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On the Well-Posedness of Reduced 3D Primitive Geostrophic Adjustment Model with Weak Dissipation

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Abstract

In this paper we prove the local well-posedness and global well-posedness with small initial data of the strong solution to the reduced 3D primitive geostrophic adjustment model with weak dissipation. The term reduced model means that the relevant physical quantities depend only on two spatial variables. The weak dissipation helps us overcome the ill-posedness of the original model. We also prove the global well-posedness of the strong solution to the Voigt \(\alpha \)-regularization of this model, and establish the convergence of the strong solution of the Voigt \(\alpha \)-regularized model to the corresponding solution of the original model. Furthermore, we derive a criterion for existence of finite-time blow-up of the original model with weak dissipation based on Voigt \(\alpha \)-regularization.

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Acknowledgements

The work of E.S.T. was supported in part by the Einstein Stiftung/Foundation Berlin, through the Einstein Visiting Fellow Program, and by the John Simon Guggenheim Memorial Foundation.

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Authors and Affiliations

  1. Department of Mathematics, Florida International University, University Park, Miami, FL, 33199, USA

    Chongsheng Cao

  2. Department of Mathematics, Texas A&M University, College Station, TX, 77840, USA

    Quyuan Lin & Edriss S. Titi

  3. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK

    Edriss S. Titi

  4. Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, 76100, Rehovot, Israel

    Edriss S. Titi

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  1. Chongsheng Cao
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Correspondence to Quyuan Lin.

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Communicated by M. Hieber.

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Cao, C., Lin, Q. & Titi, E.S. On the Well-Posedness of Reduced 3D Primitive Geostrophic Adjustment Model with Weak Dissipation. J. Math. Fluid Mech. 22, 32 (2020). https://doi.org/10.1007/s00021-020-00495-6

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