American Institute of Mathematical Sciences

Advanced
March  2019, 18(2): 663-688. doi: 10.3934/cpaa.2019033

Infinite energy solutions for the (3+1)-dimensional Yang-Mills equation in Lorenz gauge

Hartmut Pecher

Fakultät für Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, Gaußstr. 20, 42119 Wuppertal, Germany

Received  December 2017 Revised  May 2018 Published  October 2018

We prove that the Yang-Mills equation in Lorenz gauge in the (3+1)-dimensional case is locally well-posed for data of the gauge potential in $H^s$ and the curvature in $H^r$, where $s >\frac{5}{7}$ and $r > -\frac{1}{7}$, respectively. This improves a result by Tesfahun [16]. The proof is based on the fundamental results of Klainerman-Selberg [6] and on the null structure of most of the nonlinear terms detected by Selberg-Tesfahun [14] and Tesfahun [16].

Keywords: Yang-Mills, local well-posedness, Lorenz gauge.
Mathematics Subject Classification: 35Q40, 35L70.
Citation: Hartmut Pecher. Infinite energy solutions for the (3+1)-dimensional Yang-Mills equation in Lorenz gauge. Communications on Pure & Applied Analysis, 2019, 18 (2) : 663-688. doi: 10.3934/cpaa.2019033
References:
[1]

P. d'Ancona, D. Foschi and S. Selberg, Atlas of products for wave-Sobolev spaces on $\mathbb{R}^{1+3}$, Trans. Amer. Math. Soc., 364 (2012), 31-63.  doi: 10.1090/S0002-9947-2011-05250-5.  Google Scholar

[2]

P. d'Ancona, D. Foschi and S. Selberg, Null structure and almost optimal local well-posedness of the Maxwell-Dirac system, Amer. J. Math., 132 (2010), 771-839.  doi: 10.1353/ajm.0.0118.  Google Scholar

[3]

J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal., 133 (1995), 60-68.  doi: 10.1006/jfan.1995.1119.  Google Scholar

[4]

S. Klainerman and M. Machedon, Finite energy solutions of the Yang-Mills equations in ${\mathbb R}^{3+1}$, Ann. of Math., 142 (1995), 39-119.  doi: 10.2307/2118611.  Google Scholar

[5]

S. Klainerman and M. Machedon (Appendices by J. Bougain and D. Tataru), Remark on Strichartz-type inequalities, Int. Math. Res. Not. IMRN, 5 (1996), 201-220. doi: 10.1155/S1073792896000153.  Google Scholar

[6]

S. Klainerman and S. Selberg, Bilinear estimates and applications to nonlinear wave equations, Commun. Contemp. Math., 4 (2002), 223-295.  doi: 10.1142/S0219199702000634.  Google Scholar

[7]

S. Klainerman and D. Tataru, On the optimal local regularity for the Yang-Mills equations in $\mathbb{R}^{4+1}$, J. Amer. Math. Soc., 12 (1999), 93-116.  doi: 10.1090/S0894-0347-99-00282-9.  Google Scholar

[8]

J. Krieger and J. Sterbenz, Global regularity for the Yang-Mills equations on high dimensonal Minkowski space, Mem. Amer. Math. Soc., 223 (2013), No. 1047 doi: 10.1090/S0065-9266-2012-00566-1.  Google Scholar

[9]

J. Krieger and D. Tataru, Global well-posedness for the Yang-Mills equations in 4+1 dimensions. Small energy, Ann. of Math., 185 (2017), 831-893.  doi: 10.4007/annals.2017.185.3.3.  Google Scholar

[10]

S. Oh, Gauge choice for the Yang-Mills equations using the Yang-Mills heat flow and local well-posedness in H1, J. Hyperbolic Differ. Equ., 11 (2014), 1-108.  doi: 10.1142/S0219891614500015.  Google Scholar

[11]

S. Oh, 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 (2015), 1669-1732.  doi: 10.1215/00127094-3119953.  Google Scholar

[12]

H. Pecher, Local well-posedness for the (n+1)-dimensional Yang-Mills and Yang-Mills-Higgs system in temporal gauge, NoDEA Nonlinear Differential Equations Appl., 23 (2016), 23-40.  doi: 10.1007/s00030-016-0395-9.  Google Scholar

[13]

