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.

[2]

D. Applebaum, "Lévy Processes and Stochastic Calculus,'', Cambridge University Press, (2009).

[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.

[4]

S. Chandrasekhar, "Hydrodynamic and Hydromagnetic Stability,'', Dover Publications, (1981).

[5]

E. DiBenedetto, "Degenerate Parabolic Equations,", Springer-Verlag, (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. 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, (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., 1942 (2008), 51.

[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.

[11]

A. Ichikawa, Some inequalities for martingales and stochastic convolutions,, Stoch. Anal. Appl., 4 (1986), 329.

[12]

N. Jacob, "Pseudo Differential Operators and Markov Processes,'', Vol-III, (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. doi: 10.2307/1427238.

[14]

T. Komatsu, Markov processes associated with certain integro-differential operators,, Osaka J. Math., 10 (1973), 271.

[15]

T. Komatsu, On the martingale problem for generators of stable processes with perturbations,, Osaka J. Math., 21 (1984), 113.

[16]

G. Kallianpur and J. Xiong, "Stochastic Differential Equations in Infinite Dimensional Spaces,'', Lecture Notes - Monograph Series, 26 (1995).

[17]

O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,'', Gordon and Breach, (1969).

[18]

O. A. Ladyzhenskaya, The sixth millennium problem: Navier-Stokes equations, existence and smoothness,, Russian Math. Surveys, 58 (2003), 251. 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.

[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.

[21]

M. Metivier, "Semimartingales: A Course on Stochastic Processes,'', Berlin, (1982).

[22]

M. Metivier, "Stochastic Partial Differential Equations in Infinite Dimensional Spaces,'', Scuola Normale Superiore, (1988).

[23]

M. Metivier and M. Viot, On weak solutions of stochastic partial differential equations, in "Stochastic Analysis", (Paris, 1322 (1988), 139.

[24]

Y. R. Ou and S. S. Sritharan, Analysis of regularized Navier-Stokes equations I & II,, Quart. Appl. Math., 49 (1991), 651.

[25]

K. R. Parthasarathy, "Probability Measures on Metric Spaces,'', Academic Press, (1967).

[26]

S. Peszat and J. Zabczyk, "Stochastic Partial Differential Equations with Lévy Noise,'', Cambridge University Press, (2007).

[27]

P. E. Protter, "Stochastic Integration and Differential Equations,'', Springer-Verlag, (2005).

[28]

L. Schwartz, "Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures,'', Published for the Tata Institute of Fundamental Research, (1973).

[29]

A. V. Skorohod, "Random Processes with Independent Increments,'', Kluwer Academic, (1991).

[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.

[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.

[32]

S. S. Sritharan and P. Sundar, The stochastic magneto-hydrodynamic system,, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 2 (1999), 241.

[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.

[34]

D. Stroock, Diffusion processes associated with Lévy generators,, Z. Wahrsch. Verw. Gebiete., 32 (1975), 209. 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. doi: 10.1002/cpa.3160220304.

[36]

D. Stroock and S. R. S. Varadhan, "Multidimensional Diffusion Processes,'', Springer-Verlag, (1979).

[37]

R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics,'', Springer-Verlag, (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.

[39]

V. S. Varadarajan, Measures on topological spaces,, Amer. Math. Soc. Transl. Ser. II, 48 (1965), 161.

[40]

M. Viot, Solutions faibles d'équations aux dérivées partielles non linéaires,, Thése Université Pierre et marie Curie, (1976).

[41]

M. J. Vishik and A. V. Fursikov, "Mathematical Problems in Statistical Hydromechanics,'', Kluwer, (1988).

[42]

T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations,, J. Math. Kyoto Univ., 11 (1971), 155.

show all references

References:
[1]

D. Aldous, Stopping times and tightness,, Ann. Probab., 6 (1978), 335.

[2]

D. Applebaum, "Lévy Processes and Stochastic Calculus,'', Cambridge University Press, (2009).

[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.

[4]

S. Chandrasekhar, "Hydrodynamic and Hydromagnetic Stability,'', Dover Publications, (1981).

[5]

E. DiBenedetto, "Degenerate Parabolic Equations,", Springer-Verlag, (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. 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, (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., 1942 (2008), 51.

[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.

[11]

A. Ichikawa, Some inequalities for martingales and stochastic convolutions,, Stoch. Anal. Appl., 4 (1986), 329.

[12]

N. Jacob, "Pseudo Differential Operators and Markov Processes,'', Vol-III, (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. doi: 10.2307/1427238.

[14]

T. Komatsu, Markov processes associated with certain integro-differential operators,, Osaka J. Math., 10 (1973), 271.

[15]

T. Komatsu, On the martingale problem for generators of stable processes with perturbations,, Osaka J. Math., 21 (1984), 113.

[16]

G. Kallianpur and J. Xiong, "Stochastic Differential Equations in Infinite Dimensional Spaces,'', Lecture Notes - Monograph Series, 26 (1995).

[17]

O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,'', Gordon and Breach, (1969).

[18]

O. A. Ladyzhenskaya, The sixth millennium problem: Navier-Stokes equations, existence and smoothness,, Russian Math. Surveys, 58 (2003), 251. 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.

[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.

[21]

M. Metivier, "Semimartingales: A Course on Stochastic Processes,'', Berlin, (1982).

[22]

M. Metivier, "Stochastic Partial Differential Equations in Infinite Dimensional Spaces,'', Scuola Normale Superiore, (1988).

[23]

M. Metivier and M. Viot, On weak solutions of stochastic partial differential equations, in "Stochastic Analysis", (Paris, 1322 (1988), 139.

[24]

Y. R. Ou and S. S. Sritharan, Analysis of regularized Navier-Stokes equations I & II,, Quart. Appl. Math., 49 (1991), 651.

[25]

K. R. Parthasarathy, "Probability Measures on Metric Spaces,'', Academic Press, (1967).

[26]

S. Peszat and J. Zabczyk, "Stochastic Partial Differential Equations with Lévy Noise,'', Cambridge University Press, (2007).

[27]

P. E. Protter, "Stochastic Integration and Differential Equations,'', Springer-Verlag, (2005).

[28]

L. Schwartz, "Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures,'', Published for the Tata Institute of Fundamental Research, (1973).

[29]

A. V. Skorohod, "Random Processes with Independent Increments,'', Kluwer Academic, (1991).

[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.

[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.

[32]

S. S. Sritharan and P. Sundar, The stochastic magneto-hydrodynamic system,, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 2 (1999), 241.

[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.

[34]

D. Stroock, Diffusion processes associated with Lévy generators,, Z. Wahrsch. Verw. Gebiete., 32 (1975), 209. 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. doi: 10.1002/cpa.3160220304.

[36]

D. Stroock and S. R. S. Varadhan, "Multidimensional Diffusion Processes,'', Springer-Verlag, (1979).

[37]

R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics,'', Springer-Verlag, (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.

[39]

V. S. Varadarajan, Measures on topological spaces,, Amer. Math. Soc. Transl. Ser. II, 48 (1965), 161.

[40]

M. Viot, Solutions faibles d'équations aux dérivées partielles non linéaires,, Thése Université Pierre et marie Curie, (1976).

[41]

M. J. Vishik and A. V. Fursikov, "Mathematical Problems in Statistical Hydromechanics,'', Kluwer, (1988).

[42]

T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations,, J. Math. Kyoto Univ., 11 (1971), 155.

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