Forward-backward stochastic differential systems associated to Navier-Stokes equations in the whole space. (English) Zbl 1323.35010
Two coupled forward-backward stochastic differential systems (FBSDS) in the whole space are studied such that their local existences (and uniqueness) of solution are expressed by probabilistic representations. One of them is identified with the unique strong solution to the Navier-Stokes equation, while the second one is identified with the following PDE of Burgers’ type: \(\partial_t u+\nu \Delta u+ \left( (b+\alpha u) \cdot \nabla\right) u+ cu +\phi =0\) in \(\mathbb{R}^d\times ]0,T[\), and \(u(T)=\psi\). Some considerations about the global solvability are included. Moreover, the authors prove that the velocity field of the Navier-Stokes equation may be approximated by solutions of truncated in time FBSDS. Finally, the authors derive from the probabilistic representation one formula for the velocity field of the N-S equation through which the connection with the Lagrangian approach is showed, and as consequence the strong solution to the Cauchy problem of the N-S equation is interpreted as a critical point of controlled FBSDS.
Reviewer: Luisa Consiglieri (Lisboa)
MSC:
| 35D35 | Strong solutions to PDEs |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 35Q30 | Navier-Stokes equations |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
Keywords:
forward-backward stochastic differential system; Navier-Stokes equation; Feynman-Kac formula; strong solution; Lagrangian approach; variational formulationReferences:
| [1] | Albeverio, S.; Belopolskaya, Y., Generalized solutions of the Cauchy problem for the Navier-Stokes system and diffusion processes, Cubo (Temuco), 12, 2, 77-96 (2010) · Zbl 1221.60076 |
| [2] | Antonelli, F., Backward-forward stochastic differential equations, Ann. Appl. Probab., 3, 3, 777-793 (1993) · Zbl 0780.60058 |
| [3] | Arnold, V., 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, 1, 319-361 (1966) · Zbl 0148.45301 |
| [4] | Barles, G.; Lesigne, E., SDE, BSDE and PDE, (Backward Stochastic Differential Equations. Backward Stochastic Differential Equations, Pitman Research Notes in Mathematics Series, vol. 364 (1997), Longman: Longman Harlow), 47-80 · Zbl 0886.60049 |
| [5] | Bender, C.; Zhang, J., Time discretization and Markovian iteration for coupled FBSDEs, Ann. Appl. Probab., 18, 1, 143-177 (2008) · Zbl 1142.65005 |
| [6] | Bhattacharya, R. N.; Chen, L.; Dobson, S.; Guenther, R. B.; Orum, C.; Ossiander, M.; Thomann, E.; Waymire, E. C., Majorizing kernels and stochastic cascades with applications to incompressible Navier-Stokes equations, Trans. Amer. Math. Soc., 355, 12, 5003-5040 (2003) · Zbl 1031.35115 |
| [8] | Busnello, B., A probabilistic approach to the two-dimensional Navier-Stokes equations, Ann. Probab., 27, 4, 1750-1780 (1999) · Zbl 0988.60057 |
| [9] | 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., 48, 2, 295-336 (2005) · Zbl 1075.76019 |
| [10] | Cheridito, P.; Soner, H. M.; Touzi, N.; Victoir, N., Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs, Comm. Pure Appl. Math., 60, 7, 1081-1110 (2007) · Zbl 1121.60062 |
| [11] | Chorin, A. J., Numerical study of slightly viscous flow, J. Fluid Mech., 57, 4, 785-796 (1973) |
| [12] | Chorin, A. J., Vorticity and Turbulence (1994), Springer · Zbl 0795.76002 |
| [13] | Constantin, P.; Iyer, G., A stochastic Lagrangian representation of the three-dimensional incompressible Navier-Stokes equations, Comm. Pure Appl. Math., 61, 3, 330-345 (2008) · Zbl 1156.60048 |
| [14] | Constantin, P.; Iyer, G., A stochastic Lagrangian approach to the Navier-Stokes equations in domains with boundary, Ann. Appl. Probab., 21, 4, 1466-1492 (2011) · Zbl 1246.76018 |
