Quantum Navier–Stokes Equations

  • Ansgar Jüngel
  • Josipa-Pina Milišić
Conference paper
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 17)


Compressible Navier–Stokes models for quantum fluids are reviewed. They are derived from a collisional Wigner equation by a moment method and a Chapman–Enskog expansion around the quantum equilibrium. Introducing a new velocity variable, the barotropic quantum Navier–Stokes model can be reformulated as a viscous quantum Euler system, which possesses a new Lyapunov (energy) functional. This functional provides a priori estimates which are exploited to prove the global-in-time existence of weak solutions for general initial data. Furthermore, new numerical results for the isothermal model are presented.


Stokes System Negative Differential Resistance Tunneling Diode Global Weak Solution General Initial Data 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Allemand, T.: Derivation of a two-fluids model for a Bose gas from a quantum kinetic system. Kinet. Relat. Model. 2, 379–402 (2009)Google Scholar
  2. 2.
    Antonelli, P., Marcati, P.: On the finite energy weak solutions to a system in quantum fluid dynamics. Commun. Math. Phys. 287, 657–686 (2009)Google Scholar
  3. 3.
    Arecchi, F., Bragard, J., Castellano, L.: Dissipative dynamics of an open Bose-Einstein condensate. Optics. Commun. 179, 149–156 (2000)Google Scholar
  4. 4.
    Brenner, H.: Navier-Stokes revisited. Phys. A 349, 60–132 (2005)Google Scholar
  5. 5.
    Bresch, D., Desjardins, B.: Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model. Commun. Math. Phys. 238, 211–223 (2003)Google Scholar
  6. 6.
    Bresch, D., Desjardins, B.: Some diffusive capillary models of Korteweg type. C. R. Math. Acad. Sci. Paris, Sec. Mécanique 332, 881–886 (2004)Google Scholar
  7. 7.
    Bresch, D., Desjardins, B.: On the existence of global weak solutions to the Navier–Stokes equations for viscous compressible and heat conducting fluids. J. Math. Pures Appl. 87, 57–90 (2007)Google Scholar
  8. 8.
    Bresch, D., Desjardins, B., Lin, C.-K.: On some compressible fluid models: Korteweg, lubrication and shallow water systems. Commun. Part. Diff. Eqs. 28, 1009–1037 (2003)Google Scholar
  9. 9.
    Brull, S., Méhats, F.: Derivation of viscous correction terms for the isothermal quantum Euler model. Z. Angew. Math. Mech. 90, 219–230 (2010)Google Scholar
  10. 10.
    Burger, S., Cataliotti, F., Fort, C., Minardi, F., Inguscio, M., Chiofalo, M., Tosi, M.: Superfluid and dissipative dynamics of a Bose-Einstein condensate in a periodic optimal potential. Phys. Rev. Lett. 86, 4447–4450 (2001)Google Scholar
  11. 11.
    Chen, L., Dreher, M.: The viscous model of quantum hydrodynamics in several dimensions. Math. Model. Meth. Appl. Sci. 17, 1065–1093 (2007)Google Scholar
  12. 12.
    Degond, P., Méhats, F., Ringhofer, C.: Quantum energy-transport and drift-diffusion models. J. Stat. Phys. 118, 625–665 (2005)Google Scholar
  13. 13.
    Degond, P., Ringhofer, C.: Quantum moment hydrodynamics and the entropy principle. J. Stat. Phys. 112, 587–628 (2003)Google Scholar
  14. 14.
    Dong, J.: A note on barotropic compressible quantum Navier-Stokes equations. Nonlin. Anal. 73, 854–856 (2010)Google Scholar
  15. 15.
    Feireisl, E.: Dynamics of Viscous Compressible Fluids. Oxford University Press, Oxford (2004)Google Scholar
  16. 16.
    Ferry, D., Zhou, J.-R.: Form of the quantum potential for use in hydrodynamic equations for semiconductor device modeling. Phys. Rev. B 48, 7944–7950 (1993)Google Scholar
  17. 17.
    Gamba, I., Jüngel, A., Vasseur, A.: Global existence of solutions to one-dimensional viscous quantum hydrodynamic equations. J. Diff. Eqs. 247, 3117–3135 (2009)Google Scholar
  18. 18.
    Gualdani, M., Jüngel, A.: Analysis of the viscous quantum hydrodynamic equations for semiconductors. Europ. J. Appl. Math. 15, 577–595 (2004)Google Scholar
  19. 19.
    Harvey, R.: Navier-Stokes analog of quantum mechanics. Phys. Rev. 152, 1115 (1966)Google Scholar
  20. 20.
    Jiang, F.: A remark on weak solutions to the barotropic compressible quantum Navier-Stokes equations, Nonlin. Anal. Real World Appl. 12, 1733–1735 (2011)Google Scholar
  21. 21.
    Jüngel, A.: A steady-state quantum Euler–Poisson system for semiconductors. Commun. Math. Phys. 194, 463–479 (1998)Google Scholar
  22. 22.
    Jüngel, A.: Transport Equations for Semiconductors. Lecture Notes in Physics, 773. Springer, Berlin (2009)Google Scholar
  23. 23.
    Jüngel, A.: Global weak solutions to compressible Navier-Stokes equations for quantum fluids. SIAM J. Math. Anal. 42, 1025–1045 (2010)Google Scholar
  24. 24.
    Jüngel, A., Matthes, D.: The Derrida-Lebowitz-Speer-Spohn equation: Existence, nonuniqueness, and decay rates of the solutions. SIAM J. Math. Anal. 39, 1996–2015 (2008)Google Scholar
  25. 25.
    Jüngel, A., Matthes, D., Milišić, J.-P.: Derivation of new quantum hydrodynamic equations using entropy minimization. SIAM J. Appl. Math. 67, 46–68 (2006)Google Scholar
  26. 26.
    Jüngel, A., Milišić, J.-P.: Physical and numerical viscosity for quantum hydrodynamics. Commun. Math. Sci. 5, 447–471 (2007)Google Scholar
  27. 27.
    Jüngel, A., Milišić, J.-P.: Full compressible Navier-Stokes equations for quantum fluids: derivation and numerical solution. Preprint, Vienna University of Technology, Austria (2010)Google Scholar
  28. 28.
    Levermore, C.D.: Moment closure hierarchies for kinetic theory. J. Stat. Phys. 83, 1021–1065 (1996)Google Scholar
  29. 29.
    Li, H.-L., 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)Google Scholar
  30. 30.
    Madelung, E.: Quantentheorie in hydrodynamischer Form. Z. Phys. 40, 322–326 (1927)Google Scholar
  31. 31.
    Méhats, F., Pinaud, O.: An inverse problem in quantum statistical physics. Preprint, Université de Rennes, France (2010)Google Scholar
  32. 32.
    Slavchov, R., Tsekov, R.: Quantum hydrodynamics of electron gases. J. Chem. Phys. 132, 084505 (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institute for Analysis and Scientific ComputingVienna University of TechnologyWienAustria
  2. 2.Department of Applied MathematicsUniversity of ZagrebZagrebCroatia

Personalised recommendations