\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

The stampacchia maximum principle for stochastic partial differential equations forced by lévy noise

Dedicated to Professor Tomás Caraballo on the occasion of his sixtieth birthday

Abstract Full Text(HTML) Related Papers Cited by
  • In this work, we investigate the existence of positive (martingale and pathwise) solutions of stochastic partial differential equations (SPDEs) driven by a Lévy noise. The proof relies on the use of truncation, following the Stampacchia approach to maximum principle. Among the applications, the positivity and boundedness results for the solutions of some biological systems and reaction diffusion equations are provided under suitable hypotheses, as well as some comparison theorems. This article improves the results of [15] where the authors only considered the case of the Wiener noise; even in this case we improve on [15] because the coefficients of the principal differential operator are now allowed to depend upon $ t $.

    Mathematics Subject Classification: Primary: 35R60, 60H15.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [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. BrzeźniakHa usenblas 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. Brzeniak, W. Liu and J. Zhu, Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise, Nonlinear Analysis: Real World Applications, 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, (2018), 289–373.
    [11] Justin Cyr, Phuong Nguyen, Sisi Tang and Roger Temam, Review of local and global existence results for stochastic PDEs with Lévy noise, to appear.
    [12] Justin Cyr, Phuong Nguyen, Sisi Tang and Roger Temam, Stochastic one layer shallow equation with Lévy noise, Discrete and Continuous Dynamical Systems B, Special issue in honor of Peter Kloeden, 24 (2019), 3765–3818.
    [13] Justin Cyr, Phuong Nguyen, Sisi Tang and Roger Temam, The Euler equations of an inviscid incompressible fluid driven a Lévy noise, Nonlinear Analysis: Real World Applications, 44 (2018), 173–222. doi: 10.1016/j.nonrwa.2018.04.002.
    [14] Mickaël D. Chekroun and Lionel J. Roques, Models of population dynamics under the influence of external perturbations: mathematical results, Comptes Rendus Mathematique, 343 (2006), 307–310. doi: 10.1016/j.crma.2006.07.012.
    [15] Mickaël D. Chekroun, Eunhee Park and Roger Temam, The Stampacchia maximum principle for stochastic partial differential equations and applications, J. Differential Equations, 260 (2016), 2926–2972. doi: 10.1016/j.jde.2015.10.022.
    [16] 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.
    [17] D. Appblebaum and J.-L. Wu, Stochastic partial differential equations driven by Lévy space time white noise, Random Oper. Stochastic Equations, 8 (2000), 245-259.  doi: 10.1515/rose.2000.8.3.245.
    [18] 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.
    [19] 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.
    [20] 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.
    [21] 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.
    [22] 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.
    [23] 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.
    [24] 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.
    [25] 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.
    [26] N. Glatt-Holtz and M. Ziane, Strong pathwise solutions of the stochastic Navier-Stokes system, Adv. Differential Equations, 14 (2009), 567-600. 
    [27] 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.
    [28] 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.
    [29] 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.
    [30] 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.
    [31] 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.
    [32] M. Métivier, Semimartingales: A Course on Stochastic Processes, volume 2 of de Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin-New York, 1982.
    [33] 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.
    [34] 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.
    [35] 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.
    [36] P. Orenga, Un théorème d'existence de solutions d'un problème de shallow water, Arch. Rational Mech. Anal., 130 (1995), 183-204.  doi: 10.1007/BF00375155.
    [37] 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.
    [38] M. Röckner and T. Zhang, Stochastic evolution equations of jump type: existence, uniqueness and large deviation principles, Potential Anal., 26 (2007), 255-279.  doi: 10.1007/s11118-006-9035-z.
    [39] 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.
    [40] 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.
    [41] 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.
    [42] 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.
    [43] Roger Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer Science & Business Media, Vol. 68, 2012. doi: 10.1007/978-1-4684-0313-8.
    [44] 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.
    [45] 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.
    [46] 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.
    [47] A. Truman and J.-L Wu, Stochastic Burgers equation with Lévy space-time white noise, in, Probabilistic Methods in Fluids doi: 10.1142/9789812703989_0020.
    [48] 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.
  • 加载中
SHARE

Article Metrics

HTML views(186) PDF downloads(248) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return