| [15] | Cruzeiro, A. B.; Shamarova, E., Navier-Stokes equations and forward-backward SDEs on the group of diffeomorphisms of a torus, Stochastic Process. Appl., 119, 4034-4060 (2009) · Zbl 1188.60036 |
| [16] | Cruzeiro, A. B.; Shamarova, E., On a forward-backward stochastic system associated to the burgers equation, (Stoch. Anal. Financ. Appl. (2011)), 43-59 · Zbl 1250.65012 |
| [17] | Delarue, F., On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case, Stochastic Process. Appl., 99, 2, 209-286 (2002) · Zbl 1058.60042 |
| [18] | Delarue, F.; Menozzi, S., A forward-backward stochastic algorithm for quasi-linear PDEs, Ann. Appl. Probab., 16, 1, 140-184 (2006) · Zbl 1097.65011 |
| [19] | Delarue, F.; Menozzi, S., An interpolated stochastic algorithm for quasi-linear PDEs, Math. Comp., 77, 261, 125-158 (2008) · Zbl 1131.65002 |
| [20] | DiPerna, R. J.; Lions, P.-L., Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98, 511-547 (1989) · Zbl 0696.34049 |
| [21] | Ebin, D.; Marsden, J., Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math., 92, 1, 102-163 (1970) · Zbl 0211.57401 |
| [22] | El Karoui, N.; Peng, S.; Quenez, M. C., Backward stochastic differential equations in finance, Math. Finance, 7, 1, 1-71 (1997) · Zbl 0884.90035 |
| [23] | Elworthy, K. D.; Li, X., Formulae for the derivatives of heat semigroups, J. Funct. Anal., 125, 252-286 (1994) · Zbl 0813.60049 |
| [24] | Esposito, R.; Marra, R., Three-dimensional stochastic vortex flows, Math. Methods Appl. Sci., II, 431-445 (1989) · Zbl 0688.76030 |
| [25] | Esposito, R.; Marra, R.; Pulvirenti, M.; Sciarretta, C., A stochastic Lagrangian picture for the three dimensional Navier-Stokes equations, Comm. Partial Differential Equations, 13, 12, 1601-1610 (1988) · Zbl 0656.76035 |
| [26] | Eyink, G. L., Stochastic least-action principle for the incompressible Navier-Stokes equation, Physica D, 239, 14, 1236-1240 (2010) · Zbl 1193.37117 |
| [27] | Friedman, A., Partial Differential Equations (1969), Holt, Rinehart and Winston: Holt, Rinehart and Winston New York · Zbl 0224.35002 |
| [28] | Gilbarg, D.; Trudinger, N. S., (Partial Differential Equations of Second Order. Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, vol. 224 (1983), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0562.35001 |
| [29] | Gomes, D., A variational formulation for the Navier-Stokes equation, Comm. Math. Phys., 257, 1, 227-234 (2005) · Zbl 1080.37083 |
| [30] | Hu, Y.; Peng, S., Solution of forward-backward stochastic differential equations, Probab. Theory Related Fields, 103, 273-283 (1995) · Zbl 0831.60065 |
| [31] | Inoue, A.; Funaki, T., On a new derivation of the Navier-Stokes equation, Comm. Math. Phys., 65, 1, 83-90 (1979) · Zbl 0404.35079 |
| [32] | Jan, Y. L.; Sznitman, A. S., Stochastic cascades and 3-dimensional Navier-Stokes equations, Probab. Theory Related Fields, 109, 343-366 (1997) · Zbl 0888.60072 |
| [33] | Kunita, H., Stochastic Flows and Stochastic Differential Equations (1990), Cambridge University Press, World Publishing Corp. · Zbl 0743.60052 |
| [34] | Ladyzenskaja, O. A.; Solonnikov, V. A.; Ural’ceva, N. N., Linear and Quasi-linear Equations of Parabolic Type (1968), AMS: AMS Providence · Zbl 0174.15403 |
| [35] | Ma, J.; Protter, P.; Yong, J., Solving forward-backward stochastic differential equations explicitly—a four step scheme, Probab. Theory Related Fields, 98, 339-359 (1994) · Zbl 0794.60056 |
| [36] | Majda, A. J.; Bertozzi, A. L., (Vorticity and Incompressible Flow. Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, vol. 27 (2002), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0983.76001 |
| [37] | Navier, C., Mémoire sur les lois du mouvement des fluides, Mem. Acad. Sci. Inst. France, 6, 2, 375-394 (1822) |
