Solution properties of a 3D stochastic Euler fluid equation. (English) Zbl 1433.60051

Summary: We prove local well-posedness in regular spaces and a Beale-Kato-Majda blow-up criterion for a recently derived stochastic model of the 3D Euler fluid equation for incompressible flow. This model describes incompressible fluid motions whose Lagrangian particle paths follow a stochastic process with cylindrical noise and also satisfy Newton’s second law in every Lagrangian domain.


60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35Q35 PDEs in connection with fluid mechanics
60H30 Applications of stochastic analysis (to PDEs, etc.)


Full Text: DOI arXiv


[1] Beale, JT; Kato, T.; Majda, A., Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Commun. Math. Phys., 94, 61-66, (1984) · Zbl 0573.76029
[2] Brzézniak, Z.; Capínski, M.; Flandoli, F., Stochastic partial differential equations and turbulence, Math. Models Methods Appl. Sci., 1, 41-59, (1991) · Zbl 0741.60058
[3] Constantin, P.; Fefferman, C.; Majda, A., Geometric constraints on potential singularity formulation in the 3D Euler equations, Commun. Partial Differ. Equ., 21, 559-571, (1996) · Zbl 0853.35091
[4] Constantin, P.; Iyer, G., A stochastic Lagrangian representation of the three-dimensional incompressible Navier-Stokes equations, Commun. Pure Appl. Math., 61, 330-345, (2008) · Zbl 1156.60048
[5] Cotter, C.J., Crisan, D.O., Holm, D.D., Shevschenko, I., Pan, W.: Modelling uncertainty using circulation-preserving stochastic transport noise in a 2-layer quasi-geostrophic model. arXiv:1802.05711 (2018a)
[6] Cotter, C.J., Crisan, D.O., Holm, D.D., Shevschenko, I., Pan, W.: Numerically modelling stochastic lie transport in fluid dynamics. arXiv:1801.09729 (2018b)
[7] Cotter, CJ; Gottwald, GA; Holm, DD, Stochastic partial differential fluid equations as a diffusive limit of deterministic Lagrangian multi-time dynamics, Proc. R. Soc. A, 473, 20170388, (2017) · Zbl 1402.76101
[8] Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions, 2nd edn. Cambridge University Press, Cambridge (2015) · Zbl 1140.60034
[9] Delarue, F.; Flandoli, F.; Vincenzi, D., Noise prevents collapse of Vlasov-Poisson point charges, Commun. Pure Appl. Math., 67, 1700-1736, (2014) · Zbl 1302.35366
[10] Drivas, T.D., Holm, D.D.: Circulation and energy theorem preserving stochastic fluids. arXiv:1808.05308 (2018)
[11] Ebin, DG; Marsden, JE, Groups of diffeomorphisms and the notion of an incompressible fluid, Ann. Math., 2, 102-163, (1970) · Zbl 0211.57401
[12] Flandoli, F.: Random perturbation of PDEs and fluid dynamic models. Saint Flour Summer School Lectures, 2010, Lecture Notes in Mathematics no. 2015. Springer, Berlin (2011)
[13] Flandoli, F.; Gatarek, D., Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Relat. Fields, 102, 367-391, (1995) · Zbl 0831.60072
[14] Flandoli, F.; Gubinelli, M.; Priola, E., Well-posedness of the transport equation by stochastic perturbation, Invent. Math., 180, 1-53, (2010) · Zbl 1200.35226
[15] Flandoli, F.; Gubinelli, M.; Priola, E., Full well-posedness of point vortex dynamics corresponding to stochastic 2D Euler equations, Stoch. Process. Appl., 121, 1445-1463, (2011) · Zbl 1221.60082
[16] Flandoli, F.; Maurelli, M.; Neklyudov, M., Noise prevents infinite stretching of the passive field in a stochastic vector advection equation, J. Math. Fluid Mech., 16, 805-822, (2014) · Zbl 1322.60109
[17] Frisch, U., Villone, B.: Cauchy’s almost forgotten Lagrangian formulation of the Euler equation for 3D incompressible flow. Eur. Phys. J. H 39(3), 325-351 (2014). Preprint available at https://arxiv.org/pdf/1402.4957.pdf
[18] Gay-Balmaz, F., Holm, D.D.: Stochastic geometric models with non-stationary spatial correlations in Lagrangian fluid flows. J. Nonlinear Sci. (2018) https://doi.org/10.1007/s00332-017-9431-0 · Zbl 1431.37068
[19] Gibbon, J.D.: The three-dimensional Euler equations: where do we stand? In: G. Eyink, U. Frisch, R. Moreau, A. Sobolevski (eds) Euler Equations 250 years on, Aussois, France, 18-23 June 2007, Physica D 237, 1894-1904 (2008)
