939. MR 2765423
- V. I. Bogachev, G. Da Prato, M. Röckner, 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. MR 3445207
- V. I. Bogachev, N. V. Krylov, M. Röckner, S. V. Shaposhnikov, Fokker–Planck-Kolmogorov equations, Mathematical Surveys and Monographs vol. 207, Amer. Math. Soc., Providence, RI, 2015. MR 3443169
- D. Breit, E. Feireisl, and M. Hofmanová, Stochastically forced compressible fluid flows, De Gruyter Series in Applied and Numerical Mathematics vol. 3, De Gruyter, Berlin, 2018. MR 3791804
- Z. Brzeźniak, F. Flandoli, and M. Maurelli, Existence and uniqueness for stochastic 2D Euler flows with bounded vorticity, Arch. Ration. Mech. Anal. 221 (2016), no. 1, 107–142. MR 3483892
- Z. Brzeźniak and S. Peszat, Stochastic two dimensional Euler equations, Ann. Probab. 29 (2001), no. 4, 1796–1832. MR 1880243
- F. Cipriano, The two-dimensional Euler equation: a statistical study, Comm. Math. Phys. 201 (1999), no. 1, 139–154. MR 1669421
- 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. MR 3223489
- 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. MR 2610386
- G. Da Prato, F. Flandoli, E. Priola, and M. Röckner, Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift, Ann. Probab. 41 (2013), no. 5, 3306–3344. MR 3127884
- G. Da Prato, F. Flandoli, M. Röckner, Fokker–Planck equations for SPDE with non-trace-class noise, Commun. Math. Stat. 1 (2013), no. 3, 281–304. MR 3197468
- G. Da Prato, F. Flandoli, M. Röckner, and A. Yu. Veretennikov, Strong uniqueness for SDEs in Hilbert spaces with nonregular drift, Ann. Probab. 44 (2016), no. 3, 1985–2023. MR 3502599
- 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. MR 2005200
- G. Da Prato, F. Flandoli, and M. Röckner, Continuity equation in LlogL for the 2D Euler equations under the enstrophy measure, arXiv e-prints arXiv:1711.07759, 2017.
- 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. MR 3236753
- C. De Lellis and L. Székelyhidi Jr., The Euler equations as a differential inclusion, Ann. of Math. (2) 170 (2009), no. 3, 1417–1436. MR 2600877
- C. L. Fefferman, Existence and smoothness of the Navier-Stokes equation, The millennium prize problems, Clay Math. Inst., Cambridge, MA, 2006, pp. 57–67. MR 2238274
- F. Flandoli, M. Gubinelli, and E. Priola, Well-posedness of the transport equation by stochastic perturbation, Invent. Math. 180 (2010), no. 1, 1–53. MR 2593276
- 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. MR 2802460
- F. Flandoli, Random perturbation of PDEs and fluid dynamic models, Lecture Notes in Mathematics vol. 2015, Lectures from the 40th Probability Summer School held in Saint-Flour, 2010, École d’Été de Probabilités de Saint-Flour, Saint-Flour Probability Summer School. Springer, Heidelberg, 2011. MR 2796837
- F. Flandoli, Weak vorticity formulation of 2D Euler equations with white noise initial condition, Comm. Partial Differential Equations 43 (2018), no. 7, 1102–1149. MR 3910197
- 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. MR 4026605
- F. Flandoli and D. Luo, Convergence of transport noise to Ornstein-Uhlenbeck for 2D Euler equations under the enstrophy measure, Ann. Probab. 48 (2020), no. 1, 264–295. MR 4079436
- 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. MR 3947288
- F. Flandoli and D. Luo, Energy conditional measures and 2D turbulence, J. Math. Phys. 61 (2020), no. 1, 013101, 22. MR 4047934
- F. Flandoli and D. Luo, Point vortex approximation for 2D Navier–Stokes equations driven by space-time white noise, J. Math. Anal. Appl. 493 (2021), no. 2, 124560, 21 pp. MR 4144298
- 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. MR 3267550
- 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. MR 2365480
