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May  2022, 27(5): 2683-2699. doi: 10.3934/dcdsb.2021154

Gaussian invariant measures and stationary solutions of 2D primitive equations

Francesco Grotto , and  Umberto Pappalettera

Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italia

* Corresponding author: Francesco Grotto

Received  May 2020 Revised  December 2020 Published  May 2022 Early access  June 2021

We introduce a Gaussian measure formally preserved by the 2-dimensional Primitive Equations driven by additive Gaussian noise. Under such measure the stochastic equations under consideration are singular: we propose a solution theory based on the techniques developed by Gubinelli and Jara in [15] for a hyperviscous version of the equations.

Keywords: Invariant measure, primitive equations, regularization by noise, equilibrium statistical mechanics.
Mathematics Subject Classification: Primary: 60H15, 76M35; Secondary: 35Q35, 35R60, 35Q86, 76D03, 86A05.
Citation: Francesco Grotto, Umberto Pappalettera. Gaussian invariant measures and stationary solutions of 2D primitive equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2683-2699. doi: 10.3934/dcdsb.2021154
References:
[1]

S. Albeverio and B. Ferrario, Uniqueness of solutions of the stochastic Navier-Stokes equation with invariant measure given by the enstrophy, Ann. Probab., 32 (2004), 1632-1649.  doi: 10.1214/009117904000000379.

[2]

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 pp. doi: 10.1214/19-ejp379.

[3]

D. Bresch, A. Kazhikhov and J. Lemoine, On the two-dimensional hydrostatic Navier-Stokes equations, SIAM J. Math. Anal., 36 (2004/05), 796-814.  doi: 10.1137/S0036141003422242.

[4]

A. B. Cruzeiro, Équations différentielles ordinaires: Non explosion et mesures quasi-invariantes, J. Funct. Anal., 54 (1983), 193-205.  doi: 10.1016/0022-1236(83)90054-X.

[5]

G. Da Prato and A. Debussche, Two-dimensional Navier-Stokes equations driven by a space-time white noise, J. Funct. Anal., 196 (2002), 180-210.  doi: 10.1006/jfan.2002.3919.

[6] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, volume 152 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, second edition, 2014.  doi: 10.1017/CBO9781107295513.
[7]

A. Debussche, N. Glatt-Holtz, R. Temam and M. Ziane, Global existence and regularity for the 3D stochastic primitive equations of the ocean and atmosphere with multiplicative white noise, Nonlinearity, 25 (2012), 2093-2118.  doi: 10.1088/0951-7715/25/7/2093.

[8]

F. Flandoli, M. Gubinelli and E. Priola, Well-posedness of the transport equation by stochastic perturbation, Invent. Math., 180 (2010), 1-53.  doi: 10.1007/s00222-009-0224-4.

[9]

F. Flandoli, Random Perturbation of PDEs and Fluid Dynamic Models, volume 2015 of Lecture Notes in Mathematics, 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]. doi: 10.1007/978-3-642-18231-0.

[10]

H. Gao and C. Sun, Well-posedness and large deviations for the stochastic primitive equations in two space dimensions, Commun. Math. Sci., 10 (2012), 575-593.  doi: 10.4310/CMS.2012.v10.n2.a8.

[11]

H. Gao and C. Sun, Well-posedness of stochastic primitive equations with multiplicative noise in three dimensions, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3053-3073.  doi: 10.3934/dcdsb.2016087.

[12]

N. Glatt-Holtz, I. Kukavica, V. Vicol and M. Ziane, Existence and regularity of invariant measures for the three dimensional stochastic primitive equations, J. Math. Phys., 55 (2014), 051504, 34pp. doi: 10.1063/1.4875104.

[13]

N. Glatt-Holtz and R. Temam, Pathwise solutions of the 2-D stochastic primitive equations, Appl. Math. Optim., 63 (2011), 401-433.  doi: 10.1007/s00245-010-9126-5.

[14]

N. Glatt-Holtz and M. Ziane, The stochastic primitive equations in two space dimensions with multiplicative noise, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 801-822.  doi: 10.3934/dcdsb.2008.10.801.

[15]

M. Gubinelli and M. Jara, Regularization by noise and stochastic Burgers equations, Stoch. Partial Differ. Equ. Anal. Comput., 1 (2013), 325-350.  doi: 10.1007/s40072-013-0011-5.

[16]

M. Gubinelli and M. Turra, Hyperviscous stochastic Navier-Stokes equations with white noise invariant measure, Stoch. Dyn., 20 (2020), 2040005, 39pp. doi: 10.1142/S0219493720400055.

