# 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:
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Kurtz, "Markov Processes Characterization and Convergence,'' John Wiley and Sons, Inc., New York, 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., Springer, Berlin, 1942 (2008), 51-150.  Google Scholar [10] F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), 367-391. doi: 10.1007/BF01192467.  Google Scholar [11] A. Ichikawa, Some inequalities for martingales and stochastic convolutions, Stoch. Anal. Appl., 4 (1986), 329-339.  Google Scholar [12] N. Jacob, "Pseudo Differential Operators and Markov Processes,'' Vol-III, Imperial College Press, London, 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), 20-65. doi: 10.2307/1427238.  Google Scholar [14] T. Komatsu, Markov processes associated with certain integro-differential operators, Osaka J. Math., 10 (1973), 271-303.  Google Scholar [15] T. Komatsu, On the martingale problem for generators of stable processes with perturbations, Osaka J. Math., 21 (1984), 113-132.  Google Scholar [16] G. Kallianpur and J. Xiong, "Stochastic Differential Equations in Infinite Dimensional Spaces,'' Lecture Notes - Monograph Series, 26, Institute of Mathematical Statistics, Hayward, 1995.  Google Scholar [17] O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,'' Gordon and Breach, New York, 1969.  Google Scholar [18] O. A. Ladyzhenskaya, The sixth millennium problem: Navier-Stokes equations, existence and smoothness, Russian Math. 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Appl. Math., 49 (1991), 651-685, 687-728.  Google Scholar [25] K. R. Parthasarathy, "Probability Measures on Metric Spaces,'' Academic Press, New York, 1967.  Google Scholar [26] S. Peszat and J. Zabczyk, "Stochastic Partial Differential Equations with Lévy Noise,'' Cambridge University Press, Cambridge, 2007.  Google Scholar [27] P. E. Protter, "Stochastic Integration and Differential Equations,'' Springer-Verlag, $2^{nd}$ Edition, Berlin, 2005.  Google Scholar [28] L. Schwartz, "Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures,'' Published for the Tata Institute of Fundamental Research, Oxford University Press, 1973.  Google Scholar [29] A. V. Skorohod, "Random Processes with Independent Increments,'' Kluwer Academic, Dordrecht, 1991.  Google Scholar [30] S. S. Sritharan, On the acceleration of viscous fluid through an unbounded channel, J. Math. Anal. Appl., 168 (1992), 255-283. 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), 255-308. 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), 241-265.  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), 1636-1659. 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), 209-244. 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), 345-400, 479-530. doi: 10.1002/cpa.3160220304.  Google Scholar [36] D. Stroock and S. R. S. Varadhan, "Multidimensional Diffusion Processes,'' Springer-Verlag, New York, 1979.  Google Scholar [37] R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics,'' Springer-Verlag, $2^{nd}$ Edition, New York, 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), 213-238.  Google Scholar [39] V. S. Varadarajan, Measures on topological spaces, Amer. Math. Soc. Transl. Ser. II, 48 (1965), 161-228. Google Scholar [40] M. Viot, Solutions faibles d'équations aux dérivées partielles non linéaires, Thése Université Pierre et marie Curie, Paris, 1976. Google Scholar [41] M. J. Vishik and A. V. Fursikov, "Mathematical Problems in Statistical Hydromechanics,'' Kluwer, Boston, 1988. Google Scholar [42] T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ., 11 (1971), 155-167.  Google Scholar

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##### References:
