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Nonlinear stochastic evolution equations of second order with damping

Abstract

Convergence of a full discretization of a second order stochastic evolution equation with nonlinear damping is shown and thus existence of a solution is established. The discretization scheme combines an implicit time stepping scheme with an internal approximation. Uniqueness is proved as well.

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References

  1. 1.

    Amann, H., Escher, J.: Analysis III. Birkhäuser, Basel (2009)

  2. 2.

    Anton, R., Cohen, D., Larsson, S., Wang, X.: Full discretisation of semi-linear stochastic wave equations driven by multiplicative noise. SIAM J. Numer. Anal. 54(2), 1093–1119 (2016)

  3. 3.

    Brézis, H.: Functional Analysis. Sobolev Spaces and Partial Differential Equations. Springer, New York (2010)

  4. 4.

    Carmona, R., Nualart, D.: Random nonlinear wave equations: smoothness of the solutions. Probab. Theory Relat. Fields 79(4), 469–508 (1988)

  5. 5.

    de Naurois, L.J., Jentzen, A., Welti, T.: Weak convergence rates for spatial spectral Galerkin approximations of semilinear stochastic wave equations with multiplicative noise, (2015). arXiv:1508.05168

  6. 6.

    Diestel, J., Uhl Jr., J.J.: Vector Measures. American Mathematical Society, Providence, R.I. (1977)

  7. 7.

    Emmrich, E.: Time discretisation of monotone nonlinear evolution problems by discontinuous Galerkin method. BIT Numer. Math. 51, 581–607 (2011)

  8. 8.

    Emmrich, E., Šiška, D.: Full discretization of second-order nonlinear evolution equations: strong convergence and applications. Comput. Methods Appl. Math. 11(4), 441–459 (2011)

  9. 9.

    Emmrich, E., Šiška, D.: Evolution equations of second order with nonconvex potential and linear damping: existence via convergence of a full discretization. J. Differ. Equ. 255(10), 3719–3746 (2013)

  10. 10.

    Emmrich, E., Thalhammer, M.: Convergence of a time discretisation for doubly nonlinear evolution equations of second order. Found. Comput. Math. 10(2), 171–190 (2010)

  11. 11.

    Emmrich, E., Thalhammer, M.: Doubly nonlinear evolution equations of second order: existence and fully discrete approximation. J. Differ. Equ. 251, 82–118 (2011)

  12. 12.

    Gajewski, H., Gröger, K., Zacharias, K.: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Akademie-Verlag, Berlin (1974)

  13. 13.

    Gyöngy, I.: On stochastic equations with respect to semimartingales. III. Stochastics 7(4), 231–254 (1982)

  14. 14.

    Gyöngy, I., Krylov, N.V.: On stochastics equations with respect to semimartingales. II. Itô formula in Banach spaces. Stochastics 6(3–4), 153–173 (1981/82)

  15. 15.

    Gyöngy, I., Millet, A.: On discretization schemes for stochastic evolution equations. Potential Anal. 23(2), 99–134 (2005)

  16. 16.

    Hausenblas, E.: Weak approximation of the stochastic wave equation. J. Comput. Appl. Math. 235(1), 33–58 (2010)

  17. 17.

    Kim, J.U.: On the stochastic wave equation with nonlinear damping. Appl. Math. Optim. 58(1), 29–67 (2008)

  18. 18.

    Kovács, M., Saedpanah, F., Larsson, S.: Finite element approximation of the linear stochastic wave equation with additive noise. SIAM J. Numer. Anal. 48(2), 408–427 (2010)

  19. 19.

    Krylov, N.V., Rozovskiĭ, B.L.: Stochastic evolution equations. J. Sov. Math. 16, 1233–1277 (1981)

  20. 20.

    Lions, J.L., Strauss, W.A.: Some non-linear evolution equations. Bull. Soc. Math. France 93, 43–96 (1965)

  21. 21.

    Marinelli, C., Quer-Sardanyons, L.: Existence of weak solutions for a class of semilinear stochastic wave equations. SIAM J. Math. Anal. 44(2), 906–925 (2012)

  22. 22.

    Millet, A., Morien, P.-L.: On a stochastic wave equation in two space dimensions: regularity of the solution and its density. Stochas. Process. Appl. 86(1), 141–162 (2000)

  23. 23.

    Millet, A., Sanz-Solé, M.: A stochastic wave equation in two space dimension: smoothness of the law. Ann. Probab. 27(2), 803–844 (1999)

  24. 24.

    Pardoux, E.: Equations aux derivées partielles stochastiques non lineaires monotones. Étude de solutions fortes de type Ito. PhD thesis, Univ. Paris XI, Orsay (1975)

  25. 25.

    Peszat, S., Zabczyk, J.: Nonlinear stochastic wave and heat equations. Probab. Theory Relat. Fields 116(3), 421–443 (2000)

  26. 26.

    Prévôt, C., Röckner, M.: A Concise Course on Stochastic Partial Differential Equations. Springer, New York (2007)

  27. 27.

    Quer-Sardanyons, L., Sanz-Solé, M.: Space semi-discretisations for a stochastic wave equation. Potential Anal. 24(4), 303–332 (2006)

  28. 28.

    Tessitore, G., Zabczyk, J.: Wong–Zakai approximations of stochastic evolution equations. J. Evol. Equ. 6(4), 621–655 (2006)

  29. 29.

    Walsh, J.B.: On numerical solutions of the stochastic wave equation. Illinois J. Math. 50(1–4), 991–1018 (2006)

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Acknowledgments

The authors would like to thank Raphael Kruse (Berlin) for helpful discussions and comments and to the referees for their careful reading and helpful suggestions.

Author information

Correspondence to Etienne Emmrich.

Additional information

This work has been partially supported by the Collaborative Research Center 910, which is funded by the German Science Foundation.

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Emmrich, E., Šiška, D. Nonlinear stochastic evolution equations of second order with damping. Stoch PDE: Anal Comp 5, 81–112 (2017). https://doi.org/10.1007/s40072-016-0082-1

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Keywords

  • Stochastic evolution equation of second order
  • Monotone operator
  • Full discretization
  • Convergence
  • Existence
  • Uniqueness

Mathematics Subject Classification

  • 60H15
  • 47J35
  • 60H35
  • 65M12