Source: Ann. Appl. Probab.
Volume 21, Number 4
In this paper we derive a probabilistic representation of the deterministic 3-dimensional Navier–Stokes equations in the presence of spatial boundaries. The formulation in the absence of spatial boundaries was done by the authors in [Comm. Pure Appl. Math. 61 (2008) 330–345]. While the formulation in the presence of boundaries is similar in spirit, the proof is somewhat different. One aspect highlighted by the formulation in the presence of boundaries is the nonlocal, implicit influence of the boundary vorticity on the interior fluid velocity.
 Arnold, V. (1966). Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier (Grenoble) 16 319–361.
 Beale, J. T., Kato, T. and Majda, A. (1984). Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Comm. Math. Phys. 94 61–66.
Mathematical Reviews (MathSciNet): MR763762
 Bhattacharya, R. N., Chen, L., Dobson, S., Guenther, R. B., Orum, C., Ossiander, M., Thomann, E. and Waymire, E. C. (2003). Majorizing kernels and stochastic cascades with applications to incompressible Navier–Stokes equations. Trans. Amer. Math. Soc. 355 5003–5040 (electronic).
 Bhattacharya, R., Chen, L., Guenther, R. B., Orum, C., Ossiander, M., Thomann, E. and Waymire, E. C. (2005). Semi-Markov cascade representations of local solutions to 3-D incompressible Navier–Stokes. In Probability and Partial Differential Equations in Modern Applied Mathematics (E. C. Waymire and J. Duan, eds.). IMA Vol. Math. Appl. 140 27–40. Springer, New York.
 Busnello, B. (1999). A probabilistic approach to the two-dimensional Navier–Stokes equations. Ann. Probab. 27 1750–1780.
 Busnello, B., Flandoli, F. and Romito, M. (2005). A probabilistic representation for the vorticity of a three-dimensional viscous fluid and for general systems of parabolic equations. Proc. Edinb. Math. Soc. (2) 48 295–336.
 Constantin, P. (2001). An Eulerian-Lagrangian approach for incompressible fluids: Local theory. J. Amer. Math. Soc. 14 263–278 (electronic).
 Constantin, P. (2006). Generalized relative entropies and stochastic representation. Int. Math. Res. Not. Art. ID 39487, 9.
 Constantin, P. and Fefferman, C. (1993). Direction of vorticity and the problem of global regularity for the Navier–Stokes equations. Indiana Univ. Math. J. 42 775–789.
 Constantin, P. and Foias, C. (1988). Navier–Stokes Equations. Univ. Chicago Press, Chicago, IL.
Mathematical Reviews (MathSciNet): MR972259
 Constantin, P. and Iyer, G. (2008). A stochastic Lagrangian representation of the three-dimensional incompressible Navier–Stokes equations. Comm. Pure Appl. Math. 61 330–345.
 Constantin, P. and Iyer, G. (2006). Stochastic Lagrangian transport and generalized relative entropies. Commun. Math. Sci. 4 767–777.
 Cipriano, F. and Cruzeiro, A. B. (2007). Navier-Stokes equation and diffusions on the group of homeomorphisms of the torus. Comm. Math. Phys. 275 255–269.
 Chorin, A. J. (1973). Numerical study of slightly viscous flow. J. Fluid Mech. 57 785–796.
Mathematical Reviews (MathSciNet): MR395483
 Chorin, A. J. and Marsden, J. E. (1993). A Mathematical Introduction to Fluid Mechanics, 3rd ed. Texts in Applied Mathematics 4. Springer, New York.
 Eyink, G. L. (2010). Stochastic least-action principle for the incompressible Navier–Stokes equation. Phys. D 239 1236–1240.
 Eyink, G. L. (2008). Stochastic line-motion and stochastic conservation laws for non-ideal hydromagnetic models. I. Incompressible fluids and isotropic transport coefficients. Preprint. Available at arXiv:0812.0153
 Friedman, A. (2006). Stochastic Differential Equations and Applications. Dover, Mineola, NY. (Two volumes bound as one. Reprint of the 1975 and 1976 original published in two volumes.)
