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Regularity Criteria for the 3D Dissipative System Modeling Electro-Hydrodynamics

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Abstract

In this paper, we study the three-dimensional dissipative fluid-dynamical model, which is a strongly coupled nonlinear nonlocal system characterized by the Navier–Stokes/Poisson–Nernst–Planck system. It is proved that the local smooth solution can be continued beyond the time T provided that the vorticity \(\omega \) satisfies

$$\begin{aligned} \int _{0}^{T}\frac{\Vert \omega (\cdot ,t)\Vert _{\dot{B}^{-\alpha }_{\infty ,\infty }}^{\frac{2}{2-\alpha }}}{1+\ln \left( e+\Vert \omega (\cdot ,t)\Vert _{\dot{B}^{-\alpha }_{\infty ,\infty }}\right) }\mathrm{d}t<\infty \quad \text {for} \,\, 0<\alpha <2. \end{aligned}$$

Moreover, two regularity criteria for the marginal cases \(\alpha =0\) and \(\alpha =2\) are also established, respectively.

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Acknowledgements

This paper is partially supported by the National Natural Science Foundation of China (11501453), the Fundamental Research Funds for the Central Universities (2014YB031) and the Fundamental Research Project of Natural Science Foundation of Shaanxi Province–Young Talent Project (2015JQ1004). The author is glad to acknowledge his gratefulness to Dr. Jingjing Zhang for profitable communication on the regularity criterion (1.6) in Theorem 1.2 of this paper.

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  1. Institute of Applied Mathematics, College of Science, Northwest A&F University, Yangling, 712100, Shaanxi, China

    Jihong Zhao

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  1. Jihong Zhao
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Correspondence to Jihong Zhao.

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Communicated by Yong Zhou.

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Zhao, J. Regularity Criteria for the 3D Dissipative System Modeling Electro-Hydrodynamics. Bull. Malays. Math. Sci. Soc. 42, 1101–1117 (2019). https://doi.org/10.1007/s40840-017-0537-1

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  • DOI: https://doi.org/10.1007/s40840-017-0537-1

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