Abstract
In this article we study the fractal Navier-Stokes equations by using the stochastic Lagrangian particle path approach in Constantin and Iyer (Comm Pure Appl Math LXI:330–345, 2008). More precisely, a stochastic representation for the fractal Navier-Stokes equations is given in terms of stochastic differential equations driven by Lévy processes. Based on this representation, a self-contained proof for the existence of a local unique solution for the fractal Navier-Stokes equation with initial data in \({{\mathbb W}^{1,p}}\) is provided, and in the case of two dimensions or large viscosity, the existence of global solutions is also obtained. In order to obtain the global existence in any dimensions for large viscosity, the gradient estimates for Lévy processes with time dependent and discontinuous drifts are proved.
Similar content being viewed by others
References
Ambrosio L., Lecumberry M., Maniglia S.: Lipschitz regularity and approximate differentiability of the DiPerna-Lions flow. Rend. Sem. Univ. Padova 114, 29–50 (2005)
Applebaum, D.: Lévy processes and stochastic calculus. Cambridge Studies in Advance Mathematics 93, Cambridge: Cambridge University Press, 2004
Arnold V.I.: Ordinary differential equations. Spinger-Verlag, Berlin (1992)
Biler P., Funaki T., Woyczynski W.A.: Fractal Burgers equations. J. Diff. Eq. 148, 9–46 (1998)
Busnello B.: A probabilistic approach to the two-dimensional Navier-Stokes equations. Ann. Probab. 27(4), 1750–1780 (1999)
Busnello B., Flandoli F., Romito M.: A probabilistic representation for the vorticity of a three-dimensional viscous fluid and for general systems of parabolic equations. Proc. Edinb. Math. Soc. (2) 48(2), 295–336 (2005)
Constantin P., Iyer G.: A stochsatic Lagrangian representation of the three-dimensional incompressible Navier-Stokes equations. Comm. Pure Appl. Math. LXI, 330–345 (2008)
Crippa G., De Lellis C.: Estimates and regularity results for the DiPerna-Lions flow. J. Reine Angew. Math. 616, 15–46 (2008)
Cruzeiro A.B., Shamarova E.: Navier-Stokes equations and forward-backward SDEs on the group of diffeomorphisms of a torus. Stoch. Proc. Appl. 119, 4034–4060 (2009)
DiPerna R.J., Lions P.L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989)
Esposito R., Marra R., Pulvirenti M., Sciarretta C.: A stochastic Lagrangian picture for the three-dimensional Navier-Stokes equation. Comm. Partial Diff. Eq. 13(12), 1601–1610 (1988)
Friedman A.: Stochastic differential equations and applications. Volume 1. Academic Press, New York (1975)
Portenko, N.I.: Generalized diffusion processes. Moscow: Nauka, 1982 In Russian; English translation: Provdence, RI: Amer. Math. Soc., 1990
Priola, E.: Pathwise uniqueness for singular SDEs driven by stable processes. http://arxiv.org/abs/1005.4237v2 [math.DS], 2010
Iyer G.: A stochastic Lagrangian proof of global existence of Navier-Stokes equations for flows with small Reynolds Number. Ann. Inst. H. Poincare Anal. Non Lineaire 26(1), 181–189 (2009)
Iyer G., Mattingly J.: A stochastic-Lagrangian particle system for the Navier-Stokes equations. Nolinearity 21, 2537–2553 (2008)
Jacob, N.: Pseudo differential operators, Markov processes. Vol. I, Fourier Analysis and Semigroups. Singapore: Imperical College Press, World Scientific Publishing, 2001
Kiselev A., Nazarov F., Schterenberg R.: Blow up and regularity for fractal Burgers equation. Dyn. PDE 5(3), 211–240 (2008)
Kurenok V.P.: A note on L 2-estimates for stable integrals with drift. Trans. Amer. Math. Soc. 360, 925–938 (2008)
Le Jan Y., Sznitman A.S.: Stochastic cascades and 3-dimensional Navier-Stokes equations. Proab. Th. Rela. Fields 109(3), 343–366 (1997)
Majda A.J., Bertozzi A.L.: Vorticity and incompressible flow. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)
Ren J., Zhang X.: Limit theorems for stochastic differential equations with discontinuous coefficients. SIAM J. Math. Anal. 43(1), 302–321 (2011)
Resnick, S.: Dynamical problems in nonlinear advective partial differential equations. Ph. D. Thesis, University of Chicago, 1995
Schilling, R., Sztonyk, P., Wang, J.: Coupling property and gradient estimates of Lévy processes via the symbol. http://arxiv.org/abs/1011.1067v1 [math.PR], 2010
Shlesinger, M.F., Zaslavsky, G.M., Frisch U., (eds.): Lévy filghts and related topics in physics. Lect. Notes in Physics, Vol. 450, Berlin: Springer-Verlag, 1995
Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton, NJ: Princeton University Press, 1970
Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. Amsterdam: North-Holland Publishing Company, 1978
Woyczyński, W.A.: Lévy processes in the physical sciences. Boston, MA: Birkhäuser, 2001, pp. 241–266
Wu J.: Lower bounds for an integral involving fractional Laplacians and the generalized Navier-Stokes equations in Besov spaces. Commun. Math. Phys. 263, 803–831 (2005)
Zhang X.: L p-theory of semi-linear SPDEs on general measure spaces and applications. J. Func. Anal. 239/1, 44–75 (2006)
Zhang X.: A stochastic representation for backward incompressible Navier-Stokes equations. Prob. Theory and Rel. Fields 148(1-2), 305–332 (2010)
Zhang, X.: Stochastic differential equations with Sobolev drifts and driven by Lévy processes. Ann. Inst. H. Poincare (B) Probabilités Statistiques (in press)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. Constantin
Rights and permissions
About this article
Cite this article
Zhang, X. Stochastic Lagrangian Particle Approach to Fractal Navier-Stokes Equations. Commun. Math. Phys. 311, 133–155 (2012). https://doi.org/10.1007/s00220-012-1414-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-012-1414-2