American Institute of Mathematical Sciences

December  2012, 1(2): 355-392. doi: 10.3934/eect.2012.1.355

Martingale solutions for stochastic Navier-Stokes equations driven by Lévy noise

 1 Center for Decision, Risk, Controls & Signals Intelligence, Naval Postgraduate School, Monterey, CA-93943, United States, United States

Received  July 2012 Revised  August 2012 Published  October 2012

In this paper, we establish the solvability of martingale solutions for the stochastic Navier-Stokes equations with Itô-Lévy noise in bounded and unbounded domains in $\mathbb{R} ^d$,$d=2,3.$ The tightness criteria for the laws of a sequence of semimartingales is obtained from a theorem of Rebolledo as formulated by Metivier for the Lusin space valued processes. The existence of martingale solutions (in the sense of Stroock and Varadhan) relies on a generalization of Minty-Browder technique to stochastic case obtained from the local monotonicity of the drift term.
Citation: Kumarasamy Sakthivel, Sivaguru S. Sritharan. Martingale solutions for stochastic Navier-Stokes equations driven by Lévy noise. Evolution Equations & Control Theory, 2012, 1 (2) : 355-392. doi: 10.3934/eect.2012.1.355
References:
 [1] D. Aldous, Stopping times and tightness,, Ann. Probab., 6 (1978), 335. Google Scholar [2] D. Applebaum, "Lévy Processes and Stochastic Calculus,'', Cambridge University Press, (2009). Google Scholar [3] A. Bensoussan and R. Temam, Equations stochastiques du type Navier-Stokes,, J. Functional Analysis, 13 (1973), 195. doi: 10.1016/0022-1236(73)90045-1. Google Scholar [4] S. Chandrasekhar, "Hydrodynamic and Hydromagnetic Stability,'', Dover Publications, (1981). Google Scholar [5] E. DiBenedetto, "Degenerate Parabolic Equations,", Springer-Verlag, (1993). Google Scholar [6] Z. Dong and J. Zhai, Martingale solutions and Markov selection of stochastic 3D Navier-Stokes equations with jump,, J. Differential Equations, 250 (2011), 2737. doi: 10.1016/j.jde.2011.01.018. Google Scholar [7] S. N. Ethier and T. G. Kurtz, "Markov Processes Characterization and Convergence,'', John Wiley and Sons, (1986). Google Scholar [8] B. P. W. Fernando and S. S. Sritharan, Nonlinear filtering of stochastic Navier-Stokes equation with Itô-Lévy noise,, under review., (). Google Scholar [9] F. Flandoli, An introduction to 3D stochastic fluid dynamics, SPDE in hydrodynamic: recent progress and prospects,, Lecture Notes in Math., 1942 (2008), 51. Google Scholar [10] F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations,, Probab. Theory Related Fields, 102 (1995), 367. doi: 10.1007/BF01192467. Google Scholar [11] A. Ichikawa, Some inequalities for martingales and stochastic convolutions,, Stoch. Anal. Appl., 4 (1986), 329. Google Scholar [12] N. Jacob, "Pseudo Differential Operators and Markov Processes,'', Vol-III, (2005). Google Scholar [13] A. Joffe and M. Metivier, Weak convergence of sequences of semimartingales with applications to multitype branching processes,, Adv. in Appl. Probab., 18 (1986), 20. doi: 10.2307/1427238. Google Scholar [14] T. Komatsu, Markov processes associated with certain integro-differential operators,, Osaka J. Math., 10 (1973), 271. Google Scholar [15] T. Komatsu, On the martingale problem for generators of stable processes with perturbations,, Osaka J. Math., 21 (1984), 113. Google Scholar [16] G. Kallianpur and J. Xiong, "Stochastic Differential Equations in Infinite Dimensional Spaces,'', Lecture Notes - Monograph Series, 26 (1995). Google Scholar [17] O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,'', Gordon and Breach, (1969). Google Scholar [18] O. A. Ladyzhenskaya, The sixth millennium