Abstract
The blow-up in finite time for the solutions to the initial-boundary value problem associated to the multi-dimensional quantum hydrodynamic model in a bounded domain is proved. The model consists on conservation of mass equation and a momentum balance equation equivalent to a compressible Euler equations corrected by a dispersion term of the third order in the momentum balance. The proof is based on a priori estimates for the energy functional for a new observable constructed with an auxiliary function, and it is shown that, under suitable boundary conditions and assumptions on the initial data, the solution blows up after a finite time.
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Communicated by P. Markowich.
I.M. Gamba is supported by NSF-DMS0507038. M.P. Gualdani acknowledges partial support from the Deutsche Forschungsgemeinschaft, grants JU359/5 and was partially supported under the Feodor Lynen Research fellowship. P. Zhang is partially supported by the NSF of China under Grant 10525101 and 10421101, and the innovation grant from the Chinese Academy of Sciences. Part of the work was done when P. Zhang visited the Department of Mathematics of Texas University at Austin, the author would like to thank the hospitality of the department. Support from the Institute for Computational Engineering and Sciences at the University of Texas at Austin is also gratefully acknowledged.
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Gamba, I.M., Gualdani, M.P. & Zhang, P. On the blowing up of solutions to quantum hydrodynamic models on bounded domains. Monatsh Math 157, 37–54 (2009). https://doi.org/10.1007/s00605-009-0092-4
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DOI: https://doi.org/10.1007/s00605-009-0092-4