S. Selberg and A. Tesfahun, Finite-energy global well-posedness of the Maxwell-Klein-Gordon system in Lorenz gauge, Comm. Partial Differential Equations, 35 (2010), 10290-1057.  doi: 10.1080/03605301003717100.  Google Scholar

[14]

S. Selberg and A. Tesfahun, Null structure and local well-posedness in the energy class for the Yang-Mills equations in Lorenz gauge, J. Eur. Math. Soc. (JEMS), 18 (2016), 1729-1752.  doi: 10.4171/JEMS/627.  Google Scholar

[15]

T. Tao, Local well-posedness of the Yang-Mills equation in the temporal gauge below the energy norm, J. Differential Equations, 189 (2003), 366-382.  doi: 10.1016/S0022-0396(02)00177-8.  Google Scholar

[16]

A. Tesfahun, Local well-posedness of Yang-Mills equations in Lorenz gauge below the energy norm, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 849-875.  doi: 10.1007/s00030-014-0306-x.  Google Scholar

show all references

References:
[1]

P. d'Ancona, D. Foschi and S. Selberg, Atlas of products for wave-Sobolev spaces on $\mathbb{R}^{1+3}$, Trans. Amer. Math. Soc., 364 (2012), 31-63.  doi: 10.1090/S0002-9947-2011-05250-5.  Google Scholar

[2]

P. d'Ancona, D. Foschi and S. Selberg, Null structure and almost optimal local well-posedness of the Maxwell-Dirac system, Amer. J. Math., 132 (2010), 771-839.  doi: 10.1353/ajm.0.0118.  Google Scholar

[3]

J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal., 133 (1995), 60-68.  doi: 10.1006/jfan.1995.1119.  Google Scholar

[4]

S. Klainerman and M. Machedon, Finite energy solutions of the Yang-Mills equations in ${\mathbb R}^{3+1}$, Ann. of Math., 142 (1995), 39-119.  doi: 10.2307/2118611.  Google Scholar

[5]

S. Klainerman and M. Machedon (Appendices by J. Bougain and D. Tataru), Remark on Strichartz-type inequalities, Int. Math. Res. Not. IMRN, 5 (1996), 201-220. doi: 10.1155/S1073792896000153.  Google Scholar

[6]

S. Klainerman and S. Selberg, Bilinear estimates and applications to nonlinear wave equations, Commun. Contemp. Math., 4 (2002), 223-295.  doi: 10.1142/S0219199702000634.  Google Scholar

[7]

S. Klainerman and D. Tataru, On the optimal local regularity for the Yang-Mills equations in $\mathbb{R}^{4+1}$, J. Amer. Math. Soc., 12 (1999), 93-116.  doi: 10.1090/S0894-0347-99-00282-9.  Google Scholar

[8]

J. Krieger and J. Sterbenz, Global regularity for the Yang-Mills equations on high dimensonal Minkowski space, Mem. Amer. Math. Soc., 223 (2013), No. 1047 doi: 10.1090/S0065-9266-2012-00566-1.  Google Scholar

[9]

J. Krieger and D. Tataru, Global well-posedness for the Yang-Mills equations in 4+1 dimensions. Small energy, Ann. of Math., 185 (2017), 831-893.  doi: 10.4007/annals.2017.185.3.3.  Google Scholar

[10]

S. Oh, Gauge choice for the Yang-Mills equations using the Yang-Mills heat flow and local well-posedness in H1, J. Hyperbolic Differ. Equ., 11 (2014), 1-108.  doi: 10.1142/S0219891614500015.  Google Scholar

[11]

S. Oh, 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 (2015), 1669-1732.  doi: 10.1215/00127094-3119953.  Google Scholar

[12]

H. Pecher, Local well-posedness for the (n+1)-dimensional Yang-Mills and Yang-Mills-Higgs system in temporal gauge, NoDEA Nonlinear Differential Equations Appl., 23 (2016), 23-40.  doi: 10.1007/s00030-016-0395-9.  Google Scholar

[13]

S. Selberg and A. Tesfahun, Finite-energy global well-posedness of the Maxwell-Klein-Gordon system in Lorenz gauge, Comm. Partial Differential Equations, 35 (2010), 10290-1057.  doi: 10.1080/03605301003717100.  Google Scholar

[14]