problem: Navier-Stokes equations, existence and smoothness,, Russian Math. Surveys, 58 (2003), 251. doi: 10.1070/RM2003v058n02ABEH000610. Google Scholar [19] J. Leray, Étude de diverses équations intégrales non linéaires et de quelques problémes que pose l'hydrodynamique,, J. Math. Pures Appl., 12 (1933), 1. Google Scholar [20] J. L. Menaldi and S.S. Sritharan, Stochastic 2-D Navier-Stokes equation,, Appl. Math. Optim., 46 (2002), 31. doi: 10.1007/s00245-002-0734-6. Google Scholar [21] M. Metivier, "Semimartingales: A Course on Stochastic Processes,'', Berlin, (1982). Google Scholar [22] M. Metivier, "Stochastic Partial Differential Equations in Infinite Dimensional Spaces,'', Scuola Normale Superiore, (1988). Google Scholar [23] M. Metivier and M. Viot, On weak solutions of stochastic partial differential equations, in "Stochastic Analysis", (Paris, 1322 (1988), 139. Google Scholar [24] Y. R. Ou and S. S. Sritharan, Analysis of regularized Navier-Stokes equations I & II,, Quart. Appl. Math., 49 (1991), 651. Google Scholar [25] K. R. Parthasarathy, "Probability Measures on Metric Spaces,'', Academic Press, (1967). Google Scholar [26] S. Peszat and J. Zabczyk, "Stochastic Partial Differential Equations with Lévy Noise,'', Cambridge University Press, (2007). Google Scholar [27] P. E. Protter, "Stochastic Integration and Differential Equations,'', Springer-Verlag, (2005). Google Scholar [28] L. Schwartz, "Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures,'', Published for the Tata Institute of Fundamental Research, (1973). Google Scholar [29] A. V. Skorohod, "Random Processes with Independent Increments,'', Kluwer Academic, (1991). Google Scholar [30] S. S. Sritharan, On the acceleration of viscous fluid through an unbounded channel,, J. Math. Anal. Appl., 168 (1992), 255. doi: 10.1016/0022-247X(92)90204-Q. Google Scholar [31] S. S. Sritharan, Deterministic and stochastic control of Navier-Stokes equation with linear, monotone, and hyperviscosities,, Appl. Math. Optim., 41 (2000), 255. doi: 10.1007/s0024599110140. Google Scholar [32] S. S. Sritharan and P. Sundar, The stochastic magneto-hydrodynamic system,, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 2 (1999), 241. Google Scholar [33] S. S. Sritharan, and P. Sundar, Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise,, Stochastic Process. Appl., 116 (2006), 1636. doi: 10.1016/j.spa.2006.04.001. Google Scholar [34] D. Stroock, Diffusion processes associated with Lévy generators,, Z. Wahrsch. Verw. Gebiete., 32 (1975), 209. doi: 10.1007/BF00532614. Google Scholar [35] D. Stroock and S. R. S. Varadhan, Diffusion processes with continuous coefficients I & II,, Comm. Pure Appl. Math., 22 (1969), 345. doi: 10.1002/cpa.3160220304. Google Scholar [36] D. Stroock and S. R. S. Varadhan, "Multidimensional Diffusion Processes,'', Springer-Verlag, (1979). Google Scholar [37] R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics,'', Springer-Verlag, (1997). Google Scholar [38] S. R. S. Varadhan, Limit theorems for sums of independent random variables with values in a Hilbert space,, Sankhya Ser. A, 24 (1962), 213. Google Scholar [39] V. S. Varadarajan, Measures on topological spaces,, Amer. Math. Soc. Transl. Ser. II, 48 (1965), 161. Google Scholar [40] M. Viot, Solutions faibles d'équations aux dérivées partielles non linéaires,, Thése Université Pierre et marie Curie, (1976). Google Scholar [41] M. J. Vishik and A. V. Fursikov, "Mathematical Problems in Statistical Hydromechanics,'', Kluwer, (1988). Google Scholar [42] T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations,, J. Math. Kyoto Univ., 11 (1971), 155. Google Scholar