 Gliklikh, Y. (1997). Global Analysis in Mathematical Physics: Geometric and Stochastic Methods. Applied Mathematical Sciences 122. Springer, New York. (Translated from the 1989 Russian original and with Appendix F by Viktor L. Ginzburg.)
 Iyer, G. (2006). A stochastic Lagrangian formulation of the Navier–Stokes and related transport equations. Ph.D. thesis, Univ. Chicago.
 Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
 Krylov, N. V. (1993). Quasiderivatives for solutions of Itô’s stochastic equations and their applications. In Stochastic Analysis and Related Topics (Oslo, 1992). Stochastics Monogr. 8 1–44. Gordon and Breach, Montreux.
 Krylov, N. V. (2004). Quasiderivatives and interior smoothness of harmonic functions associated with degenerate diffusion processes. Electron. J. Probab. 9 615–633 (electronic).
 Kuznetsov, E. A. and Ruban, V. P. (2000). Hamiltonian dynamics of vortex and magnetic lines in hydrodynamic type systems. Phys. Rev. E (3) 61 831–841.
 Kunita, H. (1997). Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics 24. Cambridge Univ. Press, Cambridge. (Reprint of the 1990 original.)
 Le Jan, Y. and Sznitman, A. S. (1997). Stochastic cascades and 3-dimensional Navier–Stokes equations. Probab. Theory Related Fields 109 343–366.
 McKean, H. P. Jr. (1969). Stochastic Integrals. Probability and Mathematical Statistics 5. Academic Press, New York.
Mathematical Reviews (MathSciNet): MR247684
 Majda, A. J. and Bertozzi, A. L. (2002). Vorticity and Incompressible Flow. Cambridge Texts in Applied Mathematics 27. Cambridge Univ. Press, Cambridge.
 Michel, P., Mischler, S. and Perthame, B. (2004). General entropy equations for structured population models and scattering. C. R. Math. Acad. Sci. Paris 338 697–702.
 Michel, P., Mischler, S. and Perthame, B. (2005). General relative entropy inequality: An illustration on growth models. J. Math. Pures Appl. (9) 84 1235–1260.
 Øksendal, B. (2003). Stochastic Differential Equations, 6th ed. Springer, Berlin.
 Ossiander, M. (2005). A probabilistic representation of solutions of the incompressible Navier–Stokes equations in R3. Probab. Theory Related Fields 133 267–298.
 Rozovskiĭ, B. L. (1990). Stochastic Evolution Systems: Linear Theory and Applications to Nonlinear Filtering. Mathematics and Its Applications (Soviet Series) 35. Kluwer Academic, Dordrecht. (Translated from the Russian by A. Yarkho.)
 Ruban, V. P. (1999). Motion of magnetic flux lines in magnetohydrodynamics. JETP 89 299–310.
 Thomann, E. and Ossiander, M. (2003). Stochastic cascades applied to the Navier–Stokes equations. In Probabilistic Methods in Fluids 287–297. World Scientific, River Edge, NJ.
 Waymire, E. C. (2002). Multiscale and multiplicative processes in fluid flows. In Instructional and Research Workshop on Multiplicative Processes and Fluid Flows
, Aarhus Univ., 2001). Lectures on Multiscale and Multiplicative Processes in Fluid Flows 11
. Available at http://www.maphysto.dk/cgi-bin/gp.cgi?publ=407
 Waymire, E. C. (2005). Probability & incompressible Navier–Stokes equations: An overview of some recent developments. Probab. Surv. 2 1–32 (electronic).
 Webber, W. (1968). Über eine Transformation der hydrodynamischen Gleichungen. J. Reine Angew. Math. 68 286–292.
 Zhang, X. (2010). A stochastic representation for backward incompressible Navier–Stokes equations. Probab. Theory Related Fields 148 305–332.