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On the Plate Equation with Exponentially Degenerating Stochastic Coefficients on the Torus

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Harmonic Analysis and Partial Differential Equations

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Abstract

This paper aims to investigate the plate equation with time-dependent stochastic coefficients on the torus, which is used for modeling the vibration of beams with random perturbations from various sources. We mainly study the joint influence from the exponentially degenerating and strong oscillating coefficients on the biharmonic and Laplace-Beltrami operators to explore the upper bound of loss of regularity by applying important techniques from microlocal analysis and stochastic analysis. More importantly, the critical case for loss of regularity has been deduced by the exquisite normal form diagonalization process. Furthermore, appropriate counter-examples with periodic coefficients are constructed in order to demonstrate the optimality of the estimates by the application of instability arguments.

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Acknowledgements

This project is supported by Natural Science Foundation of Jiangsu Province(BK 20191257).

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Authors and Affiliations

  1. School of Mathematics, Southeast University, Nan**g, China

    **aojun Lu

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  1. **aojun Lu
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Correspondence to **aojun Lu .

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Editors and Affiliations

  1. Department of Mathematics, Ghent University, Ghent, Belgium

    Michael Ruzhansky

  2. Department of Mathematics, University of Stuttgart, Stuttgart, Germany

    Jens Wirth

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Lu, X. (2022). On the Plate Equation with Exponentially Degenerating Stochastic Coefficients on the Torus. In: Ruzhansky, M., Wirth, J. (eds) Harmonic Analysis and Partial Differential Equations. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-24311-0_7

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