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Science China Mathematics

, Volume 62, Issue 11, pp 2271–2286 | Cite as

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

  • Zhen Lei
  • Zhouping XinEmail author
Articles
  • 53 Downloads

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.

Keywords

type-I singularity compressible Navier-Stokes equations scaling invariance blowup rate 

MSC(2010)

35Q30 

Notes

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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Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematical Sciences, LMNS and Shanghai Key Laboratory for Contemporary Applied MathematicsFudan UniversityShanghaiChina
  2. 2.The Institute of Mathematical Sciences and Department of MathematicsThe Chinese University of Hong KongHong KongChina

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