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Euler-Lagrangian approach to 3D stochastic Euler equations
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 |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
G. Iyer,
A stochastic perturbation of inviscid flows, Commun. Math. Phys., 266 (2006), 631-645.
doi: 10.1007/s00220-006-0058-5. |
[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. |
[18] |
H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge Studies in
Advanced Mathematics, 24, Cambridge University Press, Cambridge, 1990. |
[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. |
[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.
|
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
G. Iyer,
A stochastic perturbation of inviscid flows, Commun. Math. Phys., 266 (2006), 631-645.
doi: 10.1007/s00220-006-0058-5. |
[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. |
[18] |
H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge Studies in
Advanced Mathematics, 24, Cambridge University Press, Cambridge, 1990. |
[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. |
[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.
|
[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. |
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