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Hydrodynamic Models

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Quantum and Stochastic Mathematical Physics

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 377))

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Abstract

Sergio Albeverio’s research in hydrodynamics is revised, focusing on statistical analysis of the bidimensional deterministic Euler equations and the bidimensional Navier–Stokes equations with space-time white noise. Both the subjects had influence on the activity of many researchers including the present authors. The main interactions with the recent research and some (open) future problems are described.

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Notes

  1. 1.

    This is another formulation of part (ii) in Theorem 4.

References

  1. S. Albeverio, V. Barbu, B. Ferrario, Uniqueness of the generators of the 2D Euler and Navier–Stokes flows. Stochastic Process. Appl. 118(11), 2071–2084 (2008). Erratum: 120(10), 2102 (2010)

    Google Scholar 

  2. S. Albeverio, A.B. Cruzeiro, Global flows with invariant (Gibbs) measures for Euler and Navier–Stokes two dimensional fluids. Comm. Math. Phys. 129, 431–444 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  3. S. Albeverio, M. Ribeiro De Faria, R. Høegh-Krohn, Stationary measures for the periodic Euler flow in two dimensions. J. Statist. Phys. 20, 585–595 (1979)

    Article  MathSciNet  Google Scholar 

  4. S. Albeverio, B. Ferrario, Uniqueness results for the generators of the two-dimensional Euler and Navier–Stokes flows. The case of Gaussian invariant measures. J. Funct. Anal. 193(1), 77–93 (2002)

    Google Scholar 

  5. S. Albeverio, B. Ferrario, 2D vortex motion of an incompressible ideal fluid: the Koopman-von Neumann approach. Infin. Dimens. Anal. Quant. Probab. Relat. Top. 6(2), 155–165 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. S. Albeverio, B. Ferrario, Invariant measures of Lévy–Khinchine type for 2D fluids, in Probabilistic Methods in Fluids (World Scientific Publishing, River Edge, 2003), pp. 130–143

    Google Scholar 

  7. S. Albeverio, B. Ferrario, Uniqueness of solutions of the stochastic Navier–Stokes equation with invariant measure given by the enstrophy. Ann. Probab. 23(2), 1632–1649 (2004)

    MathSciNet  MATH  Google Scholar 

  8. S. Albeverio, B. Ferrario, Some methods of infinite dimensional analysis in hydrodynamics: an introduction, in Lecture Notes in Mathematics 194 “SPDE in Hydrodynamic: Recent Progress and Prospects” ed. by G. Da Prato, M. Röckner (2008), pp. 1–50

    Google Scholar 

  9. S. Albeverio, R. Høegh-Krohn, Stochastic flows with stationary distribution for two-dimensional inviscid fluids. Stochastic Process. Appl. 31(1), 1–31 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  10. V. Barbu, G. Da Prato, A. Debussche, Essential m-dissipativity of Kolmogorov operators corresponding to periodic 2D-Navier Stokes equations. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 15(1), 29–38 (2004)

    Google Scholar 

  11. L. Bergh, J. Löfström, Interpolation Spaces: An introduction (Springer, Berlin, 1976)

    Book  MATH  Google Scholar 

  12. H. Bessaih, B. Ferrario, Invariant Gibbs measures of the energy for shell models of turbulence: the inviscid and viscous cases. Nonlinearity 25, 1075–1097 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. H. Bessaih, B. Ferrario, Invariant measures of Gaussian type for 2D turbulence. J. Statist. Phys. 149(2), 259–283 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. D. Blömker, G. Cannizzaro, M. Romito, Random initial conditions for semi-linear PDEs. Proc. Royal Soc. Edinburgh Sect. A Math. 150(3), 1533–1565 (2020)

    Google Scholar 

  15. J. Bourgain, Invariant measures for the 2d-defocusing nonlinear Schrodinger equation. Comm. Math. Phys. 176, 421–445 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  16. Z. Brzezniak, F. Flandoli, M. Maurelli, Existence and uniqueness for stochastic 2D Euler flows with bounded vorticity. Arch. Rational Mech. Anal. 221, 107–142 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. N. Burq, N. Tzvetkov, Random data Cauchy theory for supercritical wave equations I. local theory. Invent. Math. 173, 449–475 (2008)

