American Institute of Mathematical Sciences

August  2016, 36(8): 4477-4493. doi: 10.3934/dcds.2016.36.4477

On formation of singularity for non-isentropic Navier-Stokes equations without heat-conductivity

 1 NCMIS, AMSS, Institute of Mathematics, Chinese Academy of Sciences, Beijing 100190, China 2 The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

Received  June 2015 Revised  October 2015 Published  March 2016

It is known that smooth solutions to the non-isentropic Navier-Stokes equations without heat-conductivity may lose their regularity in finite time in the presence of vacuum. However, in spite of the recent progress on such blowup phenomena, it remains to give a possible blowup mechanism. In this paper, we present a simple continuation principle for such system, which asserts that the concentration of the density or the temperature occurs in finite time for a large class of smooth initial data, which is responsible for the breakdown of classical solutions. It also gives an affirmative answer to a strong version of a problem proposed by J.Nash in 1950s.
Citation: Xiangdi Huang, Zhouping Xin. On formation of singularity for non-isentropic Navier-Stokes equations without heat-conductivity. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4477-4493. doi: 10.3934/dcds.2016.36.4477
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