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Local and global well-posedness of entropy-bounded solutions to the compressible Navier-Stokes equations in multi-dimensions

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Abstract

Due to the high degeneracy and singularities of the entropy equation, the physical entropy for viscous and heat conductive polytropic gases behaves singularly in the presence of vacuum and it is thus a challenge to study its dynamics. It is shown in this paper that the uniform boundedness of the entropy and the inhomogeneous Sobolev regularities of the velocity and temperature can be propagated for viscous and heat conductive gases in ℝ3, provided that the initial vacuum occurs only at far fields with suitably slow decay of the initial density. Precisely, it is proved that for any strong solution to the Cauchy problem of the heat conductive compressible Navier-Stokes equations, the corresponding entropy keeps uniformly bounded and the L2 regularities of the velocity and temperature can be propagated, up to the existing time of the solution, as long as the initial density vanishes only at far fields with a rate not faster than \(O\left( {{1 \over {{{\left| x \right|}^2}}}} \right)\). The main tools are some singularly weighted energy estimates and an elaborate De Giorgi type iteration technique. We apply the De Giorgi type iterations to different equations in establishing the lower and upper bounds of the entropy.

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Acknowledgements

This work was supported by the Key Project of National Natural Science Foundation of China (Grant No. 12131010) and the Guangdong Basic and Applied Basic Research Foundation (Grant No. 2020B1515310002). Jinkai Li was supported by National Natural Science Foundation of China (Grant Nos. 11971009 and 11871005) and the Guangdong Basic and Applied Basic Research Foundation (Grant Nos. 2019A1515011621 and 2020B1515310005). Zhouping Xin was supported by the Zheng Ge Ru Foundation and the Hong Kong RGC Earmarked Research Grants (Grant Nos. CUHK-14305315, CUHK-14300917 and CUHK-14302819)

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  1. South China Research Center for Applied Mathematics and Interdisciplinary Studies, School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

    Jinkai Li

  2. The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Hong Kong, China

    Zhouping Xin

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  1. Jinkai Li
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Correspondence to Jinkai Li.

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Li, J., Xin, Z. Local and global well-posedness of entropy-bounded solutions to the compressible Navier-Stokes equations in multi-dimensions. Sci. China Math. 66, 2219–2242 (2023). https://doi.org/10.1007/s11425-022-2047-0

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