American Institute of Mathematical Sciences

Advanced
December  2019, 11(4): 553-560. doi: 10.3934/jgm.2019027

Navier-Stokes and stochastic Navier-Stokes equations via Lagrange multipliers

Ana Bela Cruzeiro

GFMUL and Departamento de Matemática Instituto Superior Técnico, Univ. Lisbon, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

Received  March 2018 Revised  March 2019 Published  November 2019

We show that the Navier-Stokes as well as a random perturbation of this equation can be derived from a stochastic variational principle where the pressure is introduced as a Lagrange multiplier. Moreover we describe how to obtain corresponding constants of the motion.

Keywords: Navier-Stokes equation, stochastic Navier-Stokes equation, stochastic variational principles, Lagrangian stochastic flows, Lagrange multipliers for Navier-Stokes.
Mathematics Subject Classification: Primary: 70H30; Secondary: 60H30, 76D05, 60H15.
Citation: Ana Bela Cruzeiro. Navier-Stokes and stochastic Navier-Stokes equations via Lagrange multipliers. Journal of Geometric Mechanics, 2019, 11 (4) : 553-560. doi: 10.3934/jgm.2019027
References:
[1]

M. Arnaudon, X. Chen and A. B. Cruzeiro, Stochastic Euler-Poincaré reduction, J. Math. Physics, 55 (2014), 081507, 17 pp. doi: 10.1063/1.4893357.  Google Scholar

[2]

M. Arnaudon and A. B. Cruzeiro, Lagrangian Navier-Stokes diffusions on manifolds: Variational principle and stability, Bull. Sci. Math., 136 (2012), 857-881.  doi: 10.1016/j.bulsci.2012.06.007.  Google Scholar

[3]

V. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l' hydrodynamique des fluides parfaits, Ann. Inst. Fourier, 16 (1966), 316-361.  doi: 10.5802/aif.233.  Google Scholar

[4]

X. Chen, A. B. Cruzeiro and T. Ratiu, Stochastic variational principles for dissipative equations with advected quantities, arXiv: 1506.05024. Google Scholar

[5]

F. Cipriano and A. B. Cruzeiro, Navier-Stokes equations and diffusions on the group of homeomorphisms of the torus, Comm. Math. Phys., 275 (2007), 255-269.  doi: 10.1007/s00220-007-0306-3.  Google Scholar

[6]

P. Constantin, Analysis of Hydrodynamic Models, CBMS-NSF Regional Conference Series in Applied Mathematics, 90. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2017. doi: 10.1137/1.9781611974805.ch1.  Google Scholar

[7]

A. B. Cruzeiro and R. Lassalle, Symmetries and martingales in a stochastic model for the Navier-Stokes equation, From Particle Systems to Partial Differential Equations. III, Springer Proc. Math. Stat., Springer, [Cham], 162 (2016), 185-194.  doi: 10.1007/978-3-319-32144-8_9.  Google Scholar

[8]

A. B. Cruzeiro and G. P. Liu, A stochastic variational approach to the viscous Camassa-Holm and Leray-alpha equations, Stoch. Proc. and their Applic, 127 (2017), 1-19.  doi: 10.1016/j.spa.2016.05.006.  Google Scholar

[9]

D. G. Ebin and J. E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math., 92 (1970), 102-163.  doi: 10.2307/1970699.  Google Scholar

[10]

G. L. Eyink, Stochastic least-action principle for the incompressible Navier-Stokes equation, Physica D, 239 (2010), 1236-1240.  doi: 10.1016/j.physd.2008.11.011.  Google Scholar

[11]

D. D. Holm, The Euler-Poincaré variational framework for modeling fluid dynamics, Geometric Mechanics and Symmetry, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, 306 (2005), 157-209.  doi: 10.1017/CBO9780511526367.004.  Google Scholar

[12]

D. D. Holm, Variational principles for stochastic fluid dynamics, Proc. Royal Soc. A, 471 (2015), 20140963, 10 pp. doi: 10.1098/rspa.2014.0963.  Google Scholar

[13]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam-New York, Kodansha, Ltd., Tokyo, 1981.  Google Scholar

[14]

H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge Studies in Advanced Mathematics, 24. Cambridge University Press, Cambridge, 1990.  Google Scholar

[15]

T. Nakagomi, K. Yasue and J.-C. Zambrini, Stochastic variational derivation of the Navier-Stokes equation, Lett. in Math. Phys., 5 (1981), 545-552.  doi: 10.1007/BF00408137.  Google Scholar

