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

Kumarasamy Sakthivel; Sivaguru S. Sritharan

doi:10.3934/eect.2012.1.355

Article Contents

2012, Volume 1, Issue 2: 355-392. Doi: 10.3934/eect.2012.1.355 open access

This issue Previous Article Energy methods for abstract nonlinear Schrödinger equations Next Article $L_p$-theory for a Cahn-Hilliard-Gurtin system

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

Kumarasamy Sakthivel^1, and
Sivaguru S. Sritharan^1,

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

Received: June 30, 2012

Revised: July 31, 2012

Published: November 2012

Abstract Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: 35Q30, 60G44, 60H15, 60G15, 60J75.

Citation:

Related Papers

Cited by

References

[1]	D. Aldous, Stopping times and tightness, Ann. Probab., 6 (1978), 335-340.
[2]	D. Applebaum, "Lévy Processes and Stochastic Calculus,'' Cambridge University Press, $2^{nd}$ Edition, Cambridge, 2009.
[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.
[4]	S. Chandrasekhar, "Hydrodynamic and Hydromagnetic Stability,'' Dover Publications, Inc., New York, 1981.
[5]	E. DiBenedetto, "Degenerate Parabolic Equations," Springer-Verlag, New York, 1993.
[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.
[7]	S. N. Ethier and T. G. Kurtz, "Markov Processes Characterization and Convergence,'' John Wiley and Sons, Inc., New York, 1986.
[8]	B. P. W. Fernando and S. S. Sritharan, Nonlinear filtering of stochastic Navier-Stokes equation with Itô-Lévy noise, under review.
[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.
[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.
[11]	A. Ichikawa, Some inequalities for martingales and stochastic convolutions, Stoch. Anal. Appl., 4 (1986), 329-339.
[12]	N. Jacob, "Pseudo Differential Operators and Markov Processes,'' Vol-III, Imperial College Press, London, 2005.
[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.
[14]	T. Komatsu, Markov processes associated with certain integro-differential operators, Osaka J. Math., 10 (1973), 271-303.
[15]	T. Komatsu, On the martingale problem for generators of stable processes with perturbations, Osaka J. Math., 21 (1984), 113-132.
[16]	G. Kallianpur and J. Xiong, "Stochastic Differential Equations in Infinite Dimensional Spaces,'' Lecture Notes - Monograph Series, 26, Institute of Mathematical Statistics, Hayward, 1995.
[17]	O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,'' Gordon and Breach, New York, 1969.
[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.
[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.
[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.
[21]	M. Metivier, "Semimartingales: A Course on Stochastic Processes,'' Berlin, DeGruyer, 1982.
[22]	M. Metivier, "Stochastic Partial Differential Equations in Infinite Dimensional Spaces,'' Scuola Normale Superiore, Pisa, 1988.
[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.
[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.
[25]	K. R. Parthasarathy, "Probability Measures on Metric Spaces,'' Academic Press, New York, 1967.
[26]	S. Peszat and J. Zabczyk, "Stochastic Partial Differential Equations with Lévy Noise,'' Cambridge University Press, Cambridge, 2007.
[27]	P. E. Protter, "Stochastic Integration and Differential Equations,'' Springer-Verlag, $2^{nd}$ Edition, Berlin, 2005.
[28]	L. Schwartz, "Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures,'' Published for the Tata Institute of Fundamental Research, Oxford University Press, 1973.
[29]	A. V. Skorohod, "Random Processes with Independent Increments,'' Kluwer Academic, Dordrecht, 1991.
[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.
[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.
[32]	S. S. Sritharan and P. Sundar, The stochastic magneto-hydrodynamic system, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 2 (1999), 241-265.
[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.
[34]	D. Stroock, Diffusion processes associated with Lévy generators, Z. Wahrsch. Verw. Gebiete., 32 (1975), 209-244.doi: 10.1007/BF00532614.
[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.
[36]	D. Stroock and S. R. S. Varadhan, "Multidimensional Diffusion Processes,'' Springer-Verlag, New York, 1979.
[37]	R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics,'' Springer-Verlag, $2^{nd}$ Edition, New York, 1997.
[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.
[39]	V. S. Varadarajan, Measures on topological spaces, Amer. Math. Soc. Transl. Ser. II, 48 (1965), 161-228.
[40]	M. Viot, Solutions faibles d'équations aux dérivées partielles non linéaires, Thése Université Pierre et marie Curie, Paris, 1976.
[41]	M. J. Vishik and A. V. Fursikov, "Mathematical Problems in Statistical Hydromechanics,'' Kluwer, Boston, 1988.
[42]	T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ., 11 (1971), 155-167.