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Towards the Chern-Simons-Higgs equation with finite energy

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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.
    Mathematics Subject Classification: Primary: 35L15, 35L45; Secondary: 35Q40.


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  • [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.


    N. Bournaveas, Low regularity solutions of the Dirac-Klein-Gordon equations in two space dimensions, Comm. Partial Differential Equations, 26 (2001), 1345-1366.


    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.


    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.


    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.


    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.


    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.


    D. Foschi and S. Klainerman, Bilinear space-time estimates for homogeneous wave equations, Ann. Sci. École Norm. Sup., 33 (2000), 211-274.


    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.


    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.


    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.


    H. Huh, Low regularity solutions of the Chern-Simons-Higgs equations, Nonlinearity, 18 (2005), 1-9.doi: 10.1088/0951-7715/18/6/009.


    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.


    R. Jackiw and E. Weinberg, Self-dual Chern-Simons vortices, Phys. Rev. Lett., 64 (1990), 2234-2237.doi: 10.1103/PhysRevLett.64.2234.


    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.


    S. Klainerman and S. Selberg, Bilinear estimates and applications to nonlinear wave equations, Commun. Contemp. Math., 4 (2002), 223-295.doi: 10.1142/S0219199702000634.


    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.


    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.


    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.


    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.


    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.


    S. Selberg, "Multilinear Spacetime Estimates and Applications to Local Existence Theory for Nonlinear Wave Equations," Ph.D. thesis, Princeton University, 1999.


    S. Selberg, On an estimate for the wave equation and applications to nonlinear problems, Differential Integral Equations, 15 (2002), 213-236.


    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.


    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.


    E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, 30, Princeton University Press, 1970.


    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.


    G. Tarantello, Multiple condensate solutions for the Chern-Simons-Higgs theory, J. Math. Phys., 37 (1996), 3769-3796.doi: 10.1063/1.531601.


    D. Tataru, On the $X^s_\theta$ spaces and unique continuation for semilinear hyperbolic equations, Comm. Partial Differential Equations, 21 (1996), 841-887.


    R. Wang, The existence of Chern-Simons vortices, Comm. Math. Phys., 137 (1991), 587-597.doi: 10.1007/BF02100279.


    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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