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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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[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.

[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.

[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. 

[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.

[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.

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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[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.

[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.

[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. 

[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.

[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.

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