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Acta Mathematicae Applicatae Sinica, English Series 2013, Vol. 29 Issue (4) :855-860    DOI: 10.1007/s10255-013-0262-y
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On the Blowing up of Solutions to One-dimensional Quantum Navier-Stokes Equations
Jian-wei DONG1, You-lin ZHANG2, Yan-ping WANG1
1 Department of Mathematics and Physics, Zhengzhou Institute of Aeronautical Industry Management, Zhengzhou 450015, China;
2 Library, Zhengzhou Institute of Aeronautical Industry Management, Zhengzhou 450015, China
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Abstract 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 differential 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.
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Jian-wei DONG
You-lin ZHANG
Yan-ping WANG
Keywordscompressible quantum Navier-Stokes equations   smooth solutions   blow up     
Abstract: 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 differential 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.
Keywordscompressible quantum Navier-Stokes equations,   smooth solutions,   blow up     
Received: 2012-03-30;
Fund:

Supported by the Vital Science Research Foundation of Henan Province Education Department (No. 12A110024).

Cite this article:   
Jian-wei DONG, You-lin ZHANG, Yan-ping WANG .On the Blowing up of Solutions to One-dimensional Quantum Navier-Stokes Equations[J]  Acta Mathematicae Applicatae Sinica, English Serie, 2013,V29(4): 855-860
URL:  
http://www.applmath.com.cn/jweb_yysxxb_en/EN/10.1007/s10255-013-0262-y      或     http://www.applmath.com.cn/jweb_yysxxb_en/EN/Y2013/V29/I4/855
 
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