Formation of singularities in solutions to the compressible radiation hydrodynamics equations with vacuum |
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Authors: | Yachun Li Shengguo Zhu |
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Institution: | 1. Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, PR China;2. Key Lab of Scientific and Engineering Computing (MOE), Shanghai Jiao Tong University, Shanghai 200240, PR China;3. School of Mathematics, Georgia Institute of Technology, Atlanta 30332, USA |
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Abstract: | We study the Cauchy problem for multi-dimensional compressible radiation hydrodynamics equations with vacuum. First, we present some sufficient conditions on the blow-up of smooth solutions in multi-dimensional space. Then, we obtain the invariance of the support of density for the smooth solutions with compactly supported initial mass density by the property of the system under the vacuum state. Based on the above-mentioned results, we prove that we cannot get a global classical solution, no matter how small the initial data are, as long as the initial mass density is of compact support. Finally, we will see that some of the results that we obtained are still valid for the isentropic flows with degenerate viscosity coefficients as well as for one-dimensional case. |
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Keywords: | primary 35A09 35B44 35Q30 secondary 35L65 35Q35 |
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