 [1] D. Aldous, Stopping times and tightness, Ann. Probab., 6 (1978), 335-340.  Google Scholar [2] D. Applebaum, "Lévy Processes and Stochastic Calculus,'' Cambridge University Press, $2^{nd}$ Edition, Cambridge, 2009.  Google Scholar [3] A. Bensoussan and R. Temam, Equations stochastiques du type Navier-Stokes, J. Functional Analysis, 13 (1973), 195-222. doi: 10.1016/0022-1236(73)90045-1.  Google Scholar [4] S. Chandrasekhar, "Hydrodynamic and Hydromagnetic Stability,'' Dover Publications, Inc., New York, 1981. Google Scholar [5] E. DiBenedetto, "Degenerate Parabolic Equations," Springer-Verlag, New York, 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), 2737-2778. 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, Inc., New York, 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., Springer, Berlin, 1942 (2008), 51-150.  Google Scholar [10] F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), 367-391. doi: 10.1007/BF01192467.  Google Scholar [11] A. Ichikawa, Some inequalities for martingales and stochastic convolutions, Stoch. Anal. Appl., 4 (1986), 329-339.  Google Scholar [12] N. Jacob, "Pseudo Differential Operators and Markov Processes,'' Vol-III, Imperial College Press, London, 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), 20-65. doi: 10.2307/1427238.  Google Scholar [14] T. Komatsu, Markov processes associated with certain integro-differential operators, Osaka J. Math., 10 (1973), 271-303.  Google Scholar [15] T. Komatsu, On the martingale problem for generators of stable processes with perturbations, Osaka J. Math., 21 (1984), 113-132.  Google Scholar [16] G. Kallianpur and J. Xiong, "Stochastic Differential Equations in Infinite Dimensional Spaces,'' Lecture Notes - Monograph Series, 26, Institute of Mathematical Statistics, Hayward, 1995.  Google Scholar [17] O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,'' Gordon and Breach, New York, 1969.  Google Scholar [18] O. A. Ladyzhenskaya, The sixth millennium problem: Navier-Stokes equations, existence and smoothness, Russian Math. Surveys, 58 (2003), 251-286. 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), 1-82. Google Scholar [20] J. L. Menaldi and S.S. Sritharan, Stochastic 2-D Navier-Stokes equation, Appl. Math. Optim., 46 (2002), 31-53. doi: 10.1007/s00245-002-0734-6.  Google Scholar [21] M. Metivier, "Semimartingales: A Course on Stochastic Processes,'' Berlin, DeGruyer, 1982.  Google Scholar [22] M. Metivier, "Stochastic Partial Differential Equations in Infinite Dimensional Spaces,'' Scuola Normale Superiore, Pisa, 1988.  Google Scholar [23] M. Metivier and M. Viot, On weak solutions of stochastic partial differential equations, in "Stochastic Analysis" (Paris, 1987), Lecture Notes in Math., 1322, Springer, Berlin, (1988), 139-150.  Google Scholar [24] Y. R. Ou and S. S. Sritharan, Analysis of regularized Navier-Stokes equations I & II, Quart. Appl. Math., 49 (1991), 651-685, 687-728.  Google Scholar [25] K. R. Parthasarathy, "Probability Measures on Metric Spaces,'' Academic Press, New York, 1967.  Google Scholar [26] S. Peszat and J. Zabczyk, "Stochastic Partial Differential Equations with Lévy Noise,'' Cambridge University Press, Cambridge, 2007.  Google Scholar [27] P. E. Protter, "Stochastic Integration and Differential Equations,'' Springer-Verlag, $2^{nd}$ Edition, Berlin, 2005.  Google Scholar [28] L. Schwartz, "Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures,'' Published for the Tata Institute of Fundamental Research, Oxford University Press, 1973.  Google Scholar [29] A. V. Skorohod, "Random Processes with Independent Increments,'' Kluwer Academic, Dordrecht, 1991.  Google Scholar [30] S. S. Sritharan, On the acceleration of viscous fluid through an unbounded channel, J. Math. Anal. Appl., 168 (1992), 255-283. 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), 255-308. 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), 241-265.  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), 1636-1659. 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), 209-244. 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), 345-400, 479-530. doi: 10.1002/cpa.3160220304.  Google Scholar [36] D. Stroock and S. R. S. Varadhan, "Multidimensional Diffusion Processes,'' Springer-Verlag, New York, 1979.  Google Scholar [37] R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics,'' Springer-Verlag, $2^{nd}$ Edition, New York, 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), 213-238.  Google Scholar [39] V. S. Varadarajan, Measures on topological spaces, Amer. Math. Soc. Transl. Ser. II, 48 (1965), 161-228. Google Scholar [40] M. Viot, Solutions faibles d'équations aux dérivées partielles non linéaires, Thése Université Pierre et marie Curie, Paris, 1976. Google Scholar [41] M. J. Vishik and A. V. Fursikov, "Mathematical Problems in Statistical Hydromechanics,'' Kluwer, Boston, 1988. Google Scholar [42] T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ., 11 (1971), 155-167.  Google Scholar
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