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Continuous data assimilation for the 3D primitive equations of the ocean
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 |
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 [
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. |
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. |
[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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