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

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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].

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 and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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