    Google Scholar 

  18. N. Burq, N. Tzvetkov, Random data Cauchy theory for supercritical wave equations II. A global existence result. Invent. Math. 173, 477–496 (2008)

    Article  MATH  Google Scholar 

  19. J.-Y. Chemin, About Navier–Stokes system (Publication du Laboratoire d’Analyse Numérique R 96023, 1996)

    Google Scholar 

  20. F. Cipriano, The two-dimensional Euler equation: a statistical study. Comm. Math. Phys. 201(1), 139–154 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  21. G. Da Prato, A. Debussche, 2D-Navier–Stokes equations driven by a space-time white noise. J. Funct. Anal. 196(1), 180–210 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions (Encyclopedia of Mathematics and its Applications (Cambridge University Press, Cambridge, 2008)

    MATH  Google Scholar 

  23. G. Da Prato, J. Zabczyk, Ergodicity for Infinite-Dimensional Systems (Cambridge University Press, Cambridge, 1996)

    Book  MATH  Google Scholar 

  24. A. Debussche, The 2D-Navier–Stokes Equations Perturbed by a Delta Correlated Noise. Probabilistic Methods in Fluids (World Scientific Publishing, River Edge, 2003), pp. 115–129

    Google Scholar 

  25. J.-M. Delort, Existence of vortex sheets in dimension two (Existence de nappes de tourbillon en dimension deux). J. Am. Math. Soc. 4, 553–586 (1991)

    Article  MATH  Google Scholar 

  26. B. Ferrario, Well posedness of a stochastic hyperviscosity-regularized 3D Navier–Stokes equation, in Proceedings of the Conference “SPDE’s and Applications—VIII”, Levico, 6–12 Jan 2008 (Quaderni di Matematica 25—Seconda Università di Napoli, 2010), pp. 127–138

    Google Scholar 

  27. B. Ferrario, A note on a result of Liptser-Shiryaev. Stochast. Anal. Appl. 30(6), 1019–1040 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. B. Ferrario, Characterization of the law for 3D stochastic hyperviscous fluids. Electron. J. Probab. 21(26), 22 (2016)

    MathSciNet  MATH  Google Scholar 

  29. B. Ferrario, C. Olivera, 2D Navier–Stokes equation with cylindrical fractional Brownian noise. Annali Mat. Pura Appl. 198(3), 1041–1067 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  30. F. Flandoli, An introduction to 3D stochastic fluid dynamics, in SPDE in Hydrodynamic: Recent Progress and Prospects. Lecture Notes in Mathematics 1942 (Springer, Berlin, 2008), pp. 51–150

    Google Scholar 

  31. F. Flandoli, Random perturbation of PDEs and fluid dynamic models, in Lectures from the 40th Probability Summer School Held in Saint-Flour, 2010. Lecture Notes in Mathematics, 2015, École d’été de Probabilités de Saint-Flour (2011)

    Google Scholar 

  32. F. Flandoli, Weak vorticity formulation of 2D Euler equations with white noise initial condition. Comm. Partial Differ. Eq. 43, 1102–1149 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  33. F. Flandoli, F. Gozzi, Kolmogorov equation associated to a stochastic Navier–Stokes equation. J. Funct. Anal. 160(1), 312–336 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  34. F. Flandoli, D. Luo, Kolmogorov equations associated to the stochastic two dimensional Euler equations. SIAM J. Math. Anal. 51(3), 1761–1791 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  35. F. Flandoli, D. Luo, \(\rho \)-white noise solution to 2D stochastic Euler equations. Probab. Theor. Relat. Field. 175, 783–832 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  36. F. Flandoli, D. Luo, Convergence of transport noise to Ornstein-Uhlenbeck for 2D Euler equations under the enstrophy measure. Ann. Probab. 48(1), 264–295 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  37. F. Flandoli, D. Luo, Energy conditional measures and 2D turbulence. J. Math. Phys. 61(1), 013101, 22 (2020)

    Google Scholar 

  38. F. Flandoli, F. Grotto, D. Luo, Fokker-Planck Equation for Dissipative 2D Euler Equations with Cylindrical Noise. Theor. Probab. Math. Stat. 102, 117–143 (2020)

