Skip to main content
SpringerLink
Log in

On the Global Dynamics of Yang–Mills–Higgs Equations

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

Abstract

We study solutions to the Yang–Mills–Higgs equations on the maximal Cauchy development of the data given on a ball of radius R in \(\mathbb {R}^3\). The energy of the data could be infinite and the solution grows at most inverse polynomially in \(R-t\) as \(t\rightarrow R\). As applications, we derive pointwise decay estimates for Yang–Mills–Higgs fields in the future of a hyperboloid or in the Minkowski space \(\mathbb {R}^{1+3}\) for data bounded in the weighted energy space with weights \(|x|^{1+\epsilon }\). Moreover, for the abelian case of Maxwell–Klein–Gordon system, we extend the small data result of Lindblad and Sterbenz (IMRP Int Math Res Pap 109:1687-3017, 2006) to general large data (under same assumptions but without any smallness). The proof is gauge independent and it is based on the framework of Eardley and Moncrief (Commun Math Phys 83(2):171–191, 1982a, 1982b) together with the geometric Kirchhoff–Sobolev parametrix constructed by Klainerman and Rodnianski (J Hyperbolic Differ Equ 4(3):401–433, 2007). The new ingredient is a class of weighted energy estimates through backward light cones adapted to the initial data.

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

Notes

  1. We have \(A_{\mu } \cdot \phi =\beta _*(A_{\mu })(\phi )\), where \(\beta _*:\mathfrak {g}\rightarrow \mathfrak {u}(V)\) is the tangent map of \(\beta \).

  2. If we formally expand the Yang–Mills field F near the spatial infinity

    $$\begin{aligned} F=F_2+F_3+\cdots +F_k+\cdots ,\quad F_k=O(|x|^{-k}), \end{aligned}$$

    then \(F_2\), \(F_3\) and \(F_4\) vanish at any fixed time.

References

  1. Tesfahun, A.: Local well-posedness of Yang–Mills equations in Lorenz gauge below the energy norm. NoDEA Nonlinear Differ. Equ. Appl. 22(4), 849–875 (2015)

    MathSciNet  Google Scholar 

  2. Bieri, L., Miao, S., Shahshahani, S.: Asymptotic properties of solutions of the Maxwell Klein Gordon equation with small data. Commun. Anal. Geom. 25(1), 25–96 (2017)

    MathSciNet  Google Scholar 

  3. Candy, T., Kauffman, C., Lindblad, H.: Asymptotic behavior of the Maxwell–Klein–Gordon system. Commun. Math. Phys. 367(2), 683–716 (2019)

    ADS  MathSciNet  Google Scholar 

  4. Choquet-Bruhat, Y.: Global solutions of Yang–Mills field equations. Rend. Sem. Mat. Fis. Milano 52(247–259), 1982 (1985)

    MathSciNet  Google Scholar 

  5. Choquet-Bruhat, Y., Christodoulou, D.: Existence of global solutions of the Yang–Mills, Higgs and spinor field equations in \(3+1\) dimensions. Ann. Sci. École Norm. Sup. (4) 14(4), 481–506 (1981)

    MathSciNet  Google Scholar 

  6. Choquet-Bruhat, Y., Paneitz, S., Segal, I.: The Yang–Mills equations on the universal cosmos. J. Funct. Anal. 53(2), 112–150 (1983)

    MathSciNet  Google Scholar 

  7. Choquet-Bruhat, Y., Segal, I.: Solution globale des équations de Yang–Mills sur l’univers d’Einstein. C. R. Acad. Sci. Paris Sér. I Math. 294(6), 225–230 (1982)

    MathSciNet  Google Scholar 

  8. Christodoulou, D.: Global solutions of nonlinear hyperbolic equations for small initial data. Commun. Pure Appl. Math. 39(2), 267–282 (1986)

    MathSciNet  Google Scholar 

  9. Christodoulou, D.: Mathematical problems of general relativity. I. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2008)

  10. Chruściel, P., Kondracki, W.: Some global charges in classical Yang–Mills theory. Phys. Rev. D (3) 36(6), 1874–1881 (1987)

    ADS  MathSciNet  Google Scholar 

  11. Chruściel, P., Shatah, J.: Global existence of solutions of the Yang–Mills equations on globally hyperbolic four-dimensional Lorentzian manifolds. Asian J. Math. 1(3), 530–548 (1997)

    MathSciNet  Google Scholar 

  12. Eardley, D., Moncrief, V.: The global existence of Yang–Mills–Higgs fields in \(4\)-dimensional Minkowski space. I. Local existence and smoothness properties. Commun. Math. Phys. 83(2), 171–191 (1982)

