June  2013, 33(6): 2531-2546. doi: 10.3934/dcds.2013.33.2531

Global well-posedness of the Chern-Simons-Higgs equations with finite energy

1. 

Department of Mathematical Sciences, Norwegian University of Science and Technology, N-7491 Trondheim, Norway, Norway

Received  January 2012 Revised  April 2012 Published  December 2012

We prove that the Cauchy problem for the Chern-Simons-Higgs equations on the (2+1)-dimensional Minkowski space-time is globally well posed for initial data with finite energy. This improves a result of Chae and Choe, who proved global well-posedness for more regular data. Moreover, we prove local well-posedness even below the energy regularity, using the the null structure of the system in Lorenz gauge and bilinear space-time estimates for wave-Sobolev norms.
Citation: Sigmund Selberg, Achenef Tesfahun. Global well-posedness of the Chern-Simons-Higgs equations with finite energy. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2531-2546. doi: 10.3934/dcds.2013.33.2531
References:
[1]

N. Bournaveas, Low regularity solutions of the relativistic Chern-Simons-Higgs theory in the Lorentz gauge,, Electronic Journal of Differential Equations, (2009), 1. Google Scholar

[2]

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. Google Scholar

[3]

P. D'Ancona, D. Foschi and S. Selberg, Product estimates for wave-Sobolev spaces in 2+1 and 1+1 dimensions,, in, 526 (2010), 125. doi: 10.1090/conm/526/10379. Google Scholar

[4]

J. Ginibre and G. Velo, The Cauchy problem for coupled Yang-Mills and scalar fields in the temporal gauge,, Commun. Math. Phys., 82 (1981), 1. Google Scholar

[5]

J. Hong, Y. Kim and P. Y. Pac, Multivortex solutions of the abelian Chern-Simons-Higgs theory,, Phys. Rev. Lett., 64 (1990), 2230. doi: 10.1103/PhysRevLett.64.2230. Google Scholar

[6]

H. Huh, Local and global solutions of the Chern-Simons-Higgs system,, Journal of Functional Analysis, 242 (2007), 526. doi: 10.1016/j.jfa.2006.09.009. Google Scholar

[7]

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

[8]

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

[9]

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. Google Scholar

[10]

S. Selberg and A. Tesfahun, Finite-energy global well-posedness of the Maxwell-Klein-Gordon system in Lorenz gauge,, Communications in Partial Differential Equations, 35 (2010), 1029. doi: 10.1080/03605301003717100. Google Scholar

[11]

J. Yuan, Local well-posedness of Chern-Simons-Higgs system in the Lorentz gauge,, Journal of Mathematical Physics, 52 (2011). doi: 10.1063/1.3645365. Google Scholar

show all references

References:
[1]

N. Bournaveas, Low regularity solutions of the relativistic Chern-Simons-Higgs theory in the Lorentz gauge,, Electronic Journal of Differential Equations, (2009), 1. Google Scholar

[2]

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. Google Scholar

[3]

P. D'Ancona, D. Foschi and S. Selberg, Product estimates for wave-Sobolev spaces in 2+1 and 1+1 dimensions,, in, 526 (2010), 125. doi: 10.1090/conm/526/10379. Google Scholar

[4]

J. Ginibre and G. Velo, The Cauchy problem for coupled Yang-Mills and scalar fields in the temporal gauge,, Commun. Math. Phys., 82 (1981), 1. Google Scholar

[5]

J. Hong, Y. Kim and P. Y. Pac, Multivortex solutions of the abelian Chern-Simons-Higgs theory,, Phys. Rev. Lett., 64 (1990), 2230. doi: 10.1103/PhysRevLett.64.2230. Google Scholar

[6]

H. Huh, Local and global solutions of the Chern-Simons-Higgs system,, Journal of Functional Analysis, 242 (2007), 526. doi: 10.1016/j.jfa.2006.09.009. Google Scholar

[7]

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

[8]

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

[9]

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. Google Scholar

[10]

S. Selberg and A. Tesfahun, Finite-energy global well-posedness of the Maxwell-Klein-Gordon system in Lorenz gauge,, Communications in Partial Differential Equations, 35 (2010), 1029. doi: 10.1080/03605301003717100. Google Scholar

[11]

J. Yuan, Local well-posedness of Chern-Simons-Higgs system in the Lorentz gauge,, Journal of Mathematical Physics, 52 (2011). doi: 10.1063/1.3645365. Google Scholar

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