Towards the Chern-Simons-Higgs equation with finite energy

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November 2011, 30(4): 1145-1159. doi: 10.3934/dcds.2011.30.1145

Towards the Chern-Simons-Higgs equation with finite energy

Hyungjin Huh ^1,

1.	Department of Mathematics, Chung-Ang University, Seoul 156-756, South Korea

Received October 2009 Revised January 2011 Published May 2011

Abstract
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Under the Coulomb gauge condition Chern-Simons-Higgs equations are formulated in the hyperbolic system coupled with elliptic equations. We consider a solution of Chern-Simons-Higgs equations with finite energy and show how to obtain $H^1$ solution with one exceptional term $\phi\partial_t A_0$ from which the model equations (63) are proposed.

Keywords: Wente's inequality., Null form, Wave-Sobolev space $\mathcal{H}^{s, \theta}$, Chern-Simons-Higgs, Coulomb gauge.

Mathematics Subject Classification: Primary: 35L15, 35L45; Secondary: 35Q4.

Citation: Hyungjin Huh. Towards the Chern-Simons-Higgs equation with finite energy. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1145-1159. doi: 10.3934/dcds.2011.30.1145

References:

[1]	M. Beals, Self-spreading and strength of singularities for solutions to semilinear wave equations, Ann. of Math., 118 (1983), 187-214. 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), 1345-1366.
[3]	H. Brezis and J. M. Coron, Multiple solutions of H-systems and Rellich's conjecture, Comm. Pure Appl. Math., 37 (1984), 149-187. 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), 321-336. 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), 747-758. 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), 119-142. 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), 171-191. 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), 211-274.
[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), 1-28. 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), 167-204. 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), 2230-2233. doi: 10.1103/PhysRevLett.64.2230.
[12]	H. Huh, Low regularity solutions of the Chern-Simons-Higgs equations, Nonlinearity, 18 (2005), 1-9. 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), 526-549. doi: 10.1016/j.jfa.2006.09.009.
[14]	R. Jackiw and E. Weinberg, Self-dual Chern-Simons vortices, Phys. Rev. Lett., 64 (1990), 2234-2237. 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), 19-44. 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), 223-295. 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), 93-116. 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), 1279-1326. 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), 357-426. 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), 2291-2296. 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), 1752-1780. doi: 10.1002/cpa.10109.
[22]	S. Selberg, "Multilinear Spacetime Estimates and Applications to Local Existence Theory for Nonlinear Wave Equations," Ph.D. thesis, Princeton University, 1999.
[23]	S. Selberg, On an estimate for the wave equation and applications to nonlinear problems, Differential Integral Equations, 15 (2002), 213-236.
[24]	S. Selberg, Almost optimal local well-posedness of the Maxwell-Klein-Gordon equations in $1+4$ dimensions, Comm. Partial Differential Equations, 27 (2002), 1183-1227.
[25]	J. Spruck and Y. Yang, Topological solutions in the self-dual Chern-Simons theory, Ann. Inst. H. Poincaré Anal. Non Linéaire, 12 (1995), 75-97.
[26]	E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, 30, Princeton University Press, 1970.
[27]	T. Tao, Multilinear weighted convolution of $L$² functions, and applications to nonlinear dispersive equations, Amer. J. Math., 123 (2001), 839-908. doi: 10.1353/ajm.2001.0035.
[28]	G. Tarantello, Multiple condensate solutions for the Chern-Simons-Higgs theory, J. Math. Phys., 37 (1996), 3769-3796. 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), 841-887.
[30]	R. Wang, The existence of Chern-Simons vortices, Comm. Math. Phys., 137 (1991), 587-597. doi: 10.1007/BF02100279.
[31]	H. Wente, An existence theorem for surfaces of constant mean curvature, J. Math. Anal. Appl., 26 (1969), 318-344. 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), 187-214. 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), 1345-1366.
[3]	H. Brezis and J. M. Coron, Multiple solutions of H-systems and Rellich's conjecture, Comm. Pure Appl. Math., 37 (1984), 149-187. 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), 321-336. 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), 747-758. 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), 119-142. 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), 171-191. 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), 211-274.
[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), 1-28. 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), 167-204. 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), 2230-2233. doi: 10.1103/PhysRevLett.64.2230.
[12]	H. Huh, Low regularity solutions of the Chern-Simons-Higgs equations, Nonlinearity, 18 (2005), 1-9. 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), 526-549. doi: 10.1016/j.jfa.2006.09.009.
[14]	R. Jackiw and E. Weinberg, Self-dual Chern-Simons vortices, Phys. Rev. Lett., 64 (1990), 2234-2237. 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), 19-44. 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), 223-295. 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), 93-116. 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), 1279-1326. 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), 357-426. 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), 2291-2296. 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), 1752-1780. doi: 10.1002/cpa.10109.
[22]	S. Selberg, "Multilinear Spacetime Estimates and Applications to Local Existence Theory for Nonlinear Wave Equations," Ph.D. thesis, Princeton University, 1999.
[23]	S. Selberg, On an estimate for the wave equation and applications to nonlinear problems, Differential Integral Equations, 15 (2002), 213-236.
[24]	S. Selberg, Almost optimal local well-posedness of the Maxwell-Klein-Gordon equations in $1+4$ dimensions, Comm. Partial Differential Equations, 27 (2002), 1183-1227.
[25]	J. Spruck and Y. Yang, Topological solutions in the self-dual Chern-Simons theory, Ann. Inst. H. Poincaré Anal. Non Linéaire, 12 (1995), 75-97.
[26]	E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, 30, Princeton University Press, 1970.
[27]	T. Tao, Multilinear weighted convolution of $L$² functions, and applications to nonlinear dispersive equations, Amer. J. Math., 123 (2001), 839-908. doi: 10.1353/ajm.2001.0035.
[28]	G. Tarantello, Multiple condensate solutions for the Chern-Simons-Higgs theory, J. Math. Phys., 37 (1996), 3769-3796. 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), 841-887.
[30]	R. Wang, The existence of Chern-Simons vortices, Comm. Math. Phys., 137 (1991), 587-597. doi: 10.1007/BF02100279.
[31]	H. Wente, An existence theorem for surfaces of constant mean curvature, J. Math. Anal. Appl., 26 (1969), 318-344. doi: 10.1016/0022-247X(69)90156-5.

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