[20] Gyongy, I.; Krylov, N., Existence of strong solutions for Itô’s stochastic equations via approximations, Probab. Theory Relat. Fields, 105, 143-158, (1996) · Zbl 0847.60038
[21] Gyöngy, I., On the approximation of stochastic partial differential equations II, Stoch. Stoch. Rep., 26, 129-164, (1989) · Zbl 0669.60059
[22] Gyöngy, I.; Krylov, N., Stochastic partial differential equations with unbounded coefficients and applications III, Stoch. Stoch. Rep., 40, 77-115, (1992) · Zbl 0791.60045
[23] Gyöngy, I.; Krylov, N., On the splitting-up method and stochastic partial differential equations, Ann. Probab., 31, 564-591, (2003) · Zbl 1028.60058
[24] Holm, DD; Marsden, JE; Ratiu, TS, Semidirect products in continuum dynamics, Adv. Math., 137, 1-81, (1998) · Zbl 0951.37020
[25] Holm, DD, Variational principles for stochastic fluid dynamics, Proc. R. Soc. A, 471, 20140963, (2015) · Zbl 1371.35219
[26] Kato, T.; Lai, CY, Nonlinear evolution equations and the Euler flow, J. Funct. Anal., 56, 15-28, (1984) · Zbl 0545.76007
[27] Krylov, N.V., Rozovskii, B.L.: Stochastic evolution equations, (Russian) current problems in mathematics, Vol. 14, pp. 71147, 256, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow (1979)
[28] Karatzas, I., Shreve, S.E.: Brownian motion and stochastic calculus. 2nd ed. Graduate Texts in Mathematics, 113. Springer, New York (1991) · Zbl 0734.60060
[29] Kunita, H.: Stochastic differential equations and stochastic flows of diffeomorphisms, Ecole d’été de probabilités de Saint-Flour, XII-1982, 143-303. Lecture Notes in Math, vol. 1097. Springer, Berlin (1984)
[30] Kunita, H.: Stochastic Flows and Stochastic Differential Equations, Cambridge Studies in Advanced Mathematics, vol. 24. Cambridge University Press, Cambridge (1990) · Zbl 0743.60052
[31] Kim, JU, Existence of a local smooth solution in probability to the stochastic Euler equations in \(R^3\), J. Funct. Anal., 256, 3660-3687, (2009) · Zbl 1166.60036
[32] Lichtenstein, L., Uber einige Existenzprobleme der Hydrodynamik unzusamendruckbarer, reibunglosiger Flussigkeiten und die Helmholtzischen Wirbelsatze, Math. Zeit., 23, 89-154, (1925) · JFM 51.0658.01
[33] Lilly, J.M.: jLab: A data analysis package for Matlab, v. 1.6.3, (2017). http://www.jmlilly.net/jmlsoft.html
[34] Lions, P.L.: Mathematical Topics in Fluid Mechanics, Vol. 1 Incompressible Models, vol. 1. Oxford University Press, New York (1996) · Zbl 0866.76002
[35] Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, vol. 27. Cambridge University Press, Cambridge (2002)
[36] Marsden, J.E., Hughes, T.J.R.: Mathematical Foundations of Elasticity, Dover Publications, Inc. New York (1994). http://authors.library.caltech.edu/25074/1/Mathematical_Foundations_of_Elasticity.pdf · Zbl 0545.73031
[37] Mémin, E., Fluid flow dynamics under location uncertainty, Geophys. Astrophys. Fluid Dyn., 108, 119-146, (2014)
[38] Mikulevicius, R.; Rozovskii, B., Stochastic Navier-Stokes equations for turbulent flows, SIAM J. Math. Anal., 35, 1250-1310, (2004) · Zbl 1062.60061
[39] Pardoux, E.: Stochastic Partial Differential Equations. Lectures given in Fudan University, Shanghai (2007) · Zbl 0777.60054
[40] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences. Springer, New York (1983) · Zbl 0516.47023
[41] Prévôt, C., Röckner, M.: A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Mathematics, vol. 1905. Springer, Berlin (2007) · Zbl 1123.60001
[42] Resseguier, V.: Mixing and Fluid Dynamics Under Location Uncertainty. PhD Thesis, Université de Rennes
[43] Simon, J., Compact sets in the space \(L^{p}(0, T;B)\), Ann. Math. Pura Appl., 146, 65-96, (1987) · Zbl 0629.46031
[44] Schaumlöffel, K-U, White noise in space and time and the cylindrical Wiener process, Stoch. Anal. Appl., 6, 81-89, (1988) · Zbl 0658.60068
[45] Sykulski, AM; Olhede, SC; Lilly, JM; Danioux, E., Lagrangian time series models for ocean surface drifter trajectories, Appl. Statist., 65, 29-50, (2016)
[46] Temam, R.: Navier-Stokes Equations, Theory and Numerical Analysis. American Mathematical Society, North-Holland (1977) · Zbl 0383.35057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.