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

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.

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.

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

Drabek, P., Kufner, A., Nicolosi, F. On the solvability of degenerated quasilinear elliptic equations of higher order. J. Differential Equations, 109: 325-347 (1994)

[2]

Jiang, F. A remark on weak solutions to the barotropic compressible quantum Navier-Stokes equations. Nonlinear Analysis: Real World Applications, 12: 1733-1735 (2011)

[3]

Hao, X.B., Shi, D.W., Shi, D.Y. Special convergence analysis of Quasi-Wilson element. Multimedia Technology (ICMT), 2011 International Conference on, July, 6016-6018, 2011

[4]

Qin, Y.S. Empirical likelihood ratio confidence regions in a partly linear model. Chinese J. Appl. Probab. Statist., 15: 363-369 (1999)

[5]

Gan, X.Q. Existence of weak solutions for double degenerated parabolic equations with measures as data. Chinese Science Bulletin, 40(15): 1354-1356 (1995)

[6]

Cameron, P.J., Omidi, G.R., Tayfeh-Rezaie, B. 3-Designs from PGL(2, q). Electron. J. Combin., 13: 1-11 (2006)

Jüngel, A. Global weak solutions to compressible Navier-Stokes equations for quantum fluids. SIAM J. Math. Anal., 42: 1025-1045 (2010)

[10]

Schwarz, G. Estimating the dimension of a model. Ann. Statist., 6: 461-464(1978)

[11]

Chen, Z.M., Price W. Decay estimates of linearized micropolar fluid equations in R^{3} space with applications to L^{3}-strong solutions. Internat. J. Engrg. Sci., 44: 859-873 (2006)

[12]

Liu, Z.H. A class of evolution hemivariational inequalities. Nonlinear Analysis, TMA., 36(1): 91-100 (1999)

[13]

Serfling, R.J. Approximation theorems of mathematical statistics. Wiley, New York. 1980

[14]

Jüngel, A. Effective velocity in compressible Navier-Stokes equations with third-order derivatives. Nonlin. Anal., 74: 2813-2818 (2011)

[15]

Li, Z.T., Gao, S.G., Wang, Z., Bhavani, M.T., Wu, W.L. A construction of cartesian authentication code from orthogonal spaces over a finite field of odd characteristic. Discrete Math., Alg. and Appl, 1(1): 105-114 (2009)

[16]

Chao, C.Y. On the classification of symmetric graphs with a prime number of vertices. Trans. Amer. Math. Soc., 158: 247-256 (1971)

Shi, J., Lau, T.S. Empirical likelihood for partially linear models. J. Multivariate Anal., 72: 132-148 (2000)

[19]

Chen, Z.M., Xin Z. Homogeneity criterion for the Navier-Stokes equations in the whole spaces. J. Math. Fluid Mech., 3: 152-182 (2001)

[20]

Liu, Z.H. On Doubly Degenerate Quasilinear Parabolic Equations of Higher Order. Acta Mathematica Sinica, English Series, 21(1): 197-208 (2005)

[21]

Tibshirani, R. Regression shrinkage and selection via the lasso. J. Roy. Statist. Soc. B., 58: 267-288 (1996)

[22]

Liu, Z.H. Existence results for quasilinear parabolic hemivariational inequalities. J. Differential Equations, 244: 1395-1409 (2008)

[23]

Tong, H. Non-linear time series: a dynamical system approach. Oxford University Press, London. 1990

[24]

Liu, Z.H. Anti-periodic solutions to nonlinear evolution equations. Journal of Functional Analysis, 258(6): 2026-2033 (2010)

[25]

Dong, B.Q, Chen, Z.M. Regularity criterion of weak solutions to the 3D Navier-Stokes equations via two velocity components. J. Math. Anal. Appl., 338: 1-10 (2008)

[26]

Conder, M.D.E., Li, C.H., Praeger, C.E. On the Weiss conjucture for finite locally primitive graphs. Pro. Edinbrugn Math. Soc., 43: 129-138 (2000)

[27]

Simmons, G.J. Message authentication with arbitration of transmitter/receiver disputes. In: Advances in Cryptology-Eurocrypt'87, Lecture Notes in Computer Science 304, Springer-Verlag, Berlin, 1988, 151-165

[28]

Andronov, A.A., Vitt, A.A., Khaikin, S.E. Theory of Oscillators. Dover Publications, Inc., New York, 1966

