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Large deviation principle for the 2D stochastic Cahn–Hilliard–Navier–Stokes equations

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Abstract

We establish a large deviation principle for the 2D stochastic Cahn–Hilliard–Navier–Stokes equations driven by multiplicative noise in a bounded domain. This system consists of the Navier–Stokes equations for the velocity, coupled with a Cahn–Hilliard model for the order parameter. The proof is completed using a weak convergence approach based on the variational representational of functional of infinite-dimensional Brownian motion. In particular, we directly prove the existence and uniqueness of the solution of the stochastic controlled equations instead of using the Girsanov transformation.

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Acknowledgements

We thank the referees for the very helpful comments and suggestions. Z. Qiu’s research is supported by the CSC under grant No.201806160015. H. Wang’s research is supported by the National Natural Science Foundation of China (Grant No. 11901066), the Natural Science Foundation of Chongqing (Grant No. cstc2019jcyj-msxmX0167) and Project No. 2019CDXYST0015 supported by the Fundamental Research Funds for the Central Universities.

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  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China

    Zhaoyang Qiu

  2. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

    Huaqiao Wang

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  1. Zhaoyang Qiu
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Correspondence to Huaqiao Wang.

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Qiu, Z., Wang, H. Large deviation principle for the 2D stochastic Cahn–Hilliard–Navier–Stokes equations. Z. Angew. Math. Phys. 71, 88 (2020). https://doi.org/10.1007/s00033-020-01312-w

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  • DOI: https://doi.org/10.1007/s00033-020-01312-w

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