Abstract
We prove existence and regularity of the stochastic flows used in the stochastic Lagrangian formulation of the incompressible Navier-Stokes equations (with periodic boundary conditions), and consequently obtain a C k,α local existence result for the Navier-Stokes equations. Our estimates are independent of viscosity, allowing us to consider the inviscid limit. We show that as ν → 0, solutions of the stochastic Lagrangian formulation (with periodic boundary conditions) converge to solutions of the Euler equations at the rate of \(O(\sqrt{\nu t})\).
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References
Bhattacharya R.N., Chen L., Dobson S., Guenther R.B., Orum C., Ossiander M., Thomann E., Waymire E.C. (2003). Majorizing kernels and stochastic cascades with applications to incompressible Navier-Stokes equations. Trans. Amer. Math. Soc. 355(12): 5003–5040, (electronic)
Busnello, B., Flandoli, F., Romito, M., A probabilistic representation for the vorticity of a 3D viscous fluid and for general systems of parabolic equations. http://arxiv.org/list/math.PR/0306075, 2003
Chorin A., Marsden J. (2000). A Mathematical Introduction to Fluid Mechanics. Springer, Berlin Heidelberg New York
Constantin P. (2001). An Eulerian-Lagrangian approach for incompressible fluids: local theory. J. Amer. Math. Soc. 14(2): 263–278, (electronic)
Constantin P. (2001). An Eulerian-Lagrangian Approach to the Navier-Stokes equations. Commun. Math. Phys. 216(3): 663–686
Constantin P., Foias C. (1988). Navier-Stokes Equations. University of Chicago Press, Chicago, IL
Constantin, P., Iyer, G.: A stochastic Lagrangian representation of the 3-dimensional incompressible Navier-Stokes equations. http://arxiv.org/list/math.PR/0511067, 2005
Friedman A. (1975). Stochastic Differential Equations and Applications, Volume 1. Academic Press, London-New York-San Diego
Gomes D.A. (2005). A variational formulation for the Navier-Stokes equation. Commun. Math. Phys 257: 227–234
Jourdain, B., Le Bris, C., Lelièvre, T.: Coupling PDEs and SDEs: the Illustrative Example of the Multiscale Simulation of Viscoelastic Flows. Lecture Notes in Computational Science and Engineering 44, Berlin Heidelberg New York: Springer, 2005, pp. 151–170
Karatzas I., Shreve S. (1991). Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics 113. Springer, New York
Krylov, N.V.: Lectures on Elliptic and Parabolic Equations in Hölder Spaces. Graduate Studies in Mathematics 12, Providence, RI: Amer. Math. Soc, 1996
Kunita H. (1997). Stochastic flows and stochastic differential equations. Cambridge Studies in Advanced Mathematics 24. Cambridge University Press, Cambridge
Le Gall, J.: Spatial Branching Processes, Random Snakes and Partial Differential Equations. Lectures in Mathematics, Basel-Baston: Birkhäuser, 1999
Le Jan Y., Sznitman A.S. (1997). Stochastic cascades and 3-dimensional Navier-Stokes equations. Probab. Theory Related Fields 109(3): 343–366
Majda A., Bertozzi A. (2002). Vorticity and Incompressible Flow. Cambridge University Press, Cambridge
Stein E. (1970). Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, NJ
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Communicated by A. Kupiainen
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Iyer, G. A Stochastic Perturbation of Inviscid Flows. Commun. Math. Phys. 266, 631–645 (2006). https://doi.org/10.1007/s00220-006-0058-5
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DOI: https://doi.org/10.1007/s00220-006-0058-5