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On the Uniqueness of Invariant Measures for the Stochastic Infinite Darcy–Prandtl Number Model

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New Trends and Advanced Methods in Interdisciplinary Mathematical Sciences

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Abstract

The infinite Darcy–Prandtl number model is an effective reduced model for describing convection in a fluid-saturated porous medium. It is well known that the deterministic model does not possess a unique invariant measure. In this work, we study the dynamics of the infinite Darcy–Prandtl number model, under an additive stochastic forcing of its low modes. This is the so-called stochastic infinite Darcy–Prandtl number model. We prove that the stochastically forced system, does indeed possess a unique invariant measure.

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Notes

  1. 1.

    For the notation in conditions 3 and 4, please refer to Section 2.3 of [10].

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Acknowledgements

The first author would like to sincerely acknowledge the suggestions and guidance provided by his PhD advisor Dr. Xiaoming Wang. These greatly helped in completing the current work.

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Authors and Affiliations

  1. Department of Mathematics, Clarkson University, Potsdam, NY, 13699, USA

    Rana D. Parshad

  2. Department of Mathematics, Florida State University, Tallahassee, FL, 32306, USA

    Brian Ewald

Authors
  1. Rana D. Parshad
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  2. Brian Ewald
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Corresponding author

Correspondence to Rana D. Parshad .

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Editors and Affiliations

  1. Department of Mathematics & Economics, Virginia State University, Petersburg, Virginia, USA

    Bourama Toni

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Parshad, R.D., Ewald, B. (2017). On the Uniqueness of Invariant Measures for the Stochastic Infinite Darcy–Prandtl Number Model. In: Toni, B. (eds) New Trends and Advanced Methods in Interdisciplinary Mathematical Sciences. STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health. Springer, Cham. https://doi.org/10.1007/978-3-319-55612-3_9

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