In this article, we focus on the short time strong solution to a compressible quantum hydrodynamic model. We establish a blow-up criterion about the solutions of the compressible quantum hydrodynamic model in terms of the gradient of the velocity, the second spacial derivative of the square root of the density, and the first order time derivative and first order spacial derivative of the square root of the density.
Guangwu WANG
,
Boling GUO
. A BLOW-UP CRITERION OF STRONG SOLUTIONS TO THE QUANTUM HYDRODYNAMIC MODEL[J]. Acta mathematica scientia, Series B, 2020
, 40(3)
: 795
-804
.
DOI: 10.1007/s10473-020-0314-3
[1] Antonelli P, Marcati P. On the finite energy weak solutions to a system in quantum fluid dynamics. Comm Math Phys, 2009, 287(2):657-686
[2] Antonelli P, Marcati P. The quantum hydrodynamics system in two space in two space dimensions. Arch Rational Mech Anal, 2012, 203:499-527
[3] Cho Y, Choe H J, Kim H. Unique solvability of the initial boundary value problems for compressible viscous fluids. J Math Pures Appl, 2004, 83:243-275
[4] Fan J S, Jiang S. Blow-up criteria for the Navier-Stokes equations of compressible fluids. J Hyper Diff Eqns, 2008, 5:167-185
[5] Feynman R, Giorgini S, Pitaevskii L, Stringari S. Theory of Bose-Einstein condensation in trapped gases. Rev Mode Phys, 1999, 71:463-512
[6] Gamba I M, Gualdani M P, Zhang P. On the blowing up of solutions to quantum hydrodynamic models on bounded domains. Monatsh Math, 2009, 157:37-54
[7] Gamba I M, Jüngel A. Asymptotic limits in quantum trajectory models. Comm PDE, 2002, 27:669-691
[8] Gamba I M, Jüngel A. Positive solutions to singular second and third order differential equations for quantum fluids. Arch Ration Mech Anal, 2001, 156:183-203
[9] Gardner C. The quantum hydrodynamic model for semiconductors devices. SIAM J Appl Math, 1994, 54:409-427
[10] Gasser I, Markowich P. Quantum hydrodynamics, Wigner transforms and the classical limit. Asymptotic Anal, 1997, 14:97-116
[11] Gasser I, Markowich P A, Ringhofer C. Closure conditions for classical and quantum moment hierarchies in the small temperature limit. Transp Theory Stat Phys, 1996, 25:409-423
[12] Gualdani M P, Jüngel A. Analysis of the viscous quantum hydrodynamic equations for semiconductors. Eur J Appl Math, 2014, 15:577-595
[13] Guo B L, Wang G W. Blow-up of the smooth solution to quantum hydrodynamic models in Rd. J Diff Eqns, 2016, 162(7):3815-3842
[14] Guo B L, Wang G W. Blow-up of the smooth solution to quatum hydrodynamic models in half space. J Math Phys, 2017, 58(3):031505
[15] Huang F M, Li H L. Matsumura A. Existence and stability of steady-state of one-dimensional quantum hydrodynamic system for semiconductors. J Diff Eqns, 2006, 225(1):1-25
[16] Huang F M, Li H L, Matsumura A, Odanaka S. Well-posedness and stability of multi-dimensional quantum hydrodynamics for semiconductors in R3//Series in Contemporary Applied Mathematics CAM 15. Beijing:High Education Press, 2010
[17] Huang X D, Li J, Xin Z P. Blowup criterion for viscous barotropic flows with vacuum states. Comm Math Phys, 2011, 301:23-35
[18] Huang X D, Li J, Xin Z P. Serrin type criterion for the three-dimensional viscous compressible flows. SIAM J Math Anal, 2011, 43:1872-1886
[19] Huang X D, Xin Z P. A blow-up criterion for classical solutions to the compressible Navier-Stokes equations. Sci China Math, 2010, 53(3):671-686
[20] Jüngel A. A steady-state quantum Euler-Poisson system for potential flows. Comm Math Phys, 1998, 194:463-479
[21] Jüngel A, Li H L. Quantum Euler-Poisson systems:existence of stationary states. Arch Math (Brno), 2004, 40:435-456
[22] Jüngel A, Li H L. Quantum Euler-Poisson systems:global existence and exponential decay. Quart Appl Math, 2004, 62(3):569-600
[23] Jüngel A, Milisic J P. Physical and numerical viscosity for quantum hydrodynamics. Comm Math Sci, 2007, 5(2):447-471
[24] Landau L D. Theory of the superfluidity of Helium II. Phys Rev, 1941, 60:356
[25] Landau L D, Lifshitz E M. Quantum mechanics:non-relativistic theory. New York:Pergamon Press, 1977
[26] Li H L, Marcati P. Existence and asymptotic behavior of multi-dimensional quantum hydrodynamic model for semiconductors. Comm Math Phys, 2004, 245:215-247
[27] Loffredo M, Morato L. On the creation of quantum vortex lines in rotating HeII. Il Nouvo Cimento, 1993, 108B:205-215
[28] Madelung E. Quantuentheorie in hydrodynamischer form. Z Physik, 1927, 40:322
[29] Nishibata S, Suzuki M. Asymptotic stability of a stationary solution to a thermal hydrodynamic model for semiconductor. Arch Rational Mech Anal, 2009, 192(2):187-215
[30] Nishibata S, Suzuki M. Initial boundary value problems for a quantum hdyrodyanmic model of semiconductors:asymptotic behaviors and classical limits. J Diff Eqns, 2008, 244(4):836-874
[31] Sun Y Z, Wang C, Zhang Z F. A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navier-Stokes equations. J Math Pures Appl, 2011, 95:36-47
[32] Zhang B, Jerome J W. On a steady-state quantum hydrodynamic model for semiconductors. Nonlinear Anal TMA, 1996, 26(4):845-856
