# American Institute of Mathematical Sciences

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

## Towards the Chern-Simons-Higgs equation with finite energy

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

Received  October 2009 Revised  January 2011 Published  May 2011

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.
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
 [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 L2 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.
 [1] Hartmut Pecher. The Chern-Simons-Higgs and the Chern-Simons-Dirac equations in Fourier-Lebesgue spaces. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4875-4893. doi: 10.3934/dcds.2019199 [2] Nikolaos Bournaveas, Timothy Candy, Shuji Machihara. A note on the Chern-Simons-Dirac equations in the Coulomb gauge. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2693-2701. doi: 10.3934/dcds.2014.34.2693 [3] Sigmund Selberg, Achenef Tesfahun. Global well-posedness of the Chern-Simons-Higgs equations with finite energy. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2531-2546. doi: 10.3934/dcds.2013.33.2531 [4] Jianjun Yuan. On the well-posedness of Maxwell-Chern-Simons-Higgs system in the Lorenz gauge. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2389-2403. doi: 10.3934/dcds.2014.34.2389 [5] Hartmut Pecher. Local solutions with infinite energy of the Maxwell-Chern-Simons-Higgs system in Lorenz gauge. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2193-2204. doi: 10.3934/dcds.2016.36.2193 [6] Jeongho Kim, Bora Moon. Hydrodynamic limits of the nonlinear Schrödinger equation with the Chern-Simons gauge fields. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2541-2561. doi: 10.3934/dcds.2021202 [7] Rudolf Ahlswede. The final form of Tao's inequality relating conditional expectation and conditional mutual information. Advances in Mathematics of Communications, 2007, 1 (2) : 239-242. doi: 10.3934/amc.2007.1.239 [8] Agnieszka Badeńska. No entire function with real multipliers in class $\mathcal{S}$. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3321-3327. doi: 10.3934/dcds.2013.33.3321 [9] Shengxin Zhu. Summation of Gaussian shifts as Jacobi's third Theta function. Mathematical Foundations of Computing, 2020, 3 (3) : 157-163. doi: 10.3934/mfc.2020015 [10] Felipe Riquelme. Ruelle's inequality in negative curvature. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 2809-2825. doi: 10.3934/dcds.2018119 [11] S. S. Dragomir, C. E. M. Pearce. Jensen's inequality for quasiconvex functions. Numerical Algebra, Control and Optimization, 2012, 2 (2) : 279-291. doi: 10.3934/naco.2012.2.279 [12] Wen-ming He, Jun-zhi Cui. The estimate of the multi-scale homogenization method for Green's function on Sobolev space $W^{1,q}(\Omega)$. Communications on Pure and Applied Analysis, 2012, 11 (2) : 501-516. doi: 10.3934/cpaa.2012.11.501 [13] Kwangseok Choe, Hyungjin Huh. Chern-Simons gauged sigma model into $\mathbb{H}^2$ and its self-dual equations. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4613-4646. doi: 10.3934/dcds.2019189 [14] Hangzhou Hu, Yuan Li, Dun Zhao. Ground state for fractional Schrödinger-Poisson equation in Coulomb-Sobolev space. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1899-1916. doi: 10.3934/dcdss.2021064 [15] Pablo Raúl Stinga, Chao Zhang. Harnack's inequality for fractional nonlocal equations. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 3153-3170. doi: 10.3934/dcds.2013.33.3153 [16] Anne-Sophie de Suzzoni. Consequences of the choice of a particular basis of $L^2(S^3)$ for the cubic wave equation on the sphere and the Euclidean space. Communications on Pure and Applied Analysis, 2014, 13 (3) : 991-1015. doi: 10.3934/cpaa.2014.13.991 [17] Letizia Stefanelli, Ugo Locatelli. Kolmogorov's normal form for equations of motion with dissipative effects. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2561-2593. doi: 10.3934/dcdsb.2012.17.2561 [18] Frank Jochmann. A variational inequality in Bean's model for superconductors with displacement current. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 545-565. doi: 10.3934/dcds.2009.25.545 [19] Anat Amir. Sharpness of Zapolsky's inequality for quasi-states and Poisson brackets. Electronic Research Announcements, 2011, 18: 61-68. doi: 10.3934/era.2011.18.61 [20] Pierdomenico Pepe. A nonlinear version of Halanay's inequality for the uniform convergence to the origin. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021045

2020 Impact Factor: 1.392
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8434544205665588, "perplexity": 2151.603245669832}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662580803.75/warc/CC-MAIN-20220525054507-20220525084507-00638.warc.gz"}