September  2017, 6(3): 409-425. doi: 10.3934/eect.2017021

$\mathbb{L}^p-$solutions of the stochastic Navier-Stokes equations subject to Lévy noise with $\mathbb{L}^m(\mathbb{R}^m)$ initial data

1. 

Department of Mathematics and Statistics, Air Force Institute of Technology, 2950 Hobson Way, Wright Patterson Air Force Base, OH 45433, USA

2. 

Office of the Provost & Vice Chancellor, Air Force Institute of Technology, 2950 Hobson Way, Wright Patterson Air Force Base, OH 45433, USA

*Corresponding author: Sivaguru S. Sritharan

Received  February 2017 Revised  March 2017 Published  July 2017

In this work, we establish the local solvability of the stochastic Navier-Stokes equations in $\mathbb{R}^m$, $m≥ 2$, perturbed by Lévy noise in $\mathbb L^p-$spaces for $p∈[m,∞)$ with an $\mathbb L^m(\mathbb{R}^m)-$valued initial data.

Citation: Manil T. Mohan, Sivaguru S. Sritharan. $\mathbb{L}^p-$solutions of the stochastic Navier-Stokes equations subject to Lévy noise with $\mathbb{L}^m(\mathbb{R}^m)$ initial data. Evolution Equations & Control Theory, 2017, 6 (3) : 409-425. doi: 10.3934/eect.2017021
References:
[1]

A. Bensoussan and J. Frehse, Local solutions for stochastic Navier-Stokes equations, ESAIM: Mathematical Modelling and Numerical Analysis, 34 (2000), 241-273. doi: 10.1051/m2an:2000140. Google Scholar

[2]

Z. Brzeźniak and H. Long, A note on $γ-$radonifying and summing operators, Stochastic Analysis, Banach Center Publications, Institute of Mathematics, Polish Academy of Sciences, Warszawa, 105 (2015), 43-57. doi: 10.4064/bc105-0-3. Google Scholar

[3]

Z. Brzeźniak, E. Hausenblas and J. Zhu, Maximal inequality for stochastic convolutions driven by compensated Poisson random measures in Banach spaces, Ann. Inst. Henri Poincar Probab. Stat., 53 (2017), 937-956, arXiv: 1005.1600v4, 2015. doi: 10.1214/16-AIHP743. Google Scholar

[4]

G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, U. K., 1996. doi: 10.1017/CBO9780511662829. Google Scholar

[5]

E. B. FabesB. F. Jones and N. M. Riviere, The Initial value problem for the Navier-Stokes equations with data in $\mathbb L^p$, Archive for Rational Mechanics and Analysis, 45 (1972), 222-240. doi: 10.1007/BF00281533. Google Scholar

[6]

B. P. W. Fernando and S. S. Sritharan, Nonlinear filtering of stochastic Navier-Stokes equation with Itô-Lévy noise, Stochastic Analysis and Applications, 31 (2013), 381-426. doi: 10.1080/07362994.2013.759482. Google Scholar

[7]

B. P. W. FernandoB. Rüdiger and S. S. Sritharan, Mild solutions of stochastic Navier-Stokes equation with jump noise in $\mathbb L^p-$spaces, Mathematische Nachrichten, 288 (2015), 1615-1621. doi: 10.1002/mana.201300248. Google Scholar

[8]

F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probability Theory and Related Fields, 102 (1995), 367-391. doi: 10.1007/BF01192467. Google Scholar

[9]

T. Kato, Strong $\mathbb L^p-$solutions of the Navier-Stokes equation in $\mathbb{R}^m$, with applications to weak solutions, Mathematische Zeitschrift, 187 (1984), 471-480. doi: 10.1007/BF01174182. Google Scholar

[10]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Silverman and John Chu. Mathematics and its Applications, Vol. 2 Gordon and Breach, Science Publishers, New York-London-Paris, 1969. Google Scholar

[11]

J. L. Menaldi and S. S. Sritharan, Stochastic $2$-D Navier-Stokes equation, Applied Mathematics and Optimization, 46 (2002), 31-53. doi: 10.1007/s00245-002-0734-6. Google Scholar

[12]

M. T. Mohan and S. S. Sritharan, Stochastic Euler equations of fluid dynamics with Lévy noise, Asymptotic Analysis, 99 (2016), 67-103. doi: 10.3233/ASY-161376. Google Scholar

[13]

M. T. Mohan and S. S. Sritharan, Stochastic quasilinear evolution equations in UMD Banach spaces Published online in Mathematische Nachrichten, 2017. doi: 10.1002/mana.201600015. Google Scholar

[14]

J. M. A. M. van Neerven, γ−radonifying operators -a survey, In The AMSI-ANU Workshop on Spectral Theory and Harmonic Analysis, volume 44 of Proc. Centre Math. Appl. Austral. Nat. Univ. Austral. Nat. Univ., Canberra, (2010), 1-61. Google Scholar

[15]

J. M. A. M. van Neerven and J. Zhu, A maximal inequality for stochastic convolutions in $2-$smooth Banach spaces, Electronic Communications in Probability, 16 (2011), 689-705. doi: 10.1214/ECP.v16-1677. Google Scholar

[16]

G. Pisier, Probabilistic methods in the geometry of Banach space, Probability and Analysis, Volume 1206 of the series Lecture Notes in Mathematics, Springer Verlag, Berlin, 167-241, 1986. doi: 10.1007/BFb0076302. Google Scholar

[17]

K. Sakthivel and S. S. Sritharan, Martingale solutions for stochastic Navier-Stokes equations driven by Lévy noise, Evolution Equations and Control Theory, 1 (2012), 355-392. doi: 10.3934/eect.2012.1.355. Google Scholar