[16]

R. Shankar, Symmetries and conservation laws of the Euler equation in Lagrangian coordinates, J. Math. Anal. and Appl., 447 (2017), 867-881.  doi: 10.1016/j.jmaa.2016.10.057.  Google Scholar

[17]

M. Thieullen and J.-C. Zambrini, Probability and quantum symmetries Ⅰ. The theorem of Noether in Schroedinger's Euclidean quantum mechanics, Ann. Inst. Henri Poincaré, 67 (1997), 297-338.   Google Scholar

[18]

M. Thieullen and J.-C. Zambrini, Symmetries in the stochastic calculus of variations, Prob. Th. and Rel. Fields, 107 (1997), 401-427.  doi: 10.1007/s004400050091.  Google Scholar

[19]

K. Yasue, A variational principle for the Navier-Stokes equation, J. Funct. Anal., 51 (1983), 133-141.  doi: 10.1016/0022-1236(83)90021-6.  Google Scholar

show all references

References:
[1]

M. Arnaudon, X. Chen and A. B. Cruzeiro, Stochastic Euler-Poincaré reduction, J. Math. Physics, 55 (2014), 081507, 17 pp. doi: 10.1063/1.4893357.  Google Scholar

[2]

M. Arnaudon and A. B. Cruzeiro, Lagrangian Navier-Stokes diffusions on manifolds: Variational principle and stability, Bull. Sci. Math., 136 (2012), 857-881.  doi: 10.1016/j.bulsci.2012.06.007.  Google Scholar

[3]

V. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l' hydrodynamique des fluides parfaits, Ann. Inst. Fourier, 16 (1966), 316-361.  doi: 10.5802/aif.233.  Google Scholar

[4]

X. Chen, A. B. Cruzeiro and T. Ratiu, Stochastic variational principles for dissipative equations with advected quantities, arXiv: 1506.05024. Google Scholar

[5]

F. Cipriano and A. B. Cruzeiro, Navier-Stokes equations and diffusions on the group of homeomorphisms of the torus, Comm. Math. Phys., 275 (2007), 255-269.  doi: 10.1007/s00220-007-0306-3.  Google Scholar

[6]

P. Constantin, Analysis of Hydrodynamic Models, CBMS-NSF Regional Conference Series in Applied Mathematics, 90. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2017. doi: 10.1137/1.9781611974805.ch1.  Google Scholar

[7]

A. B. Cruzeiro and R. Lassalle, Symmetries and martingales in a stochastic model for the Navier-Stokes equation, From Particle Systems to Partial Differential Equations. III, Springer Proc. Math. Stat., Springer, [Cham], 162 (2016), 185-194.  doi: 10.1007/978-3-319-32144-8_9.  Google Scholar

[8]

A. B. Cruzeiro and G. P. Liu, A stochastic variational approach to the viscous Camassa-Holm and Leray-alpha equations, Stoch. Proc. and their Applic, 127 (2017), 1-19.  doi: 10.1016/j.spa.2016.05.006.  Google Scholar

[9]

D. G. Ebin and J. E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math., 92 (1970), 102-163.  doi: 10.2307/1970699.  Google Scholar

[10]

G. L. Eyink, Stochastic least-action principle for the incompressible Navier-Stokes equation, Physica D, 239 (2010), 1236-1240.  doi: 10.1016/j.physd.2008.11.011.  Google Scholar

[11]

D. D. Holm, The Euler-Poincaré variational framework for modeling fluid dynamics, Geometric Mechanics and Symmetry, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, 306 (2005), 157-209.  doi: 10.1017/CBO9780511526367.004.  Google Scholar

[12]

D. D. Holm, Variational principles for stochastic fluid dynamics, Proc. Royal Soc. A, 471 (2015), 20140963, 10 pp. doi: 10.1098/rspa.2014.0963.  Google Scholar

[13]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam-New York, Kodansha, Ltd., Tokyo, 1981.  Google Scholar

[14]

H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge Studies in Advanced Mathematics, 24. Cambridge University Press, Cambridge, 1990.  Google Scholar

[15]

T. Nakagomi, K. Yasue and J.-C. Zambrini, Stochastic variational derivation of the Navier-Stokes equation, Lett. in Math. Phys., 5 (1981), 545-552.  doi: 10.1007/BF00408137.  Google Scholar

[16]

