doi: 10.3934/dcdsb.2018331

Stochastic one layer shallow water equations with Lévy noise

Department of Mathematics and The Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN 47405, USA

* Corresponding author: Roger Temam

Dedicated to Peter Eris Kloeden on the occasion of his 70th birthday

Received  December 2017 Revised  May 2018 Published  January 2019

This work investigates the existence of both martingale and pathwise solutions of the single layer shallow water equations on a bounded domain $ \mathcal{M} \subset \mathbb{R}^2 $ perturbed by a Lévy noise which may represent bursts of surface winds. The construction of both solutions are based on some truncation, the classical Faedo-Galerkin approximation scheme, a modified version of the Skorokhod representation theorem, stopping time arguments and anisotropic estimates.

Citation: Justin Cyr, Phuong Nguyen, Roger Temam. Stochastic one layer shallow water equations with Lévy noise. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2018331
References:
[1]

D. Aldous, Stopping times and tightness, Ann. Probability, 6 (1978), 335-340. doi: 10.1214/aop/1176995579.

[2]

A. Bensoussan, Stochastic Navier-Stokes equations, Acta Appl. Math., 38 (1995), 267-304. doi: 10.1007/BF00996149.

[3]

H. BessaihE. Hausenblas and P. A. Razafimandimby, Strong solutions to stochastic hydrodynamical systems with multiplicative noise of jump type, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1661-1697. doi: 10.1007/s00030-015-0339-9.

[4]

P. Billingsley, Convergence of Probability Measures, $2^{nd}$ edition, Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons, Inc., New York, 1999. doi: 10.1002/9780470316962.

[5]

Z. BrzeźniakE. Hausenblas and J. Zhu, 2D stochastic Navier-Stokes equations driven by jump noise, Nonlinear Anal., 79 (2013), 122-139. doi: 10.1016/j.na.2012.10.011.

[6]

Z. BrzeźniakE. Hausenblas and P. A. Razafimandimby, Stochastic reaction-diffusion equations driven by jump processes, Potential Anal., 49 (2018), 131-201. doi: 10.1007/s11118-017-9651-9.

[7]

Z. BrzeźniakW. Liu and J. Zhu, Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise, Nonlinear Anal. Real World Appl., 17 (2014), 283-310. doi: 10.1016/j.nonrwa.2013.12.005.

[8]

N. Chen and A. J. Majda, Simple dynamical models capturing the key features of the Central Pacific El Niño, P. Natl. Acad. Sci. USA, 113 (2016), 11732-11737.

[9]

Q. ChenC. Miao and Z. Zhang, On the well-posedness for the viscous shallow water equations, SIAM J. Math. Anal., 40 (2008), 443-474. doi: 10.1137/060660552.

[10]

J. Cyr, S. Tang and R. Temam, A comparison of two settings for stochastic integration with respect to Lévy processes in infinite dimensions, in Trends in Applications of Mathematics to Mechanics (eds. E. Rocca, U. Stefanelli, L. Truskinovsky, A. Visintin), Springer INdAM Series, 27 (2018), 289-373.

[11]

J. CyrS. Tang and R. Temam., The Euler equations of an inviscid incompressible fluid driven by a Lévy noise, Nonlinear Anal.-Real, 44 (2018), 173-222. doi: 10.1016/j.nonrwa.2018.04.002.

[12]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, $2^{nd}$ edition, volume 152 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2014. doi: 10.1017/CBO9781107295513.

[13]

D. J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes. Vol. II: General Theory and Structure, $2^{nd}$ edition, Probability and its Applications. Springer, New York, 2008. doi: 10.1007/978-0-387-49835-5.

[14]

A. DebusscheN. Glatt-Holtz and R. Temam, Local martingale and pathwise solutions for an abstract fluids model, Phys. D, 240 (2011), 1123-1144. doi: 10.1016/j.physd.2011.03.009.

[15]

A. DebusscheN. Glatt-HoltzR. Temam and M. Ziane, Global existence and regularity for the 3D stochastic primitive equations of the ocean and atmosphere with multiplicative white noise, Nonlinearity, 25 (2012), 2093-2118. doi: 10.1088/0951-7715/25/7/2093.

