Blow-up of smooth solutions to the compressible fluid models of Korteweg type |
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Authors: | Ying Hui Zhang Zhong Tan |
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Affiliation: | (1) Department of Mathematics, Hu’nan Institute of Science and Technology, Yueyang, 414006, P. R. China;(2) School of Mathematical Sciences, Xiamen University, Xiamen, 361005, P. R. China |
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Abstract: | We show the blow-up of smooth solutions to a non-isothermal model of capillary compressible fluids in arbitrary space dimensions with initial density of compact support. This is an extension of Xin’s result [Xin, Z.: Blow-up of smooth solutions to the compressible Navier-Stokes equations with compact density. Comm. Pure Appl. Math., 51, 229–240 (1998)] to the capillary case but we do not need the condition that the entropy is bounded below. Moreover, from the proof of Theorem 1.2, we also obtain the exact relationship between the size of support of the initial density and the life span of the solutions. We also present a sufficient condition on the blow-up of smooth solutions to the compressible fluid models of Korteweg type when the initial density is positive but has a decay at infinity. |
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Keywords: | Blow-up smooth solutions capillary compressible fluids compact support |
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