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References:
 [1] D. Aldous, Stopping times and tightness,, Ann. Probab., 6 (1978), 335. Google Scholar [2] D. Applebaum, "Lévy Processes and Stochastic Calculus,'', Cambridge University Press, (2009). Google Scholar [3] A. Bensoussan and R. Temam, Equations stochastiques du type Navier-Stokes,, J. Functional Analysis, 13 (1973), 195. doi: 10.1016/0022-1236(73)90045-1. Google Scholar [4] S. Chandrasekhar, "Hydrodynamic and Hydromagnetic Stability,'', Dover Publications, (1981). Google Scholar [5] E. DiBenedetto, "Degenerate Parabolic Equations,", Springer-Verlag, (1993). Google Scholar [6] Z. Dong and J. Zhai, Martingale solutions and Markov selection of stochastic 3D Navier-Stokes equations with jump,, J. Differential Equations, 250 (2011), 2737. doi: 10.1016/j.jde.2011.01.018. Google Scholar [7] S. N. Ethier and T. G. Kurtz, "Markov Processes Characterization and Convergence,'', John Wiley and Sons, (1986). Google Scholar [8] B. P. W. Fernando and S. S. Sritharan, Nonlinear filtering of stochastic Navier-Stokes equation with Itô-Lévy noise,, under review., (). Google Scholar [9] F. Flandoli, An introduction to 3D stochastic fluid dynamics, SPDE in hydrodynamic: recent progress and prospects,, Lecture Notes in Math., 1942 (2008), 51. Google Scholar [10] F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations,, Probab. Theory Related Fields, 102 (1995), 367. doi: 10.1007/BF01192467. Google Scholar [11] A. Ichikawa, Some inequalities for martingales and stochastic convolutions,, Stoch. Anal. Appl., 4 (1986), 329. Google Scholar [12] N. Jacob, "Pseudo Differential Operators and Markov Processes,'', Vol-III, (2005). Google Scholar [13] A. Joffe and M. Metivier, Weak convergence of sequences of semimartingales with applications to multitype branching processes,, Adv. in Appl. Probab., 18 (1986), 20. doi: 10.2307/1427238. Google Scholar [14] T. Komatsu, Markov processes associated with certain integro-differential operators,, Osaka J. Math., 10 (1973), 271. Google Scholar [15] T. Komatsu, On the martingale problem for generators of stable processes with perturbations,, Osaka J. Math., 21 (1984), 113. Google Scholar [16] G. Kallianpur and J. Xiong, "Stochastic Differential Equations in Infinite Dimensional Spaces,'', Lecture Notes - Monograph Series, 26 (1995). Google Scholar [17] O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,'', Gordon and Breach, (1969). Google Scholar [18] O. A. Ladyzhenskaya, The sixth millennium problem: Navier-Stokes equations, existence and smoothness,, Russian Math. Surveys, 58 (2003), 251. doi: 10.1070/RM2003v058n02ABEH000610. Google Scholar [19] J. Leray, Étude de diverses équations intégrales non linéaires et de quelques problémes que pose l'hydrodynamique,, J. Math. Pures Appl., 12 (1933), 1. Google Scholar [20] J. L. Menaldi and S.S. Sritharan, Stochastic 2-D Navier-Stokes equation,, Appl. Math. Optim., 46 (2002), 31. doi: 10.1007/s00245-002-0734-6. Google Scholar [21] M. Metivier, "Semimartingales: A Course on Stochastic Processes,'', Berlin, (1982). Google Scholar [22] M. Metivier, "Stochastic Partial Differential Equations in Infinite Dimensional Spaces,'', Scuola Normale Superiore, (1988). Google Scholar [23] M. Metivier and M. Viot, On weak solutions of stochastic partial differential equations, in "Stochastic Analysis", (Paris, 1322 (1988), 139. Google Scholar [24] Y. R. Ou and S. S. Sritharan, Analysis of regularized Navier-Stokes equations I & II,, Quart. Appl. Math., 49 (1991), 651. Google Scholar [25] K. R. Parthasarathy, "Probability Measures on Metric Spaces,'', Academic Press, (1967). Google Scholar [26] S. Peszat and J. Zabczyk, "Stochastic Partial Differential Equations with Lévy Noise,'', Cambridge University Press, (2007). Google Scholar [27] P. E. Protter, "Stochastic Integration and Differential Equations,'', Springer-Verlag, (2005). Google Scholar [28] L. Schwartz, "Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures,'', Published for the Tata Institute of Fundamental Research, (1973). Google Scholar [29] A. V. Skorohod, "Random Processes with Independent Increments,'', Kluwer Academic, (1991). Google Scholar [30] S. S. Sritharan, On the acceleration of viscous fluid through an unbounded channel,, J. Math. Anal. Appl., 168 (1992), 255. doi: 10.1016/0022-247X(92)90204-Q. Google Scholar [31] S. S. Sritharan, Deterministic and stochastic control of Navier-Stokes equation with linear, monotone, and hyperviscosities,, Appl. Math. Optim., 41 (2000), 255. doi: 10.1007/s0024599110140. Google Scholar [32] S. S. Sritharan and P. Sundar, The stochastic magneto-hydrodynamic system,, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 2 (1999), 241. Google Scholar [33] S. S. Sritharan, and P. Sundar, Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise,, Stochastic Process. Appl., 116 (2006), 1636. doi: 10.1016/j.spa.2006.04.001. Google Scholar [34] D. Stroock, Diffusion processes associated with Lévy generators,, Z. Wahrsch. Verw. Gebiete., 32 (1975), 209. doi: 10.1007/BF00532614. Google Scholar [35] D. Stroock and S. R. S. Varadhan, Diffusion processes with continuous coefficients I & II,, Comm. Pure Appl. Math., 22 (1969), 345. doi: 10.1002/cpa.3160220304. Google Scholar [36] D. Stroock and S. R. S. Varadhan, "Multidimensional Diffusion Processes,'', Springer-Verlag, (1979). Google Scholar [37] R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics,'', Springer-Verlag, (1997). Google Scholar [38] S. R. S. Varadhan, Limit theorems for sums of independent random variables with values in a Hilbert space,, Sankhya Ser. A, 24 (1962), 213. Google Scholar [39] V. S. Varadarajan, Measures on topological spaces,, Amer. Math. Soc. Transl. Ser. II, 48 (1965), 161. Google Scholar [40] M. Viot, Solutions faibles d'équations aux dérivées partielles non linéaires,, Thése Université Pierre et marie Curie, (1976). Google Scholar [41] M. J. Vishik and A. V. Fursikov, "Mathematical Problems in Statistical Hydromechanics,'', Kluwer, (1988). Google Scholar [42] T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations,, J. Math. Kyoto Univ., 11 (1971), 155. Google Scholar
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