[16]

B. Di MartinoF. J. Chatelon and P. Orenga, The nonlinear Galerkin method applied to shallow water equations, Math. Models Methods Appl. Sci., 9 (1999), 825-854. doi: 10.1142/S0218202599000397.

[17]

Z. Dong and Y. Xie, Global solutions of stochastic 2D Navier-Stokes equations with Lévy noise, Sci. China Ser. A, 52 (2009), 1497-1524. doi: 10.1007/s11425-009-0124-5.

[18]

S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. doi: 10.1002/9780470316658.

[19]

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

[20]

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

[21]

N. Glatt-Holtz and M. Ziane, Strong pathwise solutions of the stochastic Navier-Stokes system, Adv. Differential Equations, 14 (2009), 567-600.

[22]

I. Gyöngy and N. Krylov, Existence of strong solutions for Itô's stochastic equations via approximations, Probab. Theory Related Fields, 105 (1996), 143-158. doi: 10.1007/BF01203833.

[23]

D. J. Higham and P. E. Kloeden, Strong convergence rates for backward Euler on a class of nonlinear jump-diffusion problems, J. Comput. Appl. Math., 205 (2007), 949-956. doi: 10.1016/j.cam.2006.03.039.

[24]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, $2^{nd}$ edition, volume 24 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989.

[25]

T. G. Kurtz, The Yamada-Watanabe-Engelbert theorem for general stochastic equations and inequalities, Electron. J. Probab., 12 (2007), 951-965. doi: 10.1214/EJP.v12-431.

[26]

J. LinkP. Nguyen and R. Temam, Local martingale solutions to the stochastic one layer shallow water equations, J. Math. Anal. Appl., 448 (2017), 93-139. doi: 10.1016/j.jmaa.2016.10.036.

[27]

M. Métivier, Semimartingales: A Course on Stochastic Processes, volume 2 of de Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin-New York, 1982.

[28]

M. Métivier, Stochastic Partial Differential Equations in Infinite-dimensional Spaces, Scuola Normale Superiore di Pisa. Quaderni. [Publications of the Scuola Normale Superiore of Pisa]. Scuola Normale Superiore, Pisa, 1988.

[29]

E. Motyl, Stochastic Navier-Stokes equations driven by Lévy noise in unbounded 3D domains, Potential Anal., 38 (2013), 863-912. doi: 10.1007/s11118-012-9300-2.

[30]

E. Motyl, Stochastic hydrodynamic-type evolution equations driven by Lévy noise in 3D unbounded domains—abstract framework and applications, Stochastic Process. Appl., 124 (2014), 2052-2097. doi: 10.1016/j.spa.2014.01.009.

[31]

P. Orenga, Un théorème d'existence de solutions d’un problème de shallow water, (French) [A theorem on the existence of solutions of a shallow-water problem], Arch. Rational Mech. Anal., 130 (1995), 183-204. doi: 10.1007/BF00375155.

[32]

S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations With Lévy Noise: An Evolution Equation Approach, volume 113 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9780511721373.

[33]

D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, $3^{rd}$ edition, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-662-06400-9.

[34]

B. Rüdiger, Stochastic integration with respect to compensated Poisson random measures on separable Banach spaces, Stoch. Stoch. Rep., 76 (2004), 213-242. doi: 10.1080/10451120410001704081.

[35]

L. Sundbye, Global existence for the Cauchy problem for the viscous shallow water equations, Rocky Mountain J. Math., 28 (1998), 1135-1152. doi: 10.1216/rmjm/1181071760.

[36]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, $2^{nd}$ edition, volume 66 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, first edition, 1983; second edition, 1995. doi: 10.1137/1.9781611970050.

[37]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001. doi: 10.1090/chel/343.

[38]

S. ThualA. J. MajdaN. Chen and S. N. Stechmann, Simple stochastic model for El Niño with westerly wind bursts, P. Natl. Acad. Sci. USA, 113 (2016), 10245-10250. doi: 10.1073/pnas.1612002113.

