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On scaling invariance and type-I singularities for the compressible Navier-Stokes equations

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Abstract

We find a new scaling invariance of the barotropic compressible Navier-Stokes equations. Then it is shown that type-I singularities of solutions with

$$\mathop {\lim \sup }\limits_{t \nearrow T} |div(t,x)|(T - t) \leqslant \kappa $$

can never happen at time T for all adiabatic number γ > 1. Here κ > 0 does not depend on the initial data. This is achieved by proving the regularity of solutions under

$$\rho (t,x) \leqslant \frac{M}{{{{(T - t)}^\kappa }}},M < \infty .$$

This new scaling invariance also motivates us to construct an explicit type-II blowup solution for γ > 1.

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Acknowledgements

The first author was supported by National Natural Science Foundation of China (Grant No. 11725102), National Support Program for Young Top-Notch Talents, and SGST 09DZ2272900 from Shanghai Key Laboratory for Contemporary Applied Mathematics. The second author was supported by Zheng Ge Ru Foundation, Hong Kong RGC Earmarked Research Grants (Grant Nos. CUHK-14305315, CUHK-14300917 and CUHK-14302917), NSFC/RGC Joint Research Scheme Grant (Grant No. N-CUHK 443-14), and a Focus Area Grant from the Chinese University of Hong Kong.

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Authors and Affiliations

  1. School of Mathematical Sciences, LMNS and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai, 200433, China

    Zhen Lei

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

    Zhouping Xin

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  1. Zhen Lei
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  2. Zhouping Xin
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Correspondence to Zhouping Xin.

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Dedicated to Professor Lo Yang on the Occasion of His 80th Birthday

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Lei, Z., Xin, Z. On scaling invariance and type-I singularities for the compressible Navier-Stokes equations. Sci. China Math. 62, 2271–2286 (2019). https://doi.org/10.1007/s11425-018-9363-1

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