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June  2019, 11(2): 153-165. doi: 10.3934/jgm.2019008

Euler-Lagrangian approach to 3D stochastic Euler equations

Franco Flandoli 1,, and  Dejun Luo 2,3,

1. 

Scuola Normale Superiore of Pisa, Piazza dei Cavalieri, 7, 56124 Pisa, Italy
2. 

Key Laboratory of Random Complex Structures and Data Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
3. 

School of Mathematical Sciences, University of the Chinese Academy of Sciences, Beijing 100049, China

* Corresponding author: Franco Flandoli

Received  March 2018 Revised  February 2019 Published  May 2019

Fund Project: The second author is supported by the National Natural Science Foundation of China (Nos. 11431014, 11571347) and the Youth Innovation Promotion Association, CAS (2017003).

3D stochastic Euler equations with a special form of multiplicative noise are considered. A Constantin-Iyer type representation in Euler-Lagrangian form is given, based on stochastic characteristics. Local existence and uniqueness of solutions in suitable Hölder spaces is proved from the Euler-Lagrangian formulation.

Keywords: Stochastic Euler equations, Euler-Lagrangian formulation, local existence, Hölder space, vorticity.
Mathematics Subject Classification: Primary: 35Q31; Secondary: 76B03.
Citation: Franco Flandoli, Dejun Luo. Euler-Lagrangian approach to 3D stochastic Euler equations. Journal of Geometric Mechanics, 2019, 11 (2) : 153-165. doi: 10.3934/jgm.2019008
References:
[1]

J. M. Bismut and D. Michel, Diffusions conditionelles. I. Hypoellipticité partielle, J. Funct. Anal., 44 (1981), 174-211.  doi: 10.1016/0022-1236(81)90010-0.  Google Scholar

[2]

Z. Brzeniak, M. Capinski and F. Flandoli, Stochastic Navier-Stokes equations with multiplicative noise, Stoch. Anal. Appl., 10 (1992), 523-532.  doi: 10.1080/07362999208809288.  Google Scholar

[3]

P. Constantin, An Euler–Lagrangian approach for incompressible fluids: local theory, J. Amer. Math. Soc., 14 (2001), 263-278.  doi: 10.1090/S0894-0347-00-00364-7.  Google Scholar

[4]

P. Constantin and G. Iyer, A stochastic Lagrangian representation of the three-dimensional incompressible Navier–Stokes equations, Comm. Pure Appl. Math., 61 (2008), 330-345.  doi: 10.1002/cpa.20192.  Google Scholar

[5]

D. Crisan, F. Flandoli and D. D. Holm, Solution properties of a 3D stochastic Euler fluid equation, J Nonlinear Sci, (2018), 1–58, https://doi.org/10.1007/s00332-018-9506-6. doi: 10.1007/s00332-018-9506-6.  Google Scholar

[6]

R. Duboscq and A. Réveillac, Stochastic regularization effects of semi-martingales on random functions, J. Math. Pures Appl. (9), 106 (2016), 1141–1173. doi: 10.1016/j.matpur.2016.04.004.  Google Scholar

[7]

G. Falkovich, K. Gawedzki and M. Vergassola, Particles and fields in fluid turbulence, Rev. Modern Phys., 73 (2001), 913-975.  doi: 10.1103/RevModPhys.73.913.  Google Scholar

[8]

S. Fang and D. Luo, Constantin and Iyer's representation formula for the Navier-Stokes equations on manifolds, Potential Anal., 48 (2018), 181-206.  doi: 10.1007/s11118-017-9631-0.  Google Scholar

[9]

E. Fedrizzi and F. Flandoli, Noise prevents singularities in linear transport equations, J. Funct. Anal., 264 (2013), 1329-1354.  doi: 10.1016/j.jfa.2013.01.003.  Google Scholar

[10]

F. Flandoli, Random Perturbation of PDEs and Fluid Dynamic Models, École d'été de Saint Flour 2010, Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-642-18231-0.  Google Scholar

[11]

F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), 367-391.  doi: 10.1007/BF01192467.  Google Scholar

[12]

F. Flandoli, M. Maurelli and M. Neklyudov, Noise prevents infinite stretching of the passive field in a stochastic vector advection equation, J. Math. Fluid Mech., 16 (2014), 805-822.  doi: 10.1007/s00021-014-0187-0.  Google Scholar

[13]

F. Flandoli and C. Olivera, Well-posedness of the vector advection equations by stochastic perturbation, J. Evol. Equ., 18 (2018), 277-301.  doi: 10.1007/s00028-017-0401-7.  Google Scholar

[14]

F. Gay-Balmaz and D. D. Holm, Stochastic geometric models with non-stationary spatial correlations in Lagrangian fluid flows, J. Nonlinear Sci., 28 (2018), 873-904.  doi: 10.1007/s00332-017-9431-0.  Google Scholar

[15]

D. D. Holm, Variational principles for stochastic fluid dynamics, Proceedings of the Royal Society A, 471 (2015), 20140963, 19pp. doi: 10.1098/rspa.2014.0963.  Google Scholar

[16]

G. Iyer, A stochastic perturbation of inviscid flows, Commun. Math. Phys., 266 (2006), 631-645.  doi: 10.1007/s00220-006-0058-5.  Google Scholar

[17]

H. Kunita, Stochastic differential equations and stochastic flows of diffeomorphisms, École d'été de Probabilités de Saint Flour, 1982, 143–303, Lecture Notes in Math., 1097, Springer, Berlin, 1984. doi: 10.1007/BFb0099433.  Google Scholar

