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

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  • 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].

    Mathematics Subject Classification: 35Q40, 35L70.

    Citation:

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