- P. Gassiat and B. Gess, Regularization by noise for stochastic Hamilton-Jacobi equations, Probab. Theory Related Fields 173 (2019), no. 3-4, 1063–1098. MR 3936151
- B. Gess and M. Maurelli, Well-posedness by noise for scalar conservation laws, Comm. Partial Differential Equations 43 (2018), no. 12, 1702–1736. MR 3927370
- N. Glatt-Holtz, V. Šverák, 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. MR 3355006
- 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. MR 3161482
- F. Grotto, Stationary Solutions of Damped Stochastic 2-dimensional Euler’s Equation, Electron. J. Probab. 25 (2020), No. 69, 1–24. MR 4119115
- F. Grotto and M. Romito, A Central Limit Theorem for Gibbsian Invariant Measures of 2D Euler Equation, Commun. Math. Phys. 376 (2020), n. 3, 2197–2228. MR 4104546
- S. Janson, Gaussian Hilbert spaces, volume 129 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1997. MR 1474726
- V. I. Judovič, Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat. i Mat. Fiz. 3 (1963), 1032–1066. MR 158189
- R. H. Kraichnan, Remarks on turbulence theory, Advances in Math. 16 (1975), 305–331. MR 371254
- N. V. Krylov and M. Röckner, Strong solutions of stochastic equations with singular time dependent drift, Probab. Theory Related Fields 131 (2005), no. 2, 154–196. MR 2117951
- S. Kuksin and A. Shirikyan, Mathematics of two-dimensional turbulence, Cambridge Tracts in Mathematics vol. 194, Cambridge University Press, Cambridge, 2012. MR 3443633
- S. B. Kuksin, The Eulerian limit for 2D statistical hydrodynamics, J. Statist. Phys. 115 (2004), no. 1-2, 469–492. MR 2070104
- S. B. Kuksin, Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2006. MR 2225710
- S. Kullback, A lower bound for discrimination in terms of variation, IEEE Trans. Inform. Theory 16 (1967), 126–127.
- P.-L. Lions, Mathematical topics in fluid 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. MR 1637634
- A. J. Majda and A. L. Bertozzi, Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics vol. 27, Cambridge University Press, Cambridge, 2002. MR 1867882
- M. Maurelli, Wiener chaos and uniqueness for stochastic transport equation, C. R. Math. Acad. Sci. Paris 349 (2011), no. 11-12, 669–672. MR 2817388
- 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. MR 1326916
- J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4) 146 (1987), 65–96. MR 916688
- A. Ju. Veretennikov, Strong solutions and explicit formulas for solutions of stochastic integral equations, Mat. Sb. (N.S.) 111(153) (1980), no. 3, 434–452, 480. MR 568986
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Additional Information
Franco Flandoli
Affiliation:
Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italia
Email:
franco.flandoli@sns.it
Francesco Grotto
Affiliation:
Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italia
Email:
francesco.grotto@sns.it
Dejun Luo
Affiliation:
Key Laboratory of RCSDS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China; and School of Mathematical Sciences, University of the Chinese Academy of Sciences, Beijing 100049, People’s Repbulic of China
MR Author ID:
822886
Email:
luodj@amss.ac.cn
Keywords:
2D Euler equations,
Fokker–Planck equation,
cylindrical noise,
vorticity,
energy-enstrophy measure.
Received by editor(s):
July 3, 2019
Published electronically:
March 29, 2021
Additional Notes:
The third author is grateful to the financial supports of the grant “Stochastic models with spatial structure” at the Scuola Normale Superiore di Pisa, the National Natural Science Foundation of China (Nos. 11571347, 11688101), and the Youth Innovation Promotion Association, CAS (2017003).
Article copyright:
© Copyright 2020
Taras Shevchenko National University of Kyiv
