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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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References:

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