Abstract
In this chapter, we will explain how the Brenier’s relaxed variational principle for Euler equation makes involved the ordinary differential equations with Sobolev coefficients and how the investigation on stochastic differential equations (SDE) with Sobolev coefficients is useful to establish variational principles for Navier–Stokes equations. We will survey recent results on this topic.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Ambrosio, L.: Transport equation and Cauchy problem for BV vector fields. Invent. Math. 158, 227–260 (2004)
Ambrosio, L., Figalli, A.: Geodesics in the space of measure-preserving maps and plans. Arch. Rat. Mech. Anal. 194, 421–462 (2009)
Antoniouk, A., Arnaudon, M., Cruzeiro, A.B.: Generalized stochastic flows and applications to incompressible viscous fluids. Bulletin des Sciences Mathématiques, (2013 in Press)
Arnaudon, M., Cruzeiro, A.B.: Stochastic Lagrangian flows and Navier–Stokes equations (2012, preprint)
Aryasova, O.V., Pilipenko, A.Y.: On properties of a flow generated by a SDE with discontinuous drift. Electron. J. Probab. 17(106), 1–20 (2012)
Bismut, J.M.: Large Deviation and Malliavin Calculus. Progress in Mathematics, vol. 45. Birkhäuser, Boston (1984)
Bouleau, N., Hirsch, F.: Dirichlet Forms and Analysis on Wiener Space. De Gruyter Studies in Mathematics, vol. 14. De Gruyter, Berlin (1991)
Brenier, Y.: The least action principle and the related concept of generalized flows for incompressible perfect fluids. J. Am. Math. Soc. 2, 225–255 (1989)
Chemin, J.Y.: Fluides Parfaits Incompressibles. Astérisque, vol. 230. Société Mathématique de France, Paris (1995)
Crippa, G., De Lellis, C.: Estimates and regularity results for the Di Perna-Lions flows. J. Reine Angew. Math. 616, 183–201 (2008)
Cruzeiro, A.B.: Equations différentielles sur l’espace de Wiener et formules de cameron-Martin non linéaires. J. Funct. Anal. 54, 206–227 (1983)
Di Perna, R.J., Lions, P.L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989)
Driver, B.: Integration by parts for heat kernel measures revisited. J. Math. Pures Appl. 76, 703–737 (1997)
Fang, S.: Introduction to Malliavin Calculus. Mathematical Series for Graduate Students, vol. 3. Tsinghua University Press, Springer, Beijing (2005)
Fang, S., Luo, D., Thalmarier, A.: Stochastic differential equations with coefficients in Sobolev spaces. J. Funct. Anal. 259, 1129–1168 (2010)
Fang, S., Li, H., Luo, D.: Heat semi-group and generalized flows on complete Riemannian manifolds. Bull. Sci. Math. 135, 565–600 (2011)
Le Jan, Y., Raimond, O.: Integration of Brownian vector fields. Ann. Probab. 30, 826–873 (2002)
Le Jan, Y., Raimond, O.: Flows, coalescence and noise. Ann. Probab. 32, 1247–1315 (2004)
Zhang, X.: Stochastic flows of SDEs with irregular coefficients and stochastic transport equations. Bull. Sci. Math. 134, 340–378 (2010)
Acknowledgements
The author is grateful to Professor Yuliia Mishura for inviting him to attend the conference MSTA3 at Kiev in September 2012. He also thanks the referee for his careful reading and suggestions.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Fang, S. (2014). Hydrodynamics and Stochastic Differential Equation with Sobolev Coefficients. In: Korolyuk, V., Limnios, N., Mishura, Y., Sakhno, L., Shevchenko, G. (eds) Modern Stochastics and Applications. Springer Optimization and Its Applications, vol 90. Springer, Cham. https://doi.org/10.1007/978-3-319-03512-3_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-03512-3_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-03511-6
Online ISBN: 978-3-319-03512-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)