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Towards the Chern-Simons-Higgs equation with finite energy
1. | Department of Mathematics, Chung-Ang University, Seoul 156-756, South Korea |
References:
[1] |
M. Beals, Self-spreading and strength of singularities for solutions to semilinear wave equations,, Ann. of Math., 118 (1983), 187.
doi: 10.2307/2006959. |
[2] |
N. Bournaveas, Low regularity solutions of the Dirac-Klein-Gordon equations in two space dimensions,, Comm. Partial Differential Equations, 26 (2001), 1345.
|
[3] |
H. Brezis and J. M. Coron, Multiple solutions of H-systems and Rellich's conjecture,, Comm. Pure Appl. Math., 37 (1984), 149.
doi: 10.1002/cpa.3160370202. |
[4] |
L. A. Caffarelli and Y. Yang, Vortex condensation in Chern-Simons-Higgs model: An existence theorem,, Comm. Math. Phys., 168 (1995), 321.
doi: 10.1007/BF02101552. |
[5] |
D. Chae and K. Choe, Global existence in the Cauchy problem of the relativistic Chern-Simons-Higgs theory,, Nonlinearity, 15 (2002), 747.
doi: 10.1088/0951-7715/15/3/314. |
[6] |
D. Chae and O. Yu. Imanuvilov, The existence of non-topological multivortex solutions in the relativistic self-dual Chern-Simons theory,, Comm. Math. Phys., 215 (2000), 119.
doi: 10.1007/s002200000302. |
[7] |
D. M. Eardley and V. Moncrief, The global existence of Yang-Mills-Higgs fields in 4-dimensional Minkowski space,, Comm. Math. Phys., 83 (1982), 171.
doi: 10.1007/BF01976040. |
[8] |
D. Foschi and S. Klainerman, Bilinear space-time estimates for homogeneous wave equations,, Ann. Sci. École Norm. Sup., 33 (2000), 211.
|
[9] |
J. Ginibre and G. Velo, The Cauchy problem for coupled Yang-Mills and Scalar fields in the temporal gauge,, Comm. Math. Phys., 82 (1981), 1.
doi: 10.1007/BF01206943. |
[10] |
J. Han and N. Kim, Nonself-dual Chern-Simons and Maxwell-Chern-Simons vortices on bounded domains,, J. Funct. Anal., 221 (2005), 167.
doi: 10.1016/j.jfa.2004.09.012. |
[11] |
J. Hong, P. Kim and P. Pac, Multivortex solutions of the abelian Chern-Simons-Higgs theory,, Phys. Rev. Lett., 64 (1990), 2230.
doi: 10.1103/PhysRevLett.64.2230. |
[12] |
H. Huh, Low regularity solutions of the Chern-Simons-Higgs equations,, Nonlinearity, 18 (2005), 1.
doi: 10.1088/0951-7715/18/6/009. |
[13] |
H. Huh, Local and global solutions of the Chern-Simons-Higgs system,, J. Funct. Anal., 242 (2007), 526.
doi: 10.1016/j.jfa.2006.09.009. |
[14] |
R. Jackiw and E. Weinberg, Self-dual Chern-Simons vortices,, Phys. Rev. Lett., 64 (1990), 2234.
doi: 10.1103/PhysRevLett.64.2234. |
[15] |
S. Klainerman and M. Machedon, On the Maxwell-Klein-Gordon equation with finite energy,, Duke Math. J., 74 (1994), 19.
doi: 10.1215/S0012-7094-94-07402-4. |
[16] |
S. Klainerman and S. Selberg, Bilinear estimates and applications to nonlinear wave equations,, Commun. Contemp. Math., 4 (2002), 223.
doi: 10.1142/S0219199702000634. |
[17] |
S. Klainerman and D. Tataru, On the optimal local regularity for Yang-Mills equations in $\mathbbR$${4+1}$,, J. Amer. Math. Soc., 12 (1999), 93.
doi: 10.1090/S0894-0347-99-00282-9. |
[18] |
S. Lee and A. Vargas, Sharp null form estimates for the wave equation,, Amer. J. Math., 130 (2008), 1279.
doi: 10.1353/ajm.0.0024. |
[19] |
H. Lindblad and C. Sogge, On existence and scattering with minimal regularity for semilinear wave equations,, J. Funct. Anal., 130 (1995), 357.