| [38] | Nirenberg, L., On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa, 13, 115-162 (1959) · Zbl 0088.07601 |
| [39] | Ossiander, M., A probabilistic representation of solutions of the incompressible Navier-Stokes equations in \(R^3\), Probab. Theory Related Fields, 133, 267-298 (2005) · Zbl 1077.35107 |
| [40] | Pardoux, E.; Peng, S., Adapted solution of a backward stochastic differential equation, Systems Control Lett., 14, 1, 55-61 (1990) · Zbl 0692.93064 |
| [41] | Pardoux, E.; Peng, S., Backward stochastic differential equations and quasilinear parabolic partial differential equations, (Rozovskii, B. L.; Sowers, R. S., Stochastic Partial Differential Equations and their Applications. Stochastic Partial Differential Equations and their Applications, Lect. Notes Control Inf. Sci., vol. 176 (1992), Springer: Springer Berlin, Heidelberg, New York), 200-217 · Zbl 0766.60079 |
| [42] | Pardoux, E.; Tang, S., Forward-backward stochastic differential equations and quasilinear parabolic PDEs, Probab. Theory Related Fields, 114, 123-150 (1999) · Zbl 0943.60057 |
| [43] | Pardoux, E.; Zhang, S., Generalized BSDEs and nonlinear Neumann boundary value problem, Probab. Theory Related Fields, 110, 535-558 (1998) · Zbl 0909.60046 |
| [44] | Peng, S., Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stoch. Stoch. Rep., 37, 61-74 (1991) · Zbl 0739.60060 |
| [45] | Peng, S.; Shi, Y., Infinite horizon forward-backward stochastic differential equations, Stochastic Process. Appl., 85, 1, 75-92 (2000) · Zbl 0997.60062 |
| [46] | Peng, S.; Wu, Z., Fully coupled forward-backward stochastic differential equations and applications to optimal control, SIAM J. Control Optim., 37, 3, 825-843 (1999) · Zbl 0931.60048 |
| [47] | Qiu, J.; Tang, S.; You, Y., 2D backward stochastic Navier-Stokes equations with nonlinear forcing, Stochastic Process. Appl., 122, 334-356 (2012) · Zbl 1230.60067 |
| [48] | Russo, G.; Smereka, P., Impulse formulation of the Euler equations: general properties and numerical methods, J. Fluid Mech., 391, 189-209 (1999) · Zbl 0963.76019 |
| [49] | Stein, E. M., Singular Integrals and Differentiability Properties of Functions (1970), Princeton University Press: Princeton University Press Princeton · Zbl 0207.13501 |
| [50] | Stokes, G., On the theories of internal frictions of fluids in motion and of the equilibrium and motion of elastic solids, Trans. Camb. Philos. Soc., 8, 287-319 (1849) |
| [51] | Tang, S., Semi-linear systems of backward stochastic partial differential equations in \(R^n\), Chinese Ann. Math., 26B, 3, 437-456 (2005) · Zbl 1077.60047 |
| [52] | Temam, R., Navier-Stokes Equations: Theory and Numerical Analysis (1984), North-Holland: North-Holland Amsterdam, New York, Oxford · Zbl 0568.35002 |
| [53] | Temam, R., Navier-Stokes Equations and Nonlinear Functional Analysis (1995), Society for Industrial and Applied Mathematics · Zbl 0833.35110 |
| [54] | Triebel, H., Theory of Function Spaces, Monographs in Mathematics, vol. 78 (1983), Birkhäuser: Birkhäuser Basel, Boston, Stuttgart · Zbl 0546.46028 |
| [55] | Wang, J.; Zhang, X., Probabilistic approach for systems of second order quasi-linear parabolic PDEs, J. Math. Anal. Appl., 388, 2, 676-694 (2012) · Zbl 1246.35216 |
| [56] | Yasue, K., A variational principle for the Navier-Stokes equation, J. Funct. Anal., 51, 2, 133-141 (1983) · Zbl 0533.35075 |
| [57] | Yong, J., Finding adapted solutions of forward-backward stochastic differential equations: method of continuation, Probab. Theory Related Fields, 107, 537-572 (1997) · Zbl 0883.60053 |
| [58] | Zhang, X., A stochastic representation for backward incompressible Navier-Stokes equations, Probab. Theory Related Fields, 148, 305-332 (2010) · Zbl 1201.60070 |
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