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A Crank-Nicolson type minimization scheme for a hyperbolic free boundary problem
Gaussian invariant measures and stationary solutions of 2D primitive equations
Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italia |
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 [
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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