Skip to main content

The Quantum Hydrodynamics System in Two Space Dimensions


In this paper we study global existence of weak solutions for the quantum hydrodynamics system in two-dimensional energy space. We do not require any additional regularity and/or smallness assumptions on the initial data. Our approach replaces the WKB formalism with a polar decomposition theory which is not limited by the presence of vacuum regions. In this way we set up a self consistent theory, based only on particle density and current density, which does not need to define velocity fields in the nodal regions. The mathematical techniques we use in this paper are based on uniform (with respect to the approximating parameter) Strichartz estimates and the local smoothing property.

This is a preview of subscription content, access via your institution.


  1. Ancona M., Iafrate G.: Quantum correction to the equation of state of an electron gas in a semiconductor. Phys. Rev. B 39, 9536–9540 (1989)

    Article  ADS  Google Scholar 

  2. Antonelli P., Marcati P.: On the finite energy weak solutions to a system in Quantum Fluid Dynamics. Comm. Math. Phys. 287(2), 657–686 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. Aubin J.-P.: Un Théorème de compacité. C. R. Acad. Sci. 256, 5042–5044 (1963)

    MathSciNet  MATH  Google Scholar 

  4. Baccarani G., Wordeman M.R.: An investigation of steady state velocity overshoot effects in Si and GaAs devices. Solid State Electron ED-29, 970–977 (1982)

    Google Scholar 

  5. Brenier Y.: Polar factorization and monotone rearrangement of vector-valued function. Comm. Pure Appl. Math. 44, 375–417 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  6. Carlen E.: Conservative diffusions. Comm. Math. Phys. 94, 293–315 (1984)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Carlen E., Loss M.: Competing symmetries, the logarithmic HLS inequality and Onofri’s inequality on \({{\mathbb{S}^n}^n}\) . Geom. Funct. Anal. 2(1), 90–104 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cazenave T.: Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics, vol. 10. New York University, Courant Institute of Mathematical Sciences, AMS, 2003

  9. Constantin P., Saut J.-C.: Local smoothing properties of dispersive equations. J. Am. Math. Soc. 1, 413–439 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dalfovo F., Giorgini S., Pitaevskii L., Stringari S.: Theory of Bose-Einstein condensation in trapped gases. Rev. Mod. Phys. 71, 463–512 (1999)

    Article  ADS  Google Scholar 

  11. Degond P., Gallego S., Méhats F.: Isothermal quantum hydrodynamics: derivation, asymptotic analysis, and simulation. Multiscale Model. Simul. 6(1), 246–272 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Feynman R.P.: Superfluidity and superconductivity. Rev. Mod. Phys. 29(2), 205 (1957)

    Article  ADS  Google Scholar 

  13. Gardner C.: The quantum hydrodynamic model for semiconductor devices. SIAM J. Appl. Math. 54, 409–427 (1994)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. Gasser I., Markowich P.: Quantum hydrodynamics, Wigner transforms and the classical limit. Asymptot. Anal. 14(2), 97–116 (1997)

    MathSciNet  MATH  Google Scholar 

  15. Ginibre J., Velo G.: The global Cauchy problem for the nonlinear Schrödinger equations revisited. Ann. Inst. H. Poincaré Anal. Non Linéaire 2, 309–327 (1987)

    MathSciNet  Google Scholar 

  16. Jüngel A., Mariani M.C., Rial D.: Local existence of solutions to the transient quantum hydrodynamic equations. Math. Models Methods Appl. Sci. 12(4), 485–495 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jüngel A., Matthes D., Milisic J.P.: Derivation of new quantum hydrodynamic equations using entropy minimization. SIAM J. Appl. Math. 67(1), 46–68 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kadanoff L.P., Baym G.: Quantum Statistical Mechanics. Benjamin, New York (1962)

    MATH  Google Scholar 

  19. Khalatnikov I.M.: An introduction to the Theory of Superfluidity. Benjamin, New York (1965)

    Google Scholar 

  20. Keel M., Tao T.: Endpoint Strichartz estimates. Am. J. Math. 120, 955–980 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Landau L.D.: Theory of the superfluidity of helium II. Phys. Rev. 60, 356 (1941)

    Article  ADS  MATH  Google Scholar 

  22. Li H.L., Marcati P.: Existence and asymptotic behavior of multi-dimensional quantum hydrodynamic model for semiconductors. Comm. Math. Phys. 245(2), 215–247 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. Madelung E.: Quantentheorie in hydrodynamischer form. Z. Physik 40, 322 (1927)

    Article  ADS  Google Scholar 

  24. Rakotoson J.M., Temam R.: An optimal compactness theorem and application to elliptic-parabolic systems. Appl. Math. Lett. 14, 303–306 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sjölin P.: Regularity of solutions to the Schrödinger equation. Duke Math. J. 55(3), 699–715 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  26. Tao, T.: Nonlinear dispersive equations: local and global analysis.CBMS regional conference series in mathematics, vol. 106. Published for the Conference Board of the Mathematical Sciences, By the American Mathematical Society, Providence, RI, xvi+373 pp. Washington, DC, 2006

  27. Taylor M.E.: Pseudodifferential operators and nonlinear PDEs. Progress in Mathematics, vol. 100. Birkäuser, Basel (1991)

    Book  Google Scholar 

  28. Teufel S., Tumulka R.: Simple proof for global existence of bohmian trajectories. Comm. Math. Phys. 258, 349–365 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  29. Vega L.: Schrödinger equations: pointwise convergence to the initial data. Proc. AMS 102(4), 874–878 (1988)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Paolo Antonelli.

Additional information

Communicated by C. Dafermos

Paolo Antonelli is supported by Award No. KUK-I1-007-43, funded by the King Abdullah University of Science and Technology (KAUST).

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Antonelli, P., Marcati, P. The Quantum Hydrodynamics System in Two Space Dimensions. Arch Rational Mech Anal 203, 499–527 (2012).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Weak Solution
  • Polar Decomposition
  • Admissible Pair
  • Strichartz Estimate
  • Logarithmic Sobolev Inequality