[18]

P. E. Sobolevskii, An investigation of the Navier-Stokes equations by the methods of the theory of parabolic equations in Banach spaces, Dokl. Akad. Nauk SSSR (Russian), 156 (1964), 745-748; SovietMath Dokl (English), 7 (1964), 720-723. Google Scholar

[19]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, New Jersey, 1970. Google Scholar

[20]

M. E. Taylor, Partial Differential Equations Ⅲ, Nonlinear Equations, Springer, New York, 2011. doi: 10.1007/978-1-4419-7049-7. Google Scholar

[21]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. doi: 10.1090/chel/343. Google Scholar

show all references

References:
[1]

A. Bensoussan and J. Frehse, Local solutions for stochastic Navier-Stokes equations, ESAIM: Mathematical Modelling and Numerical Analysis, 34 (2000), 241-273. doi: 10.1051/m2an:2000140. Google Scholar

[2]

Z. Brzeźniak and H. Long, A note on $γ-$radonifying and summing operators, Stochastic Analysis, Banach Center Publications, Institute of Mathematics, Polish Academy of Sciences, Warszawa, 105 (2015), 43-57. doi: 10.4064/bc105-0-3. Google Scholar

[3]

Z. Brzeźniak, E. Hausenblas and J. Zhu, Maximal inequality for stochastic convolutions driven by compensated Poisson random measures in Banach spaces, Ann. Inst. Henri Poincar Probab. Stat., 53 (2017), 937-956, arXiv: 1005.1600v4, 2015. doi: 10.1214/16-AIHP743. Google Scholar

[4]

G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, U. K., 1996. doi: 10.1017/CBO9780511662829. Google Scholar

[5]

E. B. FabesB. F. Jones and N. M. Riviere, The Initial value problem for the Navier-Stokes equations with data in $\mathbb L^p$, Archive for Rational Mechanics and Analysis, 45 (1972), 222-240. doi: 10.1007/BF00281533. Google Scholar

[6]

B. P. W. Fernando and S. S. Sritharan, Nonlinear filtering of stochastic Navier-Stokes equation with Itô-Lévy noise, Stochastic Analysis and Applications, 31 (2013), 381-426. doi: 10.1080/07362994.2013.759482. Google Scholar

[7]

B. P. W. FernandoB. Rüdiger and S. S. Sritharan, Mild solutions of stochastic Navier-Stokes equation with jump noise in $\mathbb L^p-$spaces, Mathematische Nachrichten, 288 (2015), 1615-1621. doi: 10.1002/mana.201300248. Google Scholar

[8]

F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probability Theory and Related Fields, 102 (1995), 367-391. doi: 10.1007/BF01192467. Google Scholar

[9]

T. Kato, Strong $\mathbb L^p-$solutions of the Navier-Stokes equation in $\mathbb{R}^m$, with applications to weak solutions, Mathematische Zeitschrift, 187 (1984), 471-480. doi: 10.1007/BF01174182. Google Scholar

[10]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Silverman and John Chu. Mathematics and its Applications, Vol. 2 Gordon and Breach, Science Publishers, New York-London-Paris, 1969. Google Scholar

[11]

J. L. Menaldi and S. S. Sritharan, Stochastic $2$-D Navier-Stokes equation, Applied Mathematics and Optimization, 46 (2002), 31-53. doi: 10.1007/s00245-002-0734-6. Google Scholar

[12]

M. T. Mohan and S. S. Sritharan, Stochastic Euler equations of fluid dynamics with Lévy noise, Asymptotic Analysis, 99 (2016), 67-103. doi: 10.3233/ASY-161376. Google Scholar

[13]

M. T. Mohan and S. S. Sritharan, Stochastic quasilinear evolution equations in UMD Banach spaces Published online in Mathematische Nachrichten, 2017. doi: 10.1002/mana.201600015. Google Scholar

[14]

J. M. A. M. van Neerven, γ−radonifying operators -a survey, In The AMSI-ANU Workshop on Spectral Theory and Harmonic Analysis, volume 44 of Proc. Centre Math. Appl. Austral. Nat. Univ. Austral. Nat. Univ., Canberra, (2010), 1-61. Google Scholar

[15]

J. M. A. M. van Neerven and J. Zhu, A maximal inequality for stochastic convolutions in $2-$smooth Banach spaces, Electronic Communications in Probability, 16 (2011), 689-705. doi: 10.1214/ECP.v16-1677. Google Scholar

[16]

G. Pisier, Probabilistic methods in the geometry of Banach space, Probability and Analysis, Volume 1206 of the series Lecture Notes in Mathematics, Springer Verlag, Berlin, 167-241, 1986. doi: 10.1007/BFb0076302. Google Scholar

[17]

K. Sakthivel and S. S. Sritharan, Martingale solutions for stochastic Navier-Stokes equations driven by Lévy noise, Evolution Equations and Control Theory, 1 (2012), 355-392. doi: 10.3934/eect.2012.1.355. Google Scholar

[18]

P. E. Sobolevskii, An investigation of the Navier-Stokes equations by the methods of the theory of parabolic equations in Banach spaces, Dokl. Akad. Nauk SSSR (Russian), 156 (1964), 745-748; SovietMath Dokl (English), 7 (1964), 720-723. Google Scholar

[19]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, New Jersey, 1970. Google Scholar

[20]

M. E. Taylor, Partial Differential Equations Ⅲ, Nonlinear Equations, Springer, New York, 2011. doi: 10.1007/978-1-4419-7049-7. Google Scholar

[21]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. doi: 10.1090/chel/343. Google Scholar

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