[29]

Ladyzhenskaya, O.A. The Mathematical Theory of Viscous Incompressible Flows. Gordon and Breach, London, 1969

[30]

Variyath, A.M. Variable selection in generalized linear models by empirical likelihood. Ph.D. Thesis, University of Waterloo, Canada, 2006

[31]

Yin, J.X., Gao, W.J. Flows through nonhomogeneous porous media in an isolated environment. Quart. Appl. Math., 55(2): 333-346 (1997)

[32]

Lin, Q., Yan, N.N. Construction and Analysis for Effective Finite Element Methods. Hebei University Press, Baoding, 1996 (in Chinese)

[33]

Jüngel, A. López, J. L. Gámez, J. M. A new derivation of the quantum Navier-Stokes equations in the Wigner-Fokker-Planck approach. J. Stat. Phys., 145: 1661-1673 (2011)

[34]

Zeidler, E. Nonlinear Functional Analysis and Its Applications, ⅡA and ⅡB. Springer-Verlag, New York, 1990

[35]

Dong, B.Q, Chen, Z.M. Regularity criteria of weak solutions to the three-dimensional micropolar flows. J. Math. Phys., 50: 1-13 (2009)

[36]

Simmons, G.J. Authentication theory/coding theory. Advance in Cryptology: Proc. of Crypto 84, Lecture Notes in Computer Science, 196, 1985, 411-432

[37]

Stinson, D.R. The combinatorics of authentication and secrecy codes. J. Cryptology, 2: 23-49 (1990)

[38]

Zhao, J.N. Stability of solutions for a class of quasilinear degenerate parabolic equations. Northeast Math. J., 10(2): 279-284 (1994)

[39]

Wan, Z.X. Geometry of Classical Groups over Finite Fields (Second Edition). Science Press, Beijing, 2002

Jüngel, A., Miliši?, J. P. Full compressible Navier-Stokes equations for quantum fluids: derivation and numerical solution. Kinetic Related Models, 4(3): 785-807 (2011)

[42]

Variyath, A.M., Chen, J.H., Abraham, B. Empirical likelihood based variable selection. J. Statist. Plann. Inference, 140: 971-981 (2010)

[43]

Awrejcewicz, J., Holicke, M.M. Melnikov's method and stick-slip chaotic oscillations in very weakly forced mechanical systems. Internat. J. Bifur. Chaos, 9(3): 505-518 (1999)

[44]

Liu, H.P., Yan, N.N. Superconvergence analysis of the nonconforming quadrilateral linear-constant scheme for Stokes equations. Adv. Comp. Math., 29: 375-392 (2008)

[45]

Dobson, E., Witte, D. Transitive permutation groups of prime-squared degree. J. Algebraic Combin., 16: 43-69 (2002)

Wang, H., Xing, C., Safavi-Naini, R. Linear authentication codes: Bounds and constructions. IEEE Trans. Inf. Theory, 49(4): 866-872 (2003)

[48]

Liu, Y., Li, H. H1-Galerkin mixed finite element methods for pseudo-hyperbolic equations. Appl. Math. Comp., 212: 446-457 (2009)

[49]

Wang, Q.H., Jing, B.Y. Empirical likelihood for partial linear models with fixed designs. Statist. Probab. Lett., 41: 425-433 (1999)

[50]

Li, H., Li, J., Xin, Z. Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations. Commun. Math. Phys., 281: 401-444 (2008)

[51]

Dong, B.Q., Zhang, Z. The BKM criterion for the 3D Navier-Stokes equations via two velocity components. Nonliear Analysis: RWA, 11: 2415-2421 (2010)

[52]

Du, S.F., Wang, R.J., Xu, M.Y. On the normality of Cayley digraphs of order twice a prime. Australasian Journal of Combinatorics, 18: 227-234 (1998)

[53]

Liu, Y., Li, H. A new mixed finite element method for pseudo-hyperbolic equations. Math. Appl., 23: 150-157 (2010)

[54]

Wang, Q.H., Li, G. Empirical likelihood semi-parametric regression analysis under random censorship. J. Multivariate Anal., 83: 469-486 (2002)

[55]

Dong, B.Q, Zhang, Z. Global regularity for the 2D micropolar fluid flows with zero angular viscosity. J. Differential equations, 249: 200-213 (2010)

[56]

Xin, Z. Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density. Commun. Pure Appl. Math., 51: 229-240 (1998)