    ADS  MathSciNet  Google Scholar 

  13. Eardley, D., Moncrief, V.: The global existence of Yang–Mills–Higgs fields in \(4\)-dimensional Minkowski space. II. Completion of proof. Commun. Math. Phys. 83(2), 193–212 (1982)

    ADS  MathSciNet  Google Scholar 

  14. Georgiev, V., Schirmer, P.: The asymptotic behavior of Yang–Mills fields in the large. Commun. Math. Phys. 148(3), 425–444 (1992)

    ADS  MathSciNet  Google Scholar 

  15. Ghanem, S.: The global non-blow-up of the Yang–Mills curvature on curved space-times. J. Hyperbolic Differ. Equ. 13(3), 603–631 (2016)

    MathSciNet  Google Scholar 

  16. Ghanem, S., Häfner, D.: The decay of the \(SU(2)\) Yang-Mills fields on the Schwarzschild black hole for spherically symmetric small energy initial data. J. Geom. Phys. 123, 310–342 (2018)

    ADS  MathSciNet  Google Scholar 

  17. Glassey, R., Strauss, W.: Decay of a Yang–Mills field coupled to a scalar field. Commun. Math. Phys. 67(1), 51–67 (1979)

    ADS  MathSciNet  Google Scholar 

  18. Glassey, R., Strauss, W.: Decay of classical Yang–Mills fields. Commun. Math. Phys. 65(1), 1–13 (1979)

    ADS  MathSciNet  Google Scholar 

  19. Glassey, R., Strauss, W.: Some global solutions of the Yang–Mills equations in Minkowski space. Commun. Math. Phys. 81(2), 171–187 (1981)

    ADS  MathSciNet  Google Scholar 

  20. Glassey, R., Strauss, W.: The scattering of certain Yang–Mills fields. Commun. Math. Phys. 89(4), 465–482 (1983)

    ADS  MathSciNet  Google Scholar 

  21. Goganov, M., Kapitanskiĭ, L.: Global Solvability of the Cauchy Problem for Yang–Mills–Higgs Equations, vol 147, pp. 18–48, 203–204. Boundary Value Problems of Mathematical Physics and Related Problems in the Theory of Functions, No. 17 (1985)

  22. Kauffman, C.: Global Stability for Charged Scalar Fields in an Asymptotically Flat Metric in Harmonic Gauge (2018). arXiv:1801.09648

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

    MathSciNet  Google Scholar 

  24. Klainerman, S., Machedon, M.: Finite energy solutions of the Yang–Mills equations in \(\mathbb{R} ^{3+1}\). Ann. Math. (2) 142(1), 39–119 (1995)

    MathSciNet  Google Scholar 

  25. Klainerman, S., Rodnianski, I.: A Kirchoff–Sobolev parametrix for the wave equation and applications. J. Hyperbolic Differ. Equ. 4(3), 401–433 (2007)

    MathSciNet  Google Scholar 

  26. Klainerman, S., Tataru, D.: On the optimal local regularity for Yang–Mills equations in \({ R}^{4+1}\). J. Am. Math. Soc. 12(1), 93–116 (1999)

    Google Scholar 

  27. Krieger, J., Lührmann, J.: Concentration compactness for the critical Maxwell–Klein–Gordon equation. Ann. PDE 1(1), 5 (2015)

    MathSciNet  Google Scholar 

  28. Lindblad, H., Sterbenz, J.: Global stability for charged-scalar fields on Minkowski space. IMRP Int. Math. Res. Pap., pp. 1687–3017 (2006)

  29. Machedon, M., Sterbenz, J.: Almost optimal local well-posedness for the \((3+1)\)-dimensional Maxwell–Klein–Gordon equations. J. Am. Math. Soc. 17(2), 297–359 (2004)

    MathSciNet  Google Scholar 

  30. Oh, S.: Finite energy global well-posedness of the Yang–Mills equations on \(\mathbb{R} ^{1+3}\): an approach using the Yang–Mills heat flow. Duke Math. J. 164(9), 1669–1732 (2015)

    MathSciNet  Google Scholar 

  31. Oh, S., Tataru, D.: Global well-posedness and scattering of the \((4+1)\)-dimensional Maxwell–Klein–Gordon equation. Invent. Math. 205(3), 781–877 (2016)

    ADS  MathSciNet  Google Scholar 

  32. Oh, S., Tataru, D.: The threshold conjecture for the energy critical hyperbolic Yang–Mills equation. Ann. Math. (2) 194(2), 393–473 (2021)

