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**Blow-up of viscous heat-conducting compressible flows.**
*(English)*
Zbl 1121.35110

Summary: We show the blow-up of strong solutions of viscous heat-conducting flows when the initial density is compactly supported. This is an extension of Z. Xin’s result [Commun. Pure Appl. Math. 51, No. 3, 229–240 (1998; Zbl 0937.35134)] to the case of positive heat conduction coefficients but we do not need any information for the time decay of total pressure nor the lower bound of the entropy. We control the lower bound of the second moment by total energy and obtain the exact relationship between the size of support of initial density and the existence time. We also provide a sufficient condition for the blow-up in case that the initial density is positive but has a decay at infinity.

### MSC:

35Q35 | PDEs in connection with fluid mechanics |

35B40 | Asymptotic behavior of solutions to PDEs |

76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |

### Citations:

Zbl 0937.35134
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\textit{Y. Cho} and \textit{B. J. Jin}, J. Math. Anal. Appl. 320, No. 2, 819--826 (2006; Zbl 1121.35110)

### References:

[1] | Y. Cho, H. Kim, Existence results for viscous polytropic fluids with vacuum, Hokkaido University preprint series in mathematics, No. 675 |

[2] | Danchin, R.; Desjardins, B., Existence of solutions for compressible fluid models of Korteweg type, Ann. inst. H. Poincaré anal. non linéaire, 18, 97-133, (2001) · Zbl 1010.76075 |

[3] | Matsumura, A.; Nishida, T., The initial value problem for the equations of motion of viscous and heat-conductive gases, J. math. Kyoto univ., 20, 67-104, (1980) · Zbl 0429.76040 |

[4] | Sideris, T.C., Formation of singularity in three dimensional compressible fluids, Comm. math. phys., 101, 475-487, (1985) · Zbl 0606.76088 |

[5] | Xin, Z., Blow up of smooth solutions to the compressible navier – stokes equations with compact density, Comm. pure appl. math., 51, 229-240, (1998) · Zbl 0937.35134 |

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