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Well-posedness for the Cahn-Hilliard-Navier-Stokes Equations with Random Initial Data

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Abstract

We consider the almost sure well-posedness of the Cauchy problem to the Cahn-Hilliard-Navier-Stokes equations with a randomization initial data on a torus \(\mathbb {T}^{3}\). First, we prove the local existence and uniqueness of solution in a proper functional space. Furthermore, we prove the global existence and uniqueness of solution and give the relative probability estimate under the assumption of small initial data.

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Acknowledgments

The authors thank the referee for the very helpful comments and suggestions. Z.Qiu’s research is partially supported by the CSC under grant No. 201806160015. H.Wang’s research is supported by the National Natural Science Foundation of China (No. 11901066), the Natural Science Foundation of Chongqing (No. cstc2019jcyj-msxmX0167) and projects No. 2019CDXYST0015, No. 2020CDJQY-A040 supported by the Fundamental Research Funds for the Central Universities.

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Authors and Affiliations

  1. School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing, 210046, China

    Zhaoyang Qiu

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

    Yanbin Tang

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

    Huaqiao Wang

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  1. Zhaoyang Qiu
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  2. Yanbin Tang
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  3. Huaqiao Wang
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Correspondence to Huaqiao Wang.

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Qiu, Z., Tang, Y. & Wang, H. Well-posedness for the Cahn-Hilliard-Navier-Stokes Equations with Random Initial Data. Potential Anal 59, 753–770 (2023). https://doi.org/10.1007/s11118-022-10000-5

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  • DOI: https://doi.org/10.1007/s11118-022-10000-5

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