[17]

M. Gubinelli and N. Perkowski, The infinitesimal generator of the stochastic Burgers equation, Probab. Theory Related Fields, 178 (2020), 1067-1124.  doi: 10.1007/s00440-020-00996-5.

[18]

A. Hussein, Partial and full hyper-viscosity for navier-stokes and primitive equations, Journal of Differential Equations, 269 (2020), 3003-3030.  doi: 10.1016/j.jde.2020.02.019.

[19]

O. A. Ladyženskaya, On the nonstationary navier-stokes equations, Vestnik Leningrad. Univ., 13 (1958), 9-18. 

[20]

P. H. Lauritzen, Ch. Jablonowski, M. A. Taylor and R. D. Nair (Eds.), Numerical Techniques for Global Atmospheric Models, Lecture Notes in Computational Science and Engineering. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-11640-7.

[21]

J.-L. Lions, R. Temam, and S. Wang, Models for the coupled atmosphere and ocean. (CAO Ⅰ, Ⅱ), Comput. Mech. Adv., 1 (1993), 120pp.

[22]

J.-L. Lions, Quelques résultats d'existence dans des équations aux dérivées partielles non linéaires, Bulletin de la Société Mathématique de France, 87 (1959), 245-273. 

[23]

J.-L. Lions, R. Temam and S. H. Wang, New formulations of the primitive equations of atmosphere and applications, Nonlinearity, 5 (1992), 237-288. 

[24]

J.-L. Lions, R. Temam and S. H. Wang, On the equations of the large-scale ocean, Nonlinearity, 5 (1992), 1007-1053. 

[25]

N. Masmoudi and T. K. Wong, On the $H^s$ theory of hydrostatic Euler equations, Arch. Ration. Mech. Anal., 204 (2012), 231-271.  doi: 10.1007/s00205-011-0485-0.

[26]

D. Nualart, The Malliavin calculus and related topics, Probability and its Applications (New York). Springer-Verlag, Berlin, second edition, 2006.

[27]

M. Petcu, R. Temam and D. Wirosoetisno, Existence and regularity results for the primitive equations in two space dimensions, Commun. Pure Appl. Anal., 3 (2004), 115-131.  doi: 10.3934/cpaa.2004.3.115.

[28]

C. Sun, H. Gao and M. Li, Large deviation for the stochastic 2D primitive equations with additive Lévy noise, Commun. Math. Sci., 16 (2018), 165-184.  doi: 10.4310/CMS.2018.v16.n1.a8.

[29]

T. Tachim Medjo, The exponential behavior of the stochastic primitive equations in two dimensional space with multiplicative noise, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 177-197.  doi: 10.3934/dcdsb.2010.14.177.

show all references

References:
[1]

S. Albeverio and B. Ferrario, Uniqueness of solutions of the stochastic Navier-Stokes equation with invariant measure given by the enstrophy, Ann. Probab., 32 (2004), 1632-1649.  doi: 10.1214/009117904000000379.

[2]

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 pp. doi: 10.1214/19-ejp379.

[3]

D. Bresch, A. Kazhikhov and J. Lemoine, On the two-dimensional hydrostatic Navier-Stokes equations, SIAM J. Math. Anal., 36 (2004/05), 796-814.  doi: 10.1137/S0036141003422242.

[4]

A. B. Cruzeiro, Équations différentielles ordinaires: Non explosion et mesures quasi-invariantes, J. Funct. Anal., 54 (1983), 193-205.  doi: 10.1016/0022-1236(83)90054-X.

[5]

G. Da Prato and A. Debussche, Two-dimensional Navier-Stokes equations driven by a space-time white noise, J. Funct. Anal., 196 (2002), 180-210.  doi: 10.1006/jfan.2002.3919.

[6] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, volume 152 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, second edition, 2014.  doi: 10.1017/CBO9781107295513.
[7]

A. Debussche, N. Glatt-Holtz, R. Temam and M. Ziane, Global existence and regularity for the 3D stochastic primitive equations of the ocean and atmosphere with multiplicative white noise, Nonlinearity, 25 (2012), 2093-2118.  doi: 10.1088/0951-7715/25/7/2093.

[8]

F. Flandoli, M. Gubinelli and E. Priola, Well-posedness of the transport equation by stochastic perturbation, Invent. Math., 180 (2010), 1-53.  doi: 10.1007/s00222-009-0224-4.