doi: 10.1006/jfan.1995.1075. |
[20] |
V. Moncrief, Global existence of Maxwell-Klein-Gordon fields in $(2+1)$ dimensional spacetimes,, J. Math. Phys., 21 (1980), 2291.
doi: 10.1063/1.524669. |
[21] |
M. Nolasco, Non-topological N-vortex condensates for the self-dual Chern-Simons theory,, Comm. Pure Appl. Math., 56 (2003), 1752.
doi: 10.1002/cpa.10109. |
[22] |
S. Selberg, "Multilinear Spacetime Estimates and Applications to Local Existence Theory for Nonlinear Wave Equations,", Ph.D. thesis, (1999). Google Scholar |
[23] |
S. Selberg, On an estimate for the wave equation and applications to nonlinear problems,, Differential Integral Equations, 15 (2002), 213.
|
[24] |
S. Selberg, Almost optimal local well-posedness of the Maxwell-Klein-Gordon equations in $1+4$ dimensions,, Comm. Partial Differential Equations, 27 (2002), 1183.
|
[25] |
J. Spruck and Y. Yang, Topological solutions in the self-dual Chern-Simons theory,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 12 (1995), 75.
|
[26] |
E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton Mathematical Series, 30 (1970).
|
[27] |
T. Tao, Multilinear weighted convolution of $L$2 functions, and applications to nonlinear dispersive equations,, Amer. J. Math., 123 (2001), 839.
doi: 10.1353/ajm.2001.0035. |
[28] |
G. Tarantello, Multiple condensate solutions for the Chern-Simons-Higgs theory,, J. Math. Phys., 37 (1996), 3769.
doi: 10.1063/1.531601. |
[29] |
D. Tataru, On the $X^s_\theta$ spaces and unique continuation for semilinear hyperbolic equations,, Comm. Partial Differential Equations, 21 (1996), 841.
|
[30] |
R. Wang, The existence of Chern-Simons vortices,, Comm. Math. Phys., 137 (1991), 587.
doi: 10.1007/BF02100279. |
[31] |
H. Wente, An existence theorem for surfaces of constant mean curvature,, J. Math. Anal. Appl., 26 (1969), 318.
doi: 10.1016/0022-247X(69)90156-5. |
show all references
References:
[1] |
M. Beals, Self-spreading and strength of singularities for solutions to semilinear wave equations,, Ann. of Math., 118 (1983), 187.
doi: 10.2307/2006959. |
[2] |
N. Bournaveas, Low regularity solutions of the Dirac-Klein-Gordon equations in two space dimensions,, Comm. Partial Differential Equations, 26 (2001), 1345.
|
[3] |
H. Brezis and J. M. Coron, Multiple solutions of H-systems and Rellich's conjecture,, Comm. Pure Appl. Math., 37 (1984), 149.
doi: 10.1002/cpa.3160370202. |
[4] |
L. A. Caffarelli and Y. Yang, Vortex condensation in Chern-Simons-Higgs model: An existence theorem,, Comm. Math. Phys., 168 (1995), 321.
doi: 10.1007/BF02101552. |
[5] |
D. Chae and K. Choe, Global existence in the Cauchy problem of the relativistic Chern-Simons-Higgs theory,, Nonlinearity, 15 (2002), 747.
doi: 10.1088/0951-7715/15/3/314. |
[6] |
D. Chae and O. Yu. Imanuvilov, The existence of non-topological multivortex solutions in the relativistic self-dual Chern-Simons theory,, Comm. Math. Phys., 215 (2000), 119.
doi: 10.1007/s002200000302. |
[7] |
D. M. Eardley and V. Moncrief, The global existence of Yang-Mills-Higgs fields in 4-dimensional Minkowski space,, Comm. Math. Phys., 83 (1982), 171.
doi: 10.1007/BF01976040. |
[8] |
D. Foschi and S. Klainerman, Bilinear space-time estimates for homogeneous wave equations,, Ann. Sci. École Norm. Sup., 33 (2000), 211.
|
[9] |
J. Ginibre and G. Velo, The Cauchy problem for coupled Yang-Mills and Scalar fields in the temporal gauge,, Comm. Math. Phys., 82 (1981), 1.
doi: 10.1007/BF01206943. |
[10] |
J. Han and N. Kim, Nonself-dual Chern-Simons and Maxwell-Chern-Simons vortices on bounded domains,, J. Funct. Anal., 221 (2005), 167.
doi: 10.1016/j.jfa.2004.09.012. |
[11] |
J. Hong, P. Kim and P. Pac, Multivortex solutions of the abelian Chern-Simons-Higgs theory,, Phys. Rev. Lett., 64 (1990), 2230.
doi: 10.1103/PhysRevLett.64.2230. |
[12] |
H. Huh, Low regularity solutions of the Chern-Simons-Higgs equations,, Nonlinearity, 18 (2005), 1.
doi: 10.1088/0951-7715/18/6/009. |
[13] |
H. Huh, Local and global solutions of the Chern-Simons-Higgs system,, J. Funct. Anal., 242 (2007), 526.
doi: 10.1016/j.jfa.2006.09.009. |
[14] |
R. Jackiw and E. Weinberg, Self-dual Chern-Simons vortices,, Phys. Rev. Lett., 64 (1990), 2234.
doi: 10.1103/PhysRevLett.64.2234. |
[15] |
S. Klainerman and M. Machedon, On the Maxwell-Klein-Gordon equation with finite energy,, Duke Math. J., 74 (1994), 19.
doi: 10.1215/S0012-7094-94-07402-4. |
[16] |
S. Klainerman and S. Selberg, Bilinear estimates and applications to nonlinear wave equations,, Commun. Contemp. Math., 4 (2002), 223.
doi: 10.1142/S0219199702000634. |
[17] |
S. Klainerman and D. Tataru, On the optimal local regularity for Yang-Mills equations in $\mathbbR$${4+1}$,, J. Amer. Math. Soc., 12 (1999), 93.
doi: 10.1090/S0894-0347-99-00282-9. |
[18] |
S. Lee and A. Vargas, Sharp null form estimates for the wave equation,, Amer. J. Math., 130 (2008), 1279.
doi: 10.1353/ajm.0.0024. |
[19] |
H. Lindblad and C. Sogge, On existence and scattering with minimal regularity for semilinear wave equations,, J. Funct. Anal., 130 (1995), 357.
doi: 10.1006/jfan.1995.1075. |
[20] |
V. Moncrief, Global existence of Maxwell-Klein-Gordon fields in $(2+1)$ dimensional spacetimes,, J. Math. Phys., 21 (1980), 2291.
doi: 10.1063/1.524669. |
[21] |
M. Nolasco, Non-topological N-vortex condensates for the self-dual Chern-Simons theory,, Comm. Pure Appl. Math., 56 (2003), 1752.
doi: 10.1002/cpa.10109. |
[22] |
S. Selberg, "Multilinear Spacetime Estimates and Applications to Local Existence Theory for Nonlinear Wave Equations,", Ph.D. thesis, (1999). Google Scholar |
[23] |
S. Selberg, On an estimate for the wave equation and applications to nonlinear problems,, Differential Integral Equations, 15 (2002), 213.
|
[24] |
S. Selberg, Almost optimal local well-posedness of the Maxwell-Klein-Gordon equations in $1+4$ dimensions,, Comm. Partial Differential Equations, 27 (2002), 1183.
|
[25] |
J. Spruck and Y. Yang, Topological solutions in the self-dual Chern-Simons theory,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 12 (1995), 75.
|
[26] |
E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton Mathematical Series, 30 (1970).
|
[27] |
T. Tao, Multilinear weighted convolution of $L$2 functions, and applications to nonlinear dispersive equations,, Amer. J. Math., 123 (2001), 839.
doi: 10.1353/ajm.2001.0035. |
[28] |
G. Tarantello, Multiple condensate solutions for the Chern-Simons-Higgs theory,, J. Math. Phys., 37 (1996), 3769.
doi: 10.1063/1.531601. |
[29] |
D. Tataru, On the $X^s_\theta$ spaces and unique continuation for semilinear hyperbolic equations,, Comm. Partial Differential Equations, 21 (1996), 841.
|
[30] |
R. Wang, The existence of Chern-Simons vortices,, Comm. Math. Phys., 137 (1991), 587.
doi: 10.1007/BF02100279. |
[31] |
H. Wente, An existence theorem for surfaces of constant mean curvature,, J. Math. Anal. Appl., 26 (1969), 318.
doi: 10.1016/0022-247X(69)90156-5. |
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