S. Selberg and A. Tesfahun, Null structure and local well-posedness in the energy class for the Yang-Mills equations in Lorenz gauge, J. Eur. Math. Soc. (JEMS), 18 (2016), 1729-1752.  doi: 10.4171/JEMS/627.  Google Scholar

[15]

T. Tao, Local well-posedness of the Yang-Mills equation in the temporal gauge below the energy norm, J. Differential Equations, 189 (2003), 366-382.  doi: 10.1016/S0022-0396(02)00177-8.  Google Scholar

[16]

A. Tesfahun, Local well-posedness of Yang-Mills equations in Lorenz gauge below the energy norm, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 849-875.  doi: 10.1007/s00030-014-0306-x.  Google Scholar

[1]

Jianjun Yuan. On the well-posedness of Maxwell-Chern-Simons-Higgs system in the Lorenz gauge. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2389-2403. doi: 10.3934/dcds.2014.34.2389
[2]

Alberto Ibort, Amelia Spivak. Covariant Hamiltonian field theories on manifolds with boundary: Yang-Mills theories. Journal of Geometric Mechanics, 2017, 9 (1) : 47-82. doi: 10.3934/jgm.2017002
[3]

Shaoyun Shi, Wenlei Li. Non-integrability of generalized Yang-Mills Hamiltonian system. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1645-1655. doi: 10.3934/dcds.2013.33.1645
[4]

Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007
[5]

Magdalena Czubak, Nina Pikula. Low regularity well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1669-1683. doi: 10.3934/cpaa.2014.13.1669
[6]

Christopher Henderson, Stanley Snelson, Andrei Tarfulea. Local well-posedness of the Boltzmann equation with polynomially decaying initial data. Kinetic & Related Models, 2020, 13 (4) : 837-867. doi: 10.3934/krm.2020029
[7]

Yong Zhou, Jishan Fan. Local well-posedness for the ideal incompressible density dependent magnetohydrodynamic equations. Communications on Pure & Applied Analysis, 2010, 9 (3) : 813-818. doi: 10.3934/cpaa.2010.9.813
[8]

Caochuan Ma, Zaihong Jiang, Renhui Wan. Local well-posedness for the tropical climate model with fractional velocity diffusion. Kinetic & Related Models, 2016, 9 (3) : 551-570. doi: 10.3934/krm.2016006
[9]

Timur Akhunov. Local well-posedness of quasi-linear systems generalizing KdV. Communications on Pure & Applied Analysis, 2013, 12 (2) : 899-921. doi: 10.3934/cpaa.2013.12.899
[10]

Hung Luong. Local well-posedness for the Zakharov system on the background of a line soliton. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2657-2682. doi: 10.3934/cpaa.2018126
[11]

Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure & Applied Analysis, 2014, 13 (2) : 673-685. doi: 10.3934/cpaa.2014.13.673
[12]

Jae Min Lee, Stephen C. Preston. Local well-posedness of the Camassa-Holm equation on the real line. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3285-3299. doi: 10.3934/dcds.2017139
[13]

Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations I: Local well-posedness. Evolution Equations & Control Theory, 2012, 1 (1) : 195-215. doi: 10.3934/eect.2012.1.195
[14]

Yongye Zhao, Yongsheng Li, Wei Yan. Local Well-posedness and Persistence Property for the Generalized Novikov Equation. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 803-820. doi: 10.3934/dcds.2014.34.803
[15]

Alex M. Montes, Ricardo Córdoba. Local well-posedness for a class of 1D Boussinesq systems. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021030
[16]

Lassaad Aloui, Slim Tayachi. Local well-posedness for the inhomogeneous nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5409-5437. doi: 10.3934/dcds.2021082
[17]

Akansha Sanwal. Local well-posedness for the Zakharov system in dimension d ≤ 3. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021147
[18]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2699-2723. doi: 10.3934/dcds.2020382
[19]

Hartmut Pecher. Local solutions with infinite energy of the Maxwell-Chern-Simons-Higgs system in Lorenz gauge. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 2193-2204. doi: 10.3934/dcds.2016.36.2193
[20]

Kenji Nakanishi, Hideo Takaoka, Yoshio Tsutsumi. Local well-posedness in low regularity of the MKDV equation with periodic boundary condition. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1635-1654. doi: 10.3934/dcds.2010.28.1635

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (158)
  • HTML views (191)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy