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

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].

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'AnconaD. 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'AnconaD. 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'AnconaD. 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'AnconaD. 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

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