On the Blowing up of Solutions to One-dimensional Quantum Navier-Stokes Equations

The blow-up in finite time for the solutions to the initial-boundary value problem associated to the one-dimensional quantum Navier-Stokes equations in a bounded domain is proved. The model consists of the mass conservation equation and a momentum balance equation, including a nonlinear third-order...

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Bibliographic Details
Published in:Acta Mathematicae Applicatae Sinica 2013-10, Vol.29 (4), p.855-860
Main Authors: Dong, Jian-wei, Zhang, You-lin, Wang, Yan-ping
Format: Article
Language:English
Subjects:
Applications of Mathematics
Math Applications in Computer Science
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Navier-Stokes方程
Theoretical
一维
初边值问题
普朗克常数
有限时间
流体动力学模型
质量守恒方程
量子波
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Summary:The blow-up in finite time for the solutions to the initial-boundary value problem associated to the one-dimensional quantum Navier-Stokes equations in a bounded domain is proved. The model consists of the mass conservation equation and a momentum balance equation, including a nonlinear third-order differen- tial operator, with the quantum Bohm potential, and a density-dependent viscosity. It is shown that, under suitable boundary conditions and assumptions on the initial data, the solution blows up after a finite time, if the viscosity constant is not bigger than the scaled Planck constant. The proof is inspired by an observable constructed by Gamba, Gualdani and Zhang, which has been used to study the blowing up of solutions to quantum hydrodynamic models.
ISSN:0168-9673
1618-3932
DOI:10.1007/s10255-013-0262-y