Nonlinear stochastic parabolic partial differential equations with a monotone operator of the Ladyzenskaya-Smagorinsky type, driven by a Lévy noise. (English) Zbl 1522.35609
Summary: The aim of this article is to show the global existence of both martingale and pathwise solutions of stochastic equations with a monotone operator, of the Ladyzenskaya-Smagorinsky type, driven by a general Lévy noise. The classical approach based on using directly the Galerkin approximation is not valid. Instead, our approach is based on using appropriate approximations for the monotone operator, Galerkin approximations and on the theory of martingale solutions.
MSC:
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 47H05 | Monotone operators and generalizations |
| 60G51 | Processes with independent increments; Lévy processes |
Keywords:
stochastic partial differential equations; Lévy noise; Ladyzenskaya-Smagorinsky equations; monotone operatorsReferences:
| [1] | Albeverio, S.; Brzeźniak, Z.; Wu, J.-L., Existence of global solutions and invariant measures for stochastic differential equations driven by Poisson type noise with non-Lipschitz coefficients, J. Math. Anal. Appl., 371, 1, 309-322 (2010) · Zbl 1197.60050 |
| [2] | Aldous, D., Stopping times and tightness, Ann. Probab., 6, 2, 335-340 (1978) · Zbl 0391.60007 |
| [3] | Aldous, D., Stopping times and tightness. II, Ann. Probab., 17, 2, 586-595 (1989) · Zbl 0686.60036 |
| [4] | Aronson, D. G., The porous medium equation, (Nonlinear Diffusion Problems. Nonlinear Diffusion Problems, Montecatini Terme, 1985. Nonlinear Diffusion Problems. Nonlinear Diffusion Problems, Montecatini Terme, 1985, Lecture Notes in Math., vol. 1224 (1986), Springer: Springer Berlin), 1-46 · Zbl 0626.76097 |
| [5] | Bensoussan, A., Stochastic Navier-Stokes equations, Acta Appl. Math., 38, 3, 267-304 (1995) · Zbl 0836.35115 |
| [6] | Billingsley, P., Probability and Measure, Wiley Series in Probability and Mathematical Statistics (1995), John Wiley & Sons, Inc.: John Wiley & Sons, Inc. New York, A Wiley-Interscience Publication · Zbl 0822.60002 |
| [7] | Billingsley, P., Convergence of Probability Measures, Wiley Series in Probability and Statistics: Probability and Statistics (1999), John Wiley & Sons, Inc.: John Wiley & Sons, Inc. New York, A Wiley-Interscience Publication · Zbl 0172.21201 |
| [8] | Brézis, H., Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Mathematics Studies, vol. 5 (1973), North-Holland Publishing Co., American Elsevier Publishing Co., Inc.: North-Holland Publishing Co., American Elsevier Publishing Co., Inc. Amsterdam-London, New York · Zbl 0252.47055 |
| [9] | Brézis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext (2011), Springer: Springer New York · Zbl 1220.46002 |
| [10] | Browder, F. E., Pseudo-monotone operators and nonlinear elliptic boundary value problems on unbounded domains, Proc. Natl. Acad. Sci. USA, 74, 7, 2659-2661 (1977) · Zbl 0358.35034 |
| [11] | Z. Brzeźniak, E. Hausenblas, P. Razafimandimby, Martingale solutions for stochastic equation of reaction diffusion type driven by Lévy noise or Poisson random measure, ArXiv e-prints, February 2014. |
| [12] | Brzeźniak, Z.; Liu, W.; Zhu, J., Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise, Nonlinear Anal., Real World Appl., 17, 283-310 (2014) · Zbl 1310.60091 |
| [13] | Chen, N.; Majda, A. J., Simple dynamical models capturing the key features of the central Pacific El Niño, Proc. Natl. Acad. Sci., 113, 42, 11732-11737 (2016) |
| [14] | Cyr, J.; Nguyen, P.; Tang, S.; Temam, R., Review of local and global existence results for stochastic PDEs with Lévy noise, Discrete Contin. Dyn. Syst., 40, 10, 5639-5710 (2020) · Zbl 1447.35003 |
| [15] | Cyr, J.; Nguyen, P.; Temam, R., Stochastic one layer shallow water equations with Lévy noise, Discrete Contin. Dyn. Syst., Ser. B, 24, 8, 3765-3818 (2019) · Zbl 1420.35479 |
| [16] | Cyr, J.; Tang, S.; Temam, R., A comparison of two settings for stochastic integration with respect to Lévy processes in infinite dimensions, (Trends in Applications of Mathematics to Mechanics. Trends in Applications of Mathematics to Mechanics, Springer INdAM Ser., vol. 27 (2018), Springer: Springer Cham), 289-373 · Zbl 1403.60043 |