    Google Scholar 

  39. G. Gallavotti, Foundations of Fluid Dynamics. Texts and Monographs in Physics (Springer, Berlin, 2002)

    Google Scholar 

  40. M. Gubinelli, M. Jara, Regularization by noise and stochastic Burgers equations. Stoch. Partial Differ. Eqn. Anal. Comput. 1(2), 325–350 (2013)

    MathSciNet  MATH  Google Scholar 

  41. M. Gubinelli, N. Perkowski, The Infinitesimal Generator of the Stochastic Burgers Equation. Probab. Theor. Relat. Fields 178(3–4), 1067–1124 (2020)

    Google Scholar 

  42. E. Hopf, Statistical hydromechanics and functional calculus. J. Rat. Mech. Anal. 1, 87–123 (1952)

    MathSciNet  MATH  Google Scholar 

  43. S. Kuksin, A. Shirikyan, Mathematics of Two-Dimensional Turbulence. Volume 194 of Cambridge Tracts in Mathematics (Cambridge University Press, Cambridge, 2012)

    Google Scholar 

  44. C. Marchioro, M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids. Volume 96 of Applied Mathematical Sciences (Springer, New York, 1994)

    Google Scholar 

  45. T. Oh, Invariance of the white noise for KdV. Comm. Math. Phys. 292, 217–236 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  46. M. Röckner, Z. Sobol, Kolmogorov equations in infinite dimensions: well-posedness and regularity of solutions, with applications to stochastic generalized Burgers equations. Ann. Probab. 34(2), 663–727 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  47. S. Schochet, The weak vorticity formulation of the 2-D Euler equations and concentration-cancellation. Comm. Partial Diff. Eq. 20, 1077–1104 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  48. W. Stannat, \(L^1\)-uniqueness of regularized 2D-Euler and stochastic Navier–Stokes equations. J. Funct. Anal. 200(1), 101–117 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  49. W. Stannat, A new a priori estimate for the Kolmogorov operator of a 2D-stochastic Navier–Stokes equation. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10(4), 483–497 (2007)

    Google Scholar 

  50. R. Temam, Navier–Stokes equations. Theory and numerical analysis, in Studies in Mathematics and its Applications, vol. 2 (North-Holland Publishing Co., Amsterdam, 1977)

    Google Scholar 

  51. M.J. Vishik, A.V. Fursikov, Mathematical Problems of Statistical Hydromechanics. Mathematics and its Applications, vol. 9 (Kluwer Academic Publishers Group, Dordrecht, 1988)

    Google Scholar 

  52. V.I. Yudovich, Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal incompressible fluid. Math. Res. Lett. 2, 27–38 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  53. R. Zhu, X. Zhu, Three-dimensional Navier–Stokes equations driven by space-time white noise. J. Differ. Eqns. 259(9), 4443–4508 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Dipartimento di Scienze Economiche ed Aziendali, Università di Pavia, 27100, Pavia, Italy

    Benedetta Ferrario

  2. Scuola Normale Superiore, 56126, Pisa, Italy

    Franco Flandoli

Authors
  1. Benedetta Ferrario
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  2. Franco Flandoli
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Corresponding author

Correspondence to Benedetta Ferrario .

Editor information

Editors and Affiliations

  1. Faculty of Technology, Linnaeus University, Växjo, Sweden

    Astrid Hilbert

  2. Department of Economics, Insubria University, Varese, Italy

    Elisa Mastrogiacomo

  3. Department of Mathematics, University of Trento, Povo, Trento, Italy

    Sonia Mazzucchi

  4. Applied Mathematics / Stochastic, University of Wuppertal, Wuppertal, Germany

    Barbara Rüdiger

  5. Department of Mathematics, University of MIlan, Milano, Italy

    Stefania Ugolini

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Ferrario, B., Flandoli, F. (2023). Hydrodynamic Models. In: Hilbert, A., Mastrogiacomo, E., Mazzucchi, S., Rüdiger, B., Ugolini, S. (eds) Quantum and Stochastic Mathematical Physics. Springer Proceedings in Mathematics & Statistics, vol 377. Springer, Cham. https://doi.org/10.1007/978-3-031-14031-0_11

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