[18]

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

[19]

R. Mikulevicius and B. L. Rozovskii, Global $L_2$-solutions of stochastic Navier-Stokes equations, Ann. Probab., 33 (2005), 137-176.  doi: 10.1214/009117904000000630.  Google Scholar

[20]

D. Ocone and E. Pardoux, A generalized Itô–Ventzell formula. Application to a class of anticipating stochastic differential equations, Ann. Inst. Henri Poincaré, 25 (1989), 39-71.   Google Scholar

[21]

X. Zhang, A stochastic representation for backward incompressible Navier–Stokes equations, Probab. Theory Related Fields, 148 (2010), 305-332.  doi: 10.1007/s00440-009-0234-6.  Google Scholar

show all references

References:
[1]

J. M. Bismut and D. Michel, Diffusions conditionelles. I. Hypoellipticité partielle, J. Funct. Anal., 44 (1981), 174-211.  doi: 10.1016/0022-1236(81)90010-0.  Google Scholar

[2]

Z. Brzeniak, M. Capinski and F. Flandoli, Stochastic Navier-Stokes equations with multiplicative noise, Stoch. Anal. Appl., 10 (1992), 523-532.  doi: 10.1080/07362999208809288.  Google Scholar

[3]

P. Constantin, An Euler–Lagrangian approach for incompressible fluids: local theory, J. Amer. Math. Soc., 14 (2001), 263-278.  doi: 10.1090/S0894-0347-00-00364-7.  Google Scholar

[4]

P. Constantin and G. Iyer, A stochastic Lagrangian representation of the three-dimensional incompressible Navier–Stokes equations, Comm. Pure Appl. Math., 61 (2008), 330-345.  doi: 10.1002/cpa.20192.  Google Scholar

[5]

D. Crisan, F. Flandoli and D. D. Holm, Solution properties of a 3D stochastic Euler fluid equation, J Nonlinear Sci, (2018), 1–58, https://doi.org/10.1007/s00332-018-9506-6. doi: 10.1007/s00332-018-9506-6.  Google Scholar

[6]

R. Duboscq and A. Réveillac, Stochastic regularization effects of semi-martingales on random functions, J. Math. Pures Appl. (9), 106 (2016), 1141–1173. doi: 10.1016/j.matpur.2016.04.004.  Google Scholar

[7]

G. Falkovich, K. Gawedzki and M. Vergassola, Particles and fields in fluid turbulence, Rev. Modern Phys., 73 (2001), 913-975.  doi: 10.1103/RevModPhys.73.913.  Google Scholar

[8]

S. Fang and D. Luo, Constantin and Iyer's representation formula for the Navier-Stokes equations on manifolds, Potential Anal., 48 (2018), 181-206.  doi: 10.1007/s11118-017-9631-0.  Google Scholar

[9]

E. Fedrizzi and F. Flandoli, Noise prevents singularities in linear transport equations, J. Funct. Anal., 264 (2013), 1329-1354.  doi: 10.1016/j.jfa.2013.01.003.  Google Scholar

[10]

F. Flandoli, Random Perturbation of PDEs and Fluid Dynamic Models, École d'été de Saint Flour 2010, Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-642-18231-0.  Google Scholar

[11]

F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), 367-391.  doi: 10.1007/BF01192467.  Google Scholar

[12]

F. Flandoli, M. Maurelli and M. Neklyudov, Noise prevents infinite stretching of the passive field in a stochastic vector advection equation, J. Math. Fluid Mech., 16 (2014), 805-822.  doi: 10.1007/s00021-014-0187-0.  Google Scholar

[13]

F. Flandoli and C. Olivera, Well-posedness of the vector advection equations by stochastic perturbation, J. Evol. Equ., 18 (2018), 277-301.  doi: 10.1007/s00028-017-0401-7.  Google Scholar

[14]

F. Gay-Balmaz and D. D. Holm, Stochastic geometric models with non-stationary spatial correlations in Lagrangian fluid flows, J. Nonlinear Sci., 28 (2018), 873-904.  doi: 10.1007/s00332-017-9431-0.  Google Scholar

[15]

D. D. Holm, Variational principles for stochastic fluid dynamics, Proceedings of the Royal Society A, 471 (2015), 20140963, 19pp. doi: 10.1098/rspa.2014.0963.  Google Scholar

[16]

G. Iyer, A stochastic perturbation of inviscid flows, Commun. Math. Phys., 266 (2006), 631-645.  doi: 10.1007/s00220-006-0058-5.  Google Scholar

[17]

H. Kunita, Stochastic differential equations and stochastic flows of diffeomorphisms, École d'été de Probabilités de Saint Flour, 1982, 143–303, Lecture Notes in Math., 1097, Springer, Berlin, 1984. doi: 10.1007/BFb0099433.  Google Scholar

[18]

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

[19]

R. Mikulevicius and B. L. Rozovskii, Global $L_2$-solutions of stochastic Navier-Stokes equations, Ann. Probab., 33 (2005), 137-176.  doi: 10.1214/009117904000000630.  Google Scholar

[20]

D. Ocone and E. Pardoux, A generalized Itô–Ventzell formula. Application to a class of anticipating stochastic differential equations, Ann. Inst. Henri Poincaré, 25 (1989), 39-71.   Google Scholar

[21]

X. Zhang, A stochastic representation for backward incompressible Navier–Stokes equations, Probab. Theory Related Fields, 148 (2010), 305-332.  doi: 10.1007/s00440-009-0234-6.  Google Scholar

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