R. Shankar, Symmetries and conservation laws of the Euler equation in Lagrangian coordinates, J. Math. Anal. and Appl., 447 (2017), 867-881.  doi: 10.1016/j.jmaa.2016.10.057.  Google Scholar

[17]

M. Thieullen and J.-C. Zambrini, Probability and quantum symmetries Ⅰ. The theorem of Noether in Schroedinger's Euclidean quantum mechanics, Ann. Inst. Henri Poincaré, 67 (1997), 297-338.   Google Scholar

[18]

M. Thieullen and J.-C. Zambrini, Symmetries in the stochastic calculus of variations, Prob. Th. and Rel. Fields, 107 (1997), 401-427.  doi: 10.1007/s004400050091.  Google Scholar

[19]

K. Yasue, A variational principle for the Navier-Stokes equation, J. Funct. Anal., 51 (1983), 133-141.  doi: 10.1016/0022-1236(83)90021-6.  Google Scholar

[1]

Anna Amirdjanova, Jie Xiong. Large deviation principle for a stochastic navier-Stokes equation in its vorticity form for a two-dimensional incompressible flow. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 651-666. doi: 10.3934/dcdsb.2006.6.651
[2]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020241
[3]

Kuijie Li, Tohru Ozawa, Baoxiang Wang. Dynamical behavior for the solutions of the Navier-Stokes equation. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1511-1560. doi: 10.3934/cpaa.2018073
[4]

C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403
[5]

Hongyong Cui, Mirelson M. Freitas, José A. Langa. Squeezing and finite dimensionality of cocycle attractors for 2D stochastic Navier-Stokes equation with non-autonomous forcing. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1297-1324. doi: 10.3934/dcdsb.2018152
[6]

Yuri Bakhtin. Lyapunov exponents for stochastic differential equations with infinite memory and application to stochastic Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 697-709. doi: 10.3934/dcdsb.2006.6.697
[7]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602
[8]

Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149
[9]

I. Moise, Roger Temam. Renormalization group method: Application to Navier-Stokes equation. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 191-210. doi: 10.3934/dcds.2000.6.191
[10]

Igor Kukavica, Mohammed Ziane. Regularity of the Navier-Stokes equation in a thin periodic domain with large data. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 67-86. doi: 10.3934/dcds.2006.16.67
[11]

Hakima Bessaih, Benedetta Ferrario. Statistical properties of stochastic 2D Navier-Stokes equations from linear models. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 2927-2947. doi: 10.3934/dcdsb.2016080
[12]

Lihuai Du, Ting Zhang. Local and global strong solution to the stochastic 3-D incompressible anisotropic Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4745-4765. doi: 10.3934/dcds.2018209
[13]

Kumarasamy Sakthivel, Sivaguru S. Sritharan. Martingale solutions for stochastic Navier-Stokes equations driven by Lévy noise. Evolution Equations & Control Theory, 2012, 1 (2) : 355-392. doi: 10.3934/eect.2012.1.355
[14]

Takeshi Taniguchi. The existence and decay estimates of the solutions to $3$D stochastic Navier-Stokes equations with additive noise in an exterior domain. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4323-4341. doi: 10.3934/dcds.2014.34.4323
[15]

G. Deugoué, T. Tachim Medjo. The Stochastic 3D globally modified Navier-Stokes equations: Existence, uniqueness and asymptotic behavior. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2593-2621. doi: 10.3934/cpaa.2018123
[16]

G. M. de Araújo, S. B. de Menezes. On a variational inequality for the Navier-Stokes operator with variable viscosity. Communications on Pure & Applied Analysis, 2006, 5 (3) : 583-596. doi: 10.3934/cpaa.2006.5.583
[17]

Eric Blayo, Antoine Rousseau. About interface conditions for coupling hydrostatic and nonhydrostatic Navier-Stokes flows. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1565-1574. doi: 10.3934/dcdss.2016063
[18]

Misha Perepelitsa. An ill-posed problem for the Navier-Stokes equations for compressible flows. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 609-623. doi: 10.3934/dcds.2010.26.609
[19]

Hermenegildo Borges de Oliveira. Anisotropically diffused and damped Navier-Stokes equations. Conference Publications, 2015, 2015 (special) : 349-358. doi: 10.3934/proc.2015.0349
[20]

Hyukjin Kwean. Kwak transformation and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 433-446. doi: 10.3934/cpaa.2004.3.433

2019 Impact Factor: 0.649

Tools

Metrics

  • PDF downloads (102)
  • HTML views (106)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences