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Existence and uniqueness of a positive connection for the scalar viscous shallow water system in a bounded interval
Low regularity well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge
1. | Department of Mathematical Sciences, Binghamton University (SUNY), Binghamton, NY 13902-6000, United States |
2. | Department of Mathematics, University of California, San Diego (UCSD), La Jolla, CA 92093-0112, United States |
References:
[1] |
Magdalena Czubak, Local wellposedness for the $2+1$-dimensional monopole equation, Anal. PDE, 3 (2010), 151-174.
doi: 10.2140/apde.2010.3.151. |
[2] |
Piero D'Ancona, Damiano Foschi, and Sigmund Selberg, Product estimates for wave-Sobolev spaces in $2+1$ and $1+1$ dimensions, In Nonlinear partial differential equations and hyperbolic wave phenomena, volume 526 of Contemp. Math., pages 125-150. Amer. Math. Soc., Providence, RI, 2010.
doi: 10.1090/conm/526/10379. |
[3] |
Viktor Grigoryan and Andrea R. Nahmod, Almost critical well-posedness for nonlinear wave equation with $Q_{\mu\nu}$ null forms in 2D,, \arXiv{1307.6194}., ().
|
[4] |
Markus Keel, Tristan Roy, and Terence Tao, Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm, Discrete Contin. Dyn. Syst., 30 (2011), 573-621.
doi: 10.3934/dcds.2011.30.573. |
[5] |
Sergiu Klainerman and Matei Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math., 46 (1993), 1221-1268.
doi: 10.1002/cpa.3160460902. |
[6] |
Sergiu Klainerman and Matei Machedon, On the Maxwell-Klein-Gordon equation with finite energy, Duke Math. J., 74 (1994), 19-44. |
[7] |
Sergiu Klainerman, Long time behaviour of solutions to nonlinear wave equations, In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), pages 1209-1215, Warsaw, 1984. PWN. |
[8] |
Sergiu Klainerman and Matei Machedon, Estimates for null forms and the spaces $H_{s,\delta}$, Internat. Math. Res. Notices, 17 (1996), 853-865.
doi: 10.1155/S1073792896000529. |
[9] |
Sergiu Klainerman and Sigmund Selberg, Bilinear estimates and applications to nonlinear wave equations, Commun. Contemp. Math., 4 (2002), 223-295,
doi: 10.1142/S0219199702000634. |
[10] |
Sergiu Klainerman and Daniel Tataru, On the optimal local regularity for Yang-Mills equations in $R^{4+1}$, J. Amer. Math. Soc., 12 (1999), 93-116.
doi: 10.1090/S0894-0347-99-00282-9. |
[11] |
Matei Machedon and Jacob Sterbenz, Almost optimal local well-posedness for the $(3+1)$-dimensional Maxwell-Klein-Gordon equations, J. Amer. Math. Soc., 17 (2004), 297-359.
doi: 10.1090/S0894-0347-03-00445-4. |
[12] |
Vincent Moncrief, Global existence of Maxwell-Klein-Gordon fields in $(2+1)$-dimensional spacetime, J. Math. Phys., 21 (1980), 2291-2296.
doi: 10.1063/1.524669. |
[13] |
Hartmut Pecher, Low regularity local well-posedness for the Maxwell-Klein-Gordon equations in Lorenz gauge,, \arXiv{1308.1598}., ().
|
[14] |
Martin Schwarz, Jr, Global solutions of Maxwell-Higgs on Minkowski space, J. Math. Anal. Appl., 229 (1999), 426-440.
doi: 10.1006/jmaa.1998.6164. |
[15] |
Sigmund Selberg, Multilinear Spacetime Estimates and Applications to Local Existence Theory for Nonlinear Wave Equations, Ph.D. Thesis, Princeton University, 1999. |
[16] |
Sigmund Selberg, Almost optimal local well-posedness of the Maxwell-Klein-Gordon equations in $1+4$ dimensions, Comm. Partial Differential Equations, 27 (2002), 1183-1227.