[39]

K. R. Tubbs and F. T.-C. Tsai, GPU accelerated lattice Boltzmann model for shallow water flow and mass transport, Internat. J. Numer. Methods Engrg., 86 (2011), 316-334. doi: 10.1002/nme.3066.

[40]

W. Wang and C.-J. Xu, The Cauchy problem for viscous shallow water equations, Rev. Mat. Iberoamericana, 21 (2005), 1-24. doi: 10.4171/RMI/412.

show all references

References:
[1]

D. Aldous, Stopping times and tightness, Ann. Probability, 6 (1978), 335-340. doi: 10.1214/aop/1176995579.

[2]

A. Bensoussan, Stochastic Navier-Stokes equations, Acta Appl. Math., 38 (1995), 267-304. doi: 10.1007/BF00996149.

[3]

H. BessaihE. Hausenblas and P. A. Razafimandimby, Strong solutions to stochastic hydrodynamical systems with multiplicative noise of jump type, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1661-1697. doi: 10.1007/s00030-015-0339-9.

[4]

P. Billingsley, Convergence of Probability Measures, $2^{nd}$ edition, Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons, Inc., New York, 1999. doi: 10.1002/9780470316962.

[5]

Z. BrzeźniakE. Hausenblas and J. Zhu, 2D stochastic Navier-Stokes equations driven by jump noise, Nonlinear Anal., 79 (2013), 122-139. doi: 10.1016/j.na.2012.10.011.

[6]

Z. BrzeźniakE. Hausenblas and P. A. Razafimandimby, Stochastic reaction-diffusion equations driven by jump processes, Potential Anal., 49 (2018), 131-201. doi: 10.1007/s11118-017-9651-9.

[7]

Z. BrzeźniakW. Liu and J. Zhu, Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise, Nonlinear Anal. Real World Appl., 17 (2014), 283-310. doi: 10.1016/j.nonrwa.2013.12.005.

[8]

N. Chen and A. J. Majda, Simple dynamical models capturing the key features of the Central Pacific El Niño, P. Natl. Acad. Sci. USA, 113 (2016), 11732-11737.

[9]

Q. ChenC. Miao and Z. Zhang, On the well-posedness for the viscous shallow water equations, SIAM J. Math. Anal., 40 (2008), 443-474. doi: 10.1137/060660552.

[10]

J. Cyr, S. Tang and R. Temam, A comparison of two settings for stochastic integration with respect to Lévy processes in infinite dimensions, in Trends in Applications of Mathematics to Mechanics (eds. E. Rocca, U. Stefanelli, L. Truskinovsky, A. Visintin), Springer INdAM Series, 27 (2018), 289-373.

[11]

J. CyrS. Tang and R. Temam., The Euler equations of an inviscid incompressible fluid driven by a Lévy noise, Nonlinear Anal.-Real, 44 (2018), 173-222. doi: 10.1016/j.nonrwa.2018.04.002.

[12]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, $2^{nd}$ edition, volume 152 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2014. doi: 10.1017/CBO9781107295513.

[13]

D. J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes. Vol. II: General Theory and Structure, $2^{nd}$ edition, Probability and its Applications. Springer, New York, 2008. doi: 10.1007/978-0-387-49835-5.

[14]

A. DebusscheN. Glatt-Holtz and R. Temam, Local martingale and pathwise solutions for an abstract fluids model, Phys. D, 240 (2011), 1123-1144. doi: 10.1016/j.physd.2011.03.009.

[15]

A. DebusscheN. Glatt-HoltzR. Temam and M. Ziane, Global existence and regularity for the 3D stochastic primitive equations of the ocean and atmosphere with multiplicative white noise, Nonlinearity, 25 (2012), 2093-2118. doi: 10.1088/0951-7715/25/7/2093.

[16]

B. Di MartinoF. J. Chatelon and P. Orenga, The nonlinear Galerkin method applied to shallow water equations, Math. Models Methods Appl. Sci., 9 (1999), 825-854. doi: 10.1142/S0218202599000397.

