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
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show all references

References:
[1]

Ann. Probab., 6 (1978), 335-340.  Google Scholar

[2]

Cambridge University Press, $2^{nd}$ Edition, Cambridge, 2009.  Google Scholar

[3]

J. Functional Analysis, 13 (1973), 195-222. doi: 10.1016/0022-1236(73)90045-1.  Google Scholar

[4]

Dover Publications, Inc., New York, 1981. Google Scholar

[5]

Springer-Verlag, New York, 1993.  Google Scholar

[6]

J. Differential Equations, 250 (2011), 2737-2778. doi: 10.1016/j.jde.2011.01.018.  Google Scholar

[7]

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]

Lecture Notes in Math., Springer, Berlin, 1942 (2008), 51-150.  Google Scholar

[10]

Probab. Theory Related Fields, 102 (1995), 367-391. doi: 10.1007/BF01192467.  Google Scholar

[11]

Stoch. Anal. Appl., 4 (1986), 329-339.  Google Scholar

[12]

Vol-III, Imperial College Press, London, 2005.  Google Scholar

[13]

Adv. in Appl. Probab., 18 (1986), 20-65. doi: 10.2307/1427238.  Google Scholar

[14]

Osaka J. Math., 10 (1973), 271-303.  Google Scholar

[15]

Osaka J. Math., 21 (1984), 113-132.  Google Scholar

[16]

Lecture Notes - Monograph Series, 26, Institute of Mathematical Statistics, Hayward, 1995.  Google Scholar

[17]

Gordon and Breach, New York, 1969.  Google Scholar

[18]

Russian Math. Surveys, 58 (2003), 251-286. doi: 10.1070/RM2003v058n02ABEH000610.  Google Scholar

[19]

J. Math. Pures Appl., 12 (1933), 1-82. Google Scholar

[20]

Appl. Math. Optim., 46 (2002), 31-53. doi: 10.1007/s00245-002-0734-6.  Google Scholar

[21]

Berlin, DeGruyer, 1982.  Google Scholar

[22]

Scuola Normale Superiore, Pisa, 1988.  Google Scholar

[23]

(Paris, 1987), Lecture Notes in Math., 1322, Springer, Berlin, (1988), 139-150.  Google Scholar

[24]

Quart. Appl. Math., 49 (1991), 651-685, 687-728.  Google Scholar

[25]

Academic Press, New York, 1967.  Google Scholar

[26]

Cambridge University Press, Cambridge, 2007.  Google Scholar

[27]

Springer-Verlag, $2^{nd}$ Edition, Berlin, 2005.  Google Scholar

[28]

Published for the Tata Institute of Fundamental Research, Oxford University Press, 1973.  Google Scholar

[29]

Kluwer Academic, Dordrecht, 1991.  Google Scholar

[30]

J. Math. Anal. Appl., 168 (1992), 255-283. doi: 10.1016/0022-247X(92)90204-Q.  Google Scholar

[31]

Appl. Math. Optim., 41 (2000), 255-308. doi: 10.1007/s0024599110140.  Google Scholar

[32]

Infin. Dimens. Anal. Quantum Probab. Relat. Top., 2 (1999), 241-265.  Google Scholar

[33]

Stochastic Process. Appl., 116 (2006), 1636-1659. doi: 10.1016/j.spa.2006.04.001.  Google Scholar

[34]

Z. Wahrsch. Verw. Gebiete., 32 (1975), 209-244. doi: 10.1007/BF00532614.  Google Scholar

[35]

Comm. Pure Appl. Math., 22 (1969), 345-400, 479-530. doi: 10.1002/cpa.3160220304.  Google Scholar

[36]

Springer-Verlag, New York, 1979.  Google Scholar

[37]

Springer-Verlag, $2^{nd}$ Edition, New York, 1997.  Google Scholar

[38]

Sankhya Ser. A, 24 (1962), 213-238.  Google Scholar

[39]

Amer. Math. Soc. Transl. Ser. II, 48 (1965), 161-228. Google Scholar

[40]

Thése Université Pierre et marie Curie, Paris, 1976. Google Scholar

[41]

Kluwer, Boston, 1988. Google Scholar

[42]

J. Math. Kyoto Univ., 11 (1971), 155-167.  Google Scholar

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