| [17] | Cyr, J.; Tang, S.; Temam, R., The Euler equations of an inviscid incompressible fluid driven by a Lévy noise, Nonlinear Anal., Real World Appl., 44, 173-222 (2018) · Zbl 1406.35246 |
| [18] | Daley, D. J.; Vere-Jones, D., An Introduction to the Theory of Point Processes. Vol. II: General Theory and Structure, Probability and Its Applications (New York) (2008), Springer: Springer New York · Zbl 1159.60003 |
| [19] | Debussche, A.; Glatt-Holtz, N.; Temam, R., Local martingale and pathwise solutions for an abstract fluids model, Physica D, 240, 14-15, 1123-1144 (2011) · Zbl 1230.60065 |
| [20] | Debussche, A.; Högele, M.; Imkeller, P., The Dynamics of Nonlinear Reaction-Diffusion Equations with Small Lévy Noise, Lecture Notes in Mathematics, vol. 2085 (2013), Springer: Springer Cham · Zbl 1321.60004 |
| [21] | Ditlevsen, P. D., Observation of α-stable noise induced millennial climate changes from an ice-core record, Geophys. Res. Lett., 26, 10, 1441-1444 (1999) |
| [22] | Douanla, H.; Woukeng, J. L., Almost periodic homogenization of a generalized Ladyzhenkskaya model for incompressible viscous flow, Probl. Math. Anal., 189, 68, 431-458 (2013) · Zbl 1268.76012 |
| [23] | Ekeland, I.; Temam, R., Convex Analysis and Variational Problems, Series Classics in Applied Mathematics, vol. 28 (1999), Society for Industrial and Applied Mathematics, John Wiley & Sons, Inc.: Society for Industrial and Applied Mathematics, John Wiley & Sons, Inc. Philadelphia, PA, New York · Zbl 0939.49002 |
| [24] | Ewald, B. D.; Penland, C., On modelling physical systems with stochastic models: diffusion versus Lévy processes, Philos. Trans. R. Soc., Lond. Ser. A, Math. Phys. Eng. Sci., 366, 1875, 2457-2476 (2008) · Zbl 1153.60383 |
| [25] | Flandoli, F.; Gątarek, D., Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Relat. Fields, 102, 3, 367-391 (1995) · Zbl 0831.60072 |
| [26] | Glatt-Holtz, N.; Ziane, M., Strong pathwise solutions of the stochastic Navier-Stokes system, Adv. Differ. Equ., 14, 5-6, 567-600 (2009) · Zbl 1195.60090 |
| [27] | Glowinski, R.; Marrocco, A., Sur l’approximation, par éléments finis d’ordre un, et la résolution, par pénalisation-dualité, d’une classe de problèmes de Dirichlet non linéaires, Rev. Fr. Autom. Inform. Rech. Opér., Sér. Rouge Anal. Num., 9, R-2, 41-76 (1975) · Zbl 0368.65053 |
| [28] | Gyöngy, I.; Krylov, N., Existence of strong solutions for Itô’s stochastic equations via approximations, Probab. Theory Relat. Fields, 105, 2, 143-158 (1996) · Zbl 0847.60038 |
| [29] | Ikeda, N.; Nagasawa, M.; Watanabe, S., A construction of Markov processes by piecing out, Proc. Jpn. Acad., 42, 370-375 (1966) · Zbl 0178.53401 |
| [30] | Ikeda, N.; Watanabe, S., Stochastic Differential Equations and Diffusion Processes, North-Holland Mathematical Library, vol. 24 (1989), North-Holland Publishing Co., Kodansha, Ltd.: North-Holland Publishing Co., Kodansha, Ltd. Amsterdam, Tokyo · Zbl 0684.60040 |
| [31] | Krylov, N. V.; Röckner, M.; Zabczyk, J., Stochastic PDE’s and Kolmogorov equations in infinite dimensions, (Lectures Given at the 2nd C.I.M.E. Session Held in Cetraro, August 24-September 1, 1998, Edited by G. Da Prato, Fondazione CIME/CIME Foundation. Lectures Given at the 2nd C.I.M.E. Session Held in Cetraro, August 24-September 1, 1998, Edited by G. Da Prato, Fondazione CIME/CIME Foundation, Lecture Notes in Mathematics, vol. 1715 (1999), Springer-Verlag, Centro Internazionale Matematico Estivo (C.I.M.E.): Springer-Verlag, Centro Internazionale Matematico Estivo (C.I.M.E.) Berlin, Florence) · Zbl 0943.60070 |
| [32] | Krylov, N. V.; Rozovskiĭ, B. L., Stochastic Evolution Equations, Current Problems in Mathematics, vol. 14, 71-147 (1979), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii: Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii Moscow, 256 (in Russian) |
| [33] | Ladyženskaja, O. A., New equations for the description of the motions of viscous incompressible fluids, and global solvability for their boundary value problems, Tr. Mat. Inst. Steklova, 102, 85-104 (1967) · Zbl 0202.37802 |
| [34] | Ladyženskaja, O. A., Modifications of the Navier-Stokes equations for large gradients of the velocities, Zap. Naučn. Semin. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 7, 126-154 (1968) · Zbl 0195.10602 |
| [35] | Leray, J.; Lions, J.-L., Quelques résulatats de Višik sur les problèmes elliptiques nonlinéaires par les méthodes de Minty-Browder, Bull. Soc. Math. Fr., 93, 97-107 (1965) · Zbl 0132.10502 |