[9]

F. Flandoli, Random Perturbation of PDEs and Fluid Dynamic Models, volume 2015 of Lecture Notes in Mathematics, 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]. doi: 10.1007/978-3-642-18231-0.

[10]

H. Gao and C. Sun, Well-posedness and large deviations for the stochastic primitive equations in two space dimensions, Commun. Math. Sci., 10 (2012), 575-593.  doi: 10.4310/CMS.2012.v10.n2.a8.

[11]

H. Gao and C. Sun, Well-posedness of stochastic primitive equations with multiplicative noise in three dimensions, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3053-3073.  doi: 10.3934/dcdsb.2016087.

[12]

N. Glatt-Holtz, I. Kukavica, V. Vicol and M. Ziane, Existence and regularity of invariant measures for the three dimensional stochastic primitive equations, J. Math. Phys., 55 (2014), 051504, 34pp. doi: 10.1063/1.4875104.

[13]

N. Glatt-Holtz and R. Temam, Pathwise solutions of the 2-D stochastic primitive equations, Appl. Math. Optim., 63 (2011), 401-433.  doi: 10.1007/s00245-010-9126-5.

[14]

N. Glatt-Holtz and M. Ziane, The stochastic primitive equations in two space dimensions with multiplicative noise, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 801-822.  doi: 10.3934/dcdsb.2008.10.801.

[15]

M. Gubinelli and M. Jara, Regularization by noise and stochastic Burgers equations, Stoch. Partial Differ. Equ. Anal. Comput., 1 (2013), 325-350.  doi: 10.1007/s40072-013-0011-5.

[16]

M. Gubinelli and M. Turra, Hyperviscous stochastic Navier-Stokes equations with white noise invariant measure, Stoch. Dyn., 20 (2020), 2040005, 39pp. doi: 10.1142/S0219493720400055.

[17]

M. Gubinelli and N. Perkowski, The infinitesimal generator of the stochastic Burgers equation, Probab. Theory Related Fields, 178 (2020), 1067-1124.  doi: 10.1007/s00440-020-00996-5.

[18]

A. Hussein, Partial and full hyper-viscosity for navier-stokes and primitive equations, Journal of Differential Equations, 269 (2020), 3003-3030.  doi: 10.1016/j.jde.2020.02.019.

[19]

O. A. Ladyženskaya, On the nonstationary navier-stokes equations, Vestnik Leningrad. Univ., 13 (1958), 9-18. 

[20]

P. H. Lauritzen, Ch. Jablonowski, M. A. Taylor and R. D. Nair (Eds.), Numerical Techniques for Global Atmospheric Models, Lecture Notes in Computational Science and Engineering. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-11640-7.

[21]

J.-L. Lions, R. Temam, and S. Wang, Models for the coupled atmosphere and ocean. (CAO Ⅰ, Ⅱ), Comput. Mech. Adv., 1 (1993), 120pp.

[22]

J.-L. Lions, Quelques résultats d'existence dans des équations aux dérivées partielles non linéaires, Bulletin de la Société Mathématique de France, 87 (1959), 245-273. 

[23]

J.-L. Lions, R. Temam and S. H. Wang, New formulations of the primitive equations of atmosphere and applications, Nonlinearity, 5 (1992), 237-288. 

[24]

J.-L. Lions, R. Temam and S. H. Wang, On the equations of the large-scale ocean, Nonlinearity, 5 (1992), 1007-1053. 

[25]

N. Masmoudi and T. K. Wong, On the $H^s$ theory of hydrostatic Euler equations, Arch. Ration. Mech. Anal., 204 (2012), 231-271.  doi: 10.1007/s00205-011-0485-0.

[26]

D. Nualart, The Malliavin calculus and related topics, Probability and its Applications (New York). Springer-Verlag, Berlin, second edition, 2006.

[27]

M. Petcu, R. Temam and D. Wirosoetisno, Existence and regularity results for the primitive equations in two space dimensions, Commun. Pure Appl. Anal., 3 (2004), 115-131.  doi: 10.3934/cpaa.2004.3.115.

[28]

C. Sun, H. Gao and M. Li, Large deviation for the stochastic 2D primitive equations with additive Lévy noise, Commun. Math. Sci., 16 (2018), 165-184.  doi: 10.4310/CMS.2018.v16.n1.a8.

[29]

T. Tachim Medjo, The exponential behavior of the stochastic primitive equations in two dimensional space with multiplicative noise, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 177-197.  doi: 10.3934/dcdsb.2010.14.177.

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