[17]

Z. Dong and Y. Xie, Global solutions of stochastic 2D Navier-Stokes equations with Lévy noise, Sci. China Ser. A, 52 (2009), 1497-1524. doi: 10.1007/s11425-009-0124-5.

[18]

S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. doi: 10.1002/9780470316658.

[19]

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

[20]

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

[21]

N. Glatt-Holtz and M. Ziane, Strong pathwise solutions of the stochastic Navier-Stokes system, Adv. Differential Equations, 14 (2009), 567-600.

[22]

I. Gyöngy and N. Krylov, Existence of strong solutions for Itô's stochastic equations via approximations, Probab. Theory Related Fields, 105 (1996), 143-158. doi: 10.1007/BF01203833.

[23]

D. J. Higham and P. E. Kloeden, Strong convergence rates for backward Euler on a class of nonlinear jump-diffusion problems, J. Comput. Appl. Math., 205 (2007), 949-956. doi: 10.1016/j.cam.2006.03.039.

[24]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, $2^{nd}$ edition, volume 24 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989.

[25]

T. G. Kurtz, The Yamada-Watanabe-Engelbert theorem for general stochastic equations and inequalities, Electron. J. Probab., 12 (2007), 951-965. doi: 10.1214/EJP.v12-431.

[26]

J. LinkP. Nguyen and R. Temam, Local martingale solutions to the stochastic one layer shallow water equations, J. Math. Anal. Appl., 448 (2017), 93-139. doi: 10.1016/j.jmaa.2016.10.036.

[27]

M. Métivier, Semimartingales: A Course on Stochastic Processes, volume 2 of de Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin-New York, 1982.

[28]

M. Métivier, Stochastic Partial Differential Equations in Infinite-dimensional Spaces, Scuola Normale Superiore di Pisa. Quaderni. [Publications of the Scuola Normale Superiore of Pisa]. Scuola Normale Superiore, Pisa, 1988.

[29]

E. Motyl, Stochastic Navier-Stokes equations driven by Lévy noise in unbounded 3D domains, Potential Anal., 38 (2013), 863-912. doi: 10.1007/s11118-012-9300-2.

[30]

E. Motyl, Stochastic hydrodynamic-type evolution equations driven by Lévy noise in 3D unbounded domains—abstract framework and applications, Stochastic Process. Appl., 124 (2014), 2052-2097. doi: 10.1016/j.spa.2014.01.009.

[31]

P. Orenga, Un théorème d'existence de solutions d’un problème de shallow water, (French) [A theorem on the existence of solutions of a shallow-water problem], Arch. Rational Mech. Anal., 130 (1995), 183-204. doi: 10.1007/BF00375155.

[32]

S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations With Lévy Noise: An Evolution Equation Approach, volume 113 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9780511721373.

[33]

D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, $3^{rd}$ edition, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-662-06400-9.

[34]

B. Rüdiger, Stochastic integration with respect to compensated Poisson random measures on separable Banach spaces, Stoch. Stoch. Rep., 76 (2004), 213-242. doi: 10.1080/10451120410001704081.

[35]

L. Sundbye, Global existence for the Cauchy problem for the viscous shallow water equations, Rocky Mountain J. Math., 28 (1998), 1135-1152. doi: 10.1216/rmjm/1181071760.

[36]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, $2^{nd}$ edition, volume 66 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, first edition, 1983; second edition, 1995. doi: 10.1137/1.9781611970050.

[37]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001. doi: 10.1090/chel/343.

[38]

S. ThualA. J. MajdaN. Chen and S. N. Stechmann, Simple stochastic model for El Niño with westerly wind bursts, P. Natl. Acad. Sci. USA, 113 (2016), 10245-10250. doi: 10.1073/pnas.1612002113.

[39]

K. R. Tubbs and F. T.-C. Tsai, GPU accelerated lattice Boltzmann model for shallow water flow and mass transport, Internat. J. Numer. Methods Engrg., 86 (2011), 316-334. doi: 10.1002/nme.3066.

[40]

W. Wang and C.-J. Xu, The Cauchy problem for viscous shallow water equations, Rev. Mat. Iberoamericana, 21 (2005), 1-24. doi: 10.4171/RMI/412.

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