doi: 10.1081/PDE-120004899. |
[17] |
Sigmund Selberg, On an estimate for the wave equation and applications to nonlinear problems, Differential Integral Equations, 15 (2002), 213-236. |
[18] |
Yi Zhou, Local existence with minimal regularity for nonlinear wave equations, Amer. J. Math., 119 (1997), 671-703. |
show all references
References:
[1] |
Magdalena Czubak, Local wellposedness for the $2+1$-dimensional monopole equation, Anal. PDE, 3 (2010), 151-174.
doi: 10.2140/apde.2010.3.151. |
[2] |
Piero D'Ancona, Damiano Foschi, and Sigmund Selberg, Product estimates for wave-Sobolev spaces in $2+1$ and $1+1$ dimensions, In Nonlinear partial differential equations and hyperbolic wave phenomena, volume 526 of Contemp. Math., pages 125-150. Amer. Math. Soc., Providence, RI, 2010.
doi: 10.1090/conm/526/10379. |
[3] |
Viktor Grigoryan and Andrea R. Nahmod, Almost critical well-posedness for nonlinear wave equation with $Q_{\mu\nu}$ null forms in 2D,, \arXiv{1307.6194}., ().
|
[4] |
Markus Keel, Tristan Roy, and Terence Tao, Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm, Discrete Contin. Dyn. Syst., 30 (2011), 573-621.
doi: 10.3934/dcds.2011.30.573. |
[5] |
Sergiu Klainerman and Matei Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math., 46 (1993), 1221-1268.
doi: 10.1002/cpa.3160460902. |
[6] |
Sergiu Klainerman and Matei Machedon, On the Maxwell-Klein-Gordon equation with finite energy, Duke Math. J., 74 (1994), 19-44. |
[7] |
Sergiu Klainerman, Long time behaviour of solutions to nonlinear wave equations, In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), pages 1209-1215, Warsaw, 1984. PWN. |
[8] |
Sergiu Klainerman and Matei Machedon, Estimates for null forms and the spaces $H_{s,\delta}$, Internat. Math. Res. Notices, 17 (1996), 853-865.
doi: 10.1155/S1073792896000529. |
[9] |
Sergiu Klainerman and Sigmund Selberg, Bilinear estimates and applications to nonlinear wave equations, Commun. Contemp. Math., 4 (2002), 223-295,
doi: 10.1142/S0219199702000634. |
[10] |
Sergiu Klainerman and Daniel Tataru, On the optimal local regularity for Yang-Mills equations in $R^{4+1}$, J. Amer. Math. Soc., 12 (1999), 93-116.
doi: 10.1090/S0894-0347-99-00282-9. |
[11] |
Matei Machedon and Jacob Sterbenz, Almost optimal local well-posedness for the $(3+1)$-dimensional Maxwell-Klein-Gordon equations, J. Amer. Math. Soc., 17 (2004), 297-359.
doi: 10.1090/S0894-0347-03-00445-4. |
[12] |
Vincent Moncrief, Global existence of Maxwell-Klein-Gordon fields in $(2+1)$-dimensional spacetime, J. Math. Phys., 21 (1980), 2291-2296.
doi: 10.1063/1.524669. |
[13] |
Hartmut Pecher, Low regularity local well-posedness for the Maxwell-Klein-Gordon equations in Lorenz gauge,, \arXiv{1308.1598}., ().
|
[14] |
Martin Schwarz, Jr, Global solutions of Maxwell-Higgs on Minkowski space, J. Math. Anal. Appl., 229 (1999), 426-440.
doi: 10.1006/jmaa.1998.6164. |
[15] |
Sigmund Selberg, Multilinear Spacetime Estimates and Applications to Local Existence Theory for Nonlinear Wave Equations, Ph.D. Thesis, Princeton University, 1999. |
[16] |
Sigmund Selberg, Almost optimal local well-posedness of the Maxwell-Klein-Gordon equations in $1+4$ dimensions, Comm. Partial Differential Equations, 27 (2002), 1183-1227.
doi: 10.1081/PDE-120004899. |
[17] |
Sigmund Selberg, On an estimate for the wave equation and applications to nonlinear problems, Differential Integral Equations, 15 (2002), 213-236. |
[18] |
Yi Zhou, Local existence with minimal regularity for nonlinear wave equations, Amer. J. Math., 119 (1997), 671-703. |
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