    MathSciNet  Google Scholar 

  33. Pecher, H.: Infinite energy solutions for the \((3+1)\)-dimensional Yang–Mills equation in Lorenz gauge. Commun. Pure Appl. Anal. 18(2), 663–688 (2019)

    MathSciNet  Google Scholar 

  34. Pecher, H.: Low regularity well-posedness for the Yang–Mills system in Fourier–Lebesgue spaces. SIAM J. Math. Anal. 52(4), 3131–3148 (2020)

    MathSciNet  Google Scholar 

  35. Rodnianski, I., Tao, T.: Global regularity for the Maxwell–Klein–Gordon equation with small critical Sobolev norm in high dimensions. Commun. Math. Phys. 251(2), 377–426 (2004)

    ADS  MathSciNet  Google Scholar 

  36. Schirmer, P.: Decay estimates for spherically symmetric Yang–Mills fields in Minkowski space-time. Ann. Inst. H. Poincaré Anal. Non Linéaire 10(5), 481–522 (1993)

    ADS  MathSciNet  Google Scholar 

  37. Selberg, S., Tesfahun, A.: Null structure and local well-posedness in the energy class for the Yang–Mills equations in Lorenz gauge. J. Eur. Math. Soc. (JEMS) 18(8), 1729–1752 (2016)

    MathSciNet  Google Scholar 

  38. Shu, W.: Asymptotic properties of the solutions of linear and nonlinear spin field equations in Minkowski space. Commun. Math. Phys. 140(3), 449–480 (1991)

    ADS  MathSciNet  Google Scholar 

  39. Shu, W.: Global existence of Maxwell-Higgs fields. In: Nonlinear Hyperbolic Equations and Field Theory (Lake Como, 1990), volume 253 of Pitman Res. Notes Math. Ser., pp. 214–227. Longman Sci. Tech., Harlow (1992)

  40. Tao, T.: Local well-posedness of the Yang–Mills equation in the temporal gauge below the energy norm. J. Differ. Equ. 189(2), 366–382 (2003)

    ADS  MathSciNet  Google Scholar 

  41. Taujanskas, G.: Large data decay of Yang–Mills–Higgs fields on Minkowski and de Sitter spacetimes. J. Math. Phys. 60(12), 121504 (2019)

    ADS  MathSciNet  Google Scholar 

  42. Tesfahun, A.: Finite energy local well-posedness for the Yang–Mills–Higgs equations in Lorenz gauge. Int. Math. Res. Not. IMRN 13, 5140–5161 (2015)

    MathSciNet  Google Scholar 

  43. Yang, C., Mills, R.: Conservation of isotopic spin and isotopic gauge invariance. Phys. Rev. 2(96), 191–195 (1954)

    ADS  MathSciNet  Google Scholar 

  44. Yang, S.: Decay of solutions of Maxwell–Klein–Gordon equations with arbitrary Maxwell field. Anal. PDE 9(8), 1829–1902 (2016)

    MathSciNet  Google Scholar 

  45. Yang, S.: On the global behavior of solutions of the Maxwell–Klein–Gordon equations. Adv. Math. 326, 490–520 (2018)

    MathSciNet  Google Scholar 

  46. Yang, S.: Pointwise decay for semilinear wave equations in \(\mathbb{R} ^{1+3}\). J. Funct. Anal. 283(2) (2022)

  47. Yang, S., Yu, P.: On global dynamics of the Maxwell–Klein–Gordon equations. Camb. J. Math. 7(4), 365–467 (2019)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

S. Yang is supported by the National Science Foundation of China 12171011, 12141102 and the National Key R &D Program of China 2021YFA1001700. P. Yu is supported by the National Science Foundation of China 11825103, 12141102, MOST-2020YFA0713003 and Xiaomi Fellowship.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Peking University, Beijing, China

    Dongyi Wei

  2. Beijing International Center for Mathematical Research, Peking University, Beijing, China

    Shiwu Yang

  3. Department of Mathematical Sciences, Tsinghua University, Beijing, China

    Pin Yu

Authors
  1. Dongyi Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shiwu Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pin Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pin Yu.

Additional information

Communicated by A. Ionescu

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wei, D., Yang, S. & Yu, P. On the Global Dynamics of Yang–Mills–Higgs Equations. Commun. Math. Phys. 405, 4 (2024). https://doi.org/10.1007/s00220-023-04881-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00220-023-04881-9

Search

Navigation