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On the Global Well-posedness for the Boussinesq System with Horizontal Dissipation

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Abstract

In this paper, we investigate the Cauchy problem for the tridimensional Boussinesq equations with horizontal dissipation. Under the assumption that the initial data is axisymmetric without swirl, we prove the global well-posedness for this system. In the absence of vertical dissipation, there is no smoothing effect on the vertical derivatives. To make up this shortcoming, we first establish a magic relationship between \({\frac{u^{r}}{r}}\) and \({\frac{\omega_\theta}{r}}\) by taking full advantage of the structure of the axisymmetric fluid without swirl and some tricks in harmonic analysis. This together with the structure of the coupling of (1.2) entails the desired regularity.

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Authors and Affiliations

  1. Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing, 100088, P.R. China

    Changxing Miao

  2. The Graduate School of China Academy of Engineering Physics, P.O. Box 2101, Beijing, 100088, P.R. China

    Xiaoxin Zheng

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  1. Changxing Miao
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  2. Xiaoxin Zheng
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Correspondence to Changxing Miao.

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Communicated by P. Constantin

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Miao, C., Zheng, X. On the Global Well-posedness for the Boussinesq System with Horizontal Dissipation. Commun. Math. Phys. 321, 33–67 (2013). https://doi.org/10.1007/s00220-013-1721-2

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  • DOI: https://doi.org/10.1007/s00220-013-1721-2

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