| [36] | Lions, J.-L., Quelques méthodes de résolution des problèmes aux limites non linéaires (1969), Dunod, Gauthier-Villars: Dunod, Gauthier-Villars Paris · Zbl 0189.40603 |
| [37] | Lions, J.-L.; Magenes, E., Non-homogeneous Boundary Value Problems and Applications. Vol. I, Die Grundlehren der Mathematischen Wissenschaften, Band 181 (1972), Springer-Verlag: Springer-Verlag New York-Heidelberg, Translated from the French by P. Kenneth · Zbl 0223.35039 |
| [38] | Liu, Wei; Röckner, Michael, SPDE in Hilbert space with locally monotone coefficients, J. Funct. Anal., 259, 11, 2902-2922 (2010) · Zbl 1236.60064 |
| [39] | Menaldi, J.-L.; Sritharan, S. S., Stochastic 2-D Navier-Stokes equation, Appl. Math. Optim., 46, 1, 31-53 (2002) · Zbl 1016.35072 |
| [40] | Métivier, M., Semimartingales: A Course on Stochastic Processes, de Gruyter Studies in Mathematics, vol. 2 (1982), Walter de Gruyter & Co.: Walter de Gruyter & Co. Berlin-New York · Zbl 0503.60054 |
| [41] | Métivier, M., Stochastic Partial Differential Equations in Infinite-Dimensional Spaces, Scuola Normale Superiore di Pisa. Quaderni. Publications of the Scuola Normale Superiore of Pisa (1988), Scuola Normale Superiore: Scuola Normale Superiore Pisa · Zbl 0664.60062 |
| [42] | Minty, G. J., Monotone (nonlinear) operators in Hilbert space, Duke Math. J., 29, 341-346 (1962) · Zbl 0111.31202 |
| [43] | Minty, G. J., On a “monotonicity” method for the solution of non-linear equations in Banach spaces, Proc. Natl. Acad. Sci. USA, 50, 1038-1041 (1963) · Zbl 0124.07303 |
| [44] | Mohan, M. T.; Sritharan, S. S., Stochastic Euler equations of fluid dynamics with Lévy noise, Asymptot. Anal., 99, 1-2, 67-103 (2016) · Zbl 1348.35177 |
| [45] | Mohan, Manil T.; Sritharan, Sivaguru S., \( \mathbb{L}^p\)-solutions of the stochastic Navier-Stokes equations subject to Lévy noise with \(\mathbb{L}^m( \mathbb{R}^m)\) initial data, Evol. Equ. Control Theory, 6, 3, 409-425 (2017) · Zbl 1366.76022 |
| [46] | Mohan, Manil T.; Sritharan, Sivaguru S., Stochastic Navier-Stokes equations perturbed by Lévy noise with hereditary viscosity, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 22, 1, Article 1950006 pp. (2019) · Zbl 1421.35252 |
| [47] | Motyl, E., Stochastic Navier-Stokes equations driven by Lévy noise in unbounded 3D domains, Potential Anal., 38, 3, 863-912 (2013) · Zbl 1282.35282 |
| [48] | Nguyen, P.; Temam, R., The Stampacchia maximum principle for stochastic partial differential equations forced by Lévy noise, Commun. Pure Appl. Anal., 19, 4, 2289-2331 (2020) · Zbl 1446.35274 |
| [49] | Peszat, S.; Zabczyk, J., Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach, Encyclopedia of Mathematics and Its Applications, vol. 113 (2007), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1205.60122 |
| [50] | Prévôt, C.; Röckner, M., A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Mathematics, vol. 1905 (2007), Springer: Springer Berlin · Zbl 1123.60001 |
| [51] | Smagorinsky, J., General circulation experiments with the primitive equations, I. The basic experiment, Mon. Weather Rev., 91, 99-164 (1963) |
| [52] | Stechmann, S. N.; Neelin, J. D., First-passage-time prototypes for precipitation statistics, J. Atmos. Sci., 71, 9, 3269-3291 (2014) |
| [53] | Tawri, K.; Temam, R., Hilbertian approximation of monotone operators, Pure Appl. Funct. Anal. (2021), Special issue dedicated to the memory of Ciprian Foias, in press |
| [54] | Temam, R., Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 66 (1995), Society for Industrial and Applied Mathematics (SIAM): Society for Industrial and Applied Mathematics (SIAM) Philadelphia, PA · Zbl 0833.35110 |
| [55] | Temam, R., Navier-Stokes Equations: Theory and Numerical Analysis (2001), AMS Chelsea Publishing: AMS Chelsea Publishing Providence, RI, Reprint of the 1984 edition · Zbl 0981.35001 |
| [56] | Thual, S.; Majda, A. J.; Chen, N.; Stechmann, S. N., Simple stochastic model for El Niño with westerly wind bursts, Proc. Natl. Acad. Sci. USA, 113, 37, 10245-10250 (2016) · Zbl 1355.86005 |
| [57] | Watanabe, S.; Yamada, T., On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ., 11, 155-167 (1971) · Zbl 0236.60037 |
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