Stochastic Models for Nonlinear Cross-Diffusion Systems

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 231)

Abstract

Under a priori assumptions concerning existence and uniqueness of the Cauchy problem solution for a system of quasilinear parabolic equations with cross-diffusion, we treat the PDE system as an analogue of systems of forward Kolmogorov equations for some unknown stochastic processes and derive expressions for their generators. This allows to construct a stochastic representation of the required solution. We prove that introducing stochastic test function we can check that the stochastic system gives rise to the required generalized solution of the original PDE system. Next, we derive a closed stochastic system which can be treated as a stochastic counterpart of the Cauchy problem for a parabolic system with cross-diffusion.

Keywords

Stochastic flow Cross-diffusion PDE generalized solution Probabilistic representation 

Notes

Acknowledgements

The financial support of the RFBR Grant 15-01-01453 is gratefully acknowledged.

References

  1. 1.
    Belopolskaya, Ya.: Generalized solutions of nonlinear parabolic systems and vanishing viscosity method. J. Math. Sci. 133 1207–1223 (2006)Google Scholar
  2. 2.
    Belopolskaya, Ya., Woyczynski, W.: Generalized solution of the Cauchy problem for systems of nonlinear parabolic equations and diffusion processes. Stoch. Dyn. 11(1), 1–31 (2012)Google Scholar
  3. 3.
    Kunita H.: Stochastic Flows and Stochastic Differential Equations. Cambridge University Press, Cambridge (1990)Google Scholar
  4. 4.
    Kunita, H.: Stochastic flows acting on Schwartz distributions. J. Theor. Pobab. 7(2), 247–278 (1994)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Kunita, H.: Generalized solutions of stochastic partial differential equations. J. Theor. Pobab. 7(2), 279–308 (1994)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Shigesada, N., Kawasaki, K., Teramoto, E.: Spatial segregation of interacting species. J. Theor. Biol. 79, 83–99 (1979)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Jüngel, A.: Diffusive and nondiffusive population models. In: Naldi, G., Pareschi, L., Toscani, G. (eds.) Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, pp. 397–425. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  8. 8.
    Desvilletes, L., Lepoutre, Th, Moussa, A.: Entropy, duality and cross diffusion. SIAM J. Math. Anal. 46(1), 820–853 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Jüngel, A.: The boundedness-by-entropy method for cross-diffusion systems. Nonlinearity 28, 1963–2001 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Galiano, G., Selgas, V.: On a cross-diffusion segregation problem arising from a model of interacting particles. Nonlinear Anal. Real World Appl. 18, 34–49 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fontbona, J., Meleard, S.: Non local Lotka-Volterra system with cross-diffusion in an heterogeneous medium. J. Math. Biol. 70(4), 829–854 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bogachev, V., Röckner, M., Shaposhnikov, S.: On uniqueness problems related to the Fokker-Planck-Kolmogorov equation for measures. J. Math. Sci. 179(1), 7–47 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Protter, P.: Stochastic Integration and Differential Equations. Springer, Berlin (2010)Google Scholar
  14. 14.
    Belopolskaya, Ya., Dalecky, Yu.: Stochastic Equations and Differential Geometry. Kluwer Academic Publishers, Dordrecht (1990)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Saint-Petersburg State University of Architecture and Civil EngineeringSt. PetersburgRussian Federation

Personalised recommendations