Skip to main content
SpringerLink
Log in

Uniqueness of Finite Energy Solutions for Maxwell-Dirac and Maxwell-Klein-Gordon Equations

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We prove uniqueness of solutions to the Maxwell-Dirac system in the energy space, namely . We also give a proof for uniqueness of finite energy solutions to the Maxwell-Klein-Gordon equations, which is simpler than that given in [16].

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bergh, J., Löfström, J.: Interpolation spaces. An introduction. Grundlehren Math. Wiss. 223, Berlin-Heidelberg-New York: Springer, 1976

  2. Bournaveas, N.: Local existence for the Maxwell-Dirac equations in three space dimensions. Commun. Partial Diff. Eqs. 21(5–6), 693–720 (1996)

    Google Scholar 

  3. Bournaveas, N.: Local existence of energy class solutions for the Dirac-Klein-Gordon equations. Comm. Partial Differ. Eqs. 24(7–8), 1167–1193 (1999)

    Google Scholar 

  4. Cuccagna, S.: On the local existence for the Maxwell-Klein-Gordon system in R 3+1. Comm. Partial Differ. Eqs. 24(5–6), 851–867 (1999)

    Google Scholar 

  5. Dirac, P.A.M.: Principles of Quantum Mechanics. 4th ed., London: Oxford University Press, 1958

  6. Esteban, M.J., Georgiev, V., Séré, E.: Stationary solutions of the Maxwell-Dirac and the Klein- Gordon-Dirac equations, Calc. Var. 4, 265–281 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Furioli, G., Lemarié-Rieusset, P.G., Terraneo, E.: Unicité dans L 3(ℝ3) et d’autres espaces fonctionnels limites pour Navier-Stokes. Rev. Mat. Iberoamericana 16(3), 605–667 (2000)

    MATH  Google Scholar 

  8. Klainerman, S., Machedon, M.: On the Maxwell-Klein-Gordon equation with finite energy. Duke Math. J. 74(1), 19–44 (1994)

    MATH  Google Scholar 

  9. Klainerman, S., Machedon, M.: Space-time estimates for null forms and the local existence theorem. Comm. Pure Appl. Math. 46(9), 1221–1268 (1993)

    MATH  Google Scholar 

  10. Klainerman, S., Machedon, M.: Smoothing estimates for null forms and applications. Internat. Math. Res. Notices 9, 383–390 (1994)

    Article  MATH  Google Scholar 

  11. Lions, P.L., Masmoudi, N.: Uniqueness of mild solutions of the Navier-Stokes system in L N. Comm. Partial Differ. Eqs. 26(11-12), 2211–2226 (2001)

    Google Scholar 

  12. Masmoudi, N., Mauser, N.: The selfconsistent Pauli equation and its semiclassical/nonrelativistic limits. Monatsh. Math. 132(1), 19–24 (2001)

    Article  MATH  Google Scholar 

  13. Masmoudi, N., Nakanishi, K.: Nonrelativistic limit from Maxwell-Klein-Gordon and Maxwell-Dirac to Poisson-Schrödinger. In. Math. Res. No. 13, 697–734 (2003)

    Article  MATH  Google Scholar 

  14. Planchon, F.: On uniqueness for semilinear wave equations. To appear in Math. Z.

  15. Tataru, D.: The X s θ spaces and unique continuation for solutions to the semilinear wave equation. Comm. Partial Differ. Eqs. 21(5-6), 841–887 (1996)

  16. Zhou, Y.: Uniqueness of generalized solutions to nonlinear wave equations. Am. J. Math. 122(5), 939–965 (2000)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Courant Institute, New York University, New York, NY, 10012, USA

    Nader Masmoudi

  2. Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan

    Kenji Nakanishi

Authors
  1. Nader Masmoudi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kenji Nakanishi
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by H.-T. Yau

The first author is partially supported by an NSF grant and an Alfred Sloan fellowship. The second author is supported by JSPS Postdoctoral Fellowships for Research Abroad (2001–2003)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Masmoudi, N., Nakanishi, K. Uniqueness of Finite Energy Solutions for Maxwell-Dirac and Maxwell-Klein-Gordon Equations. Commun. Math. Phys. 243, 123–136 (2003). https://doi.org/10.1007/s00220-003-0951-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-003-0951-0

Keywords

Search

Navigation