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August  2019, 39(8): 4875-4893. doi: 10.3934/dcds.2019199

The Chern-Simons-Higgs and the Chern-Simons-Dirac equations in Fourier-Lebesgue spaces

Hartmut Pecher

Fakultät für Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, Gaußstr. 20, 42119 Wuppertal, Germany

Received  November 2018 Revised  January 2019 Published  May 2019

The Chern-Simons-Higgs and the Chern-Simons-Dirac systems in Lorenz gauge are locally well-posed in suitable Fourier-Lebesgue spaces $ \hat{H}^{s, r} $. Our aim is to minimize $ s = s(r) $ in the range $ 1<r \le 2 $. If $ r \to 1 $ we show that we almost reach the critical regularity dictated by scaling. In the classical case $ r = 2 $ the results are due to Huh and Oh. Crucial is the fact that the decisive quadratic nonlinearities fulfill a null condition.

Keywords: Chern-Simons-Higgs, Chern-Simons-Dirac, local well-posedness, Lorenz gauge, Fourier-Lebesgue spaces.
Mathematics Subject Classification: 35Q40, 35L70.
Citation: Hartmut Pecher. The Chern-Simons-Higgs and the Chern-Simons-Dirac equations in Fourier-Lebesgue spaces. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4875-4893. doi: 10.3934/dcds.2019199
References:
[1]

N. Bourneveas, T. Candy and S. Machihara, A note on the Chern-Simons-Dirac equations in the Coulomb gauge, Discr. Cont. Dyn. Syst., 34 (2014), 2693-2701.  doi: 10.3934/dcds.2014.34.2693.  Google Scholar

[2]

Y. M. Cho, J. W. Kim and D. H. Park, Fermionic vortex solutions in Chern-Simons electrodynamics, Phys. Rev. D, 45 (1992), 3802-3806.  doi: 10.1103/PhysRevD.45.3802.  Google Scholar

[3]

P. d'Ancona, D. Foschi and S. Selberg, Product estimates for wave-Sobolev spaces in 2+1 and 1+1 dimensions, Contemp. Math., 526 (2010), 125-150.  doi: 10.1090/conm/526/10379.  Google Scholar

[4]

P. d'Ancona, D. Foschi and S. Selberg, Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system, J. Eur. Math. Soc., 9 (2007), 877-899.  doi: 10.4171/JEMS/100.  Google Scholar

[5]

D. Foschi and S. Klainerman, Bilinear space-time estimates for homogeneous wave equations, Ann. Sc. ENS. 4. serie, 33 (2000), 211-274.  doi: 10.1016/S0012-9593(00)00109-9.  Google Scholar

[6]

V. Grigoryan and A. Nahmod, Almost critical wee-posedmess for nonlinear wave equation with $Q_{\mu \nu}$ null forms in 2D, Math. Res. Letters, 21 (2014), 313-332.  doi: 10.4310/MRL.2014.v21.n2.a9.  Google Scholar

[7]

V. Grigoryan and A. Tanguay, Improved well-poseness for the quadratic derivative nonlinear wave equation in 2D, Preprint, arXiv: 1308.1719. Google Scholar

[8]

A. Grünrock, An improved local well-posedness result for the modified KdV equation, Int. Math. Res. Not., (2004), 3287–3308. doi: 10.1155/S1073792804140981.  Google Scholar

[9]

A. Grünrock, On the wave equation with quadratic nonlinearities in three space dimensions, Hyperbolic Diff. Equ., 8 (2011), 1-8.  doi: 10.1142/S0219891611002305.  Google Scholar

[10]

A. Grünrock and L. Vega, Local well-posedness for the modified KdV equation in almost critical $\hat{H}^r_s$ -spaces, Trans. Amer. Mat. Soc., 361 (2009), 5681-5694.  doi: 10.1090/S0002-9947-09-04611-X.  Google Scholar

[11]

J. Hong, Y. Kim and P. Y. Pac, Multivortex solutions of the abelian Chern-Simons-Higgs theory, Phys. Rev. Letters, 64 (1990), 2230-2233.  doi: 10.1103/PhysRevLett.64.2230.  Google Scholar

[12]

H. Huh, Cauchy problem for the Fermion field equation coupled with the Chern-Simons gauge, Lett. Math. Phys., 79 (2007), 75-94.  doi: 10.1007/s11005-006-0118-y.  Google Scholar

[13]

H. Huh and S.-J. Oh, Low regularity solutions to the Chern-Simons-Dirac and the Chern-Simons-Higgs equations in the Lorenz gauge, Comm. PDE, 41 (2016), 375-397.  doi: 10.1080/03605302.2015.1132730.  Google Scholar

[14]

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

[15]

S. Li and R. K. Bhaduri, Planar solitons of the gauged Dirac equation, Phys. Rev. D, 43 (1991), 3573-3574.   Google Scholar

[16]

H. Pecher, Low regularity solutions for Chern-Simons-Dirac systems in the temporal and Coulomb gauge, Electron. J. Differential Equations, 2016 (2016), 1-16.   Google Scholar

[17]

H. Pecher, Global well-posedness in energy space for the Chern-Simons-Higgs system in temporal gauge, J. Hyperbolic Diff. Equ., 13 (2016), 331-351.  doi: 10.1142/S0219891616500107.  Google Scholar

[18]

S. Selberg, Bilinear Fourier restriction estimates related to the 2D wave equation, Adv. Diff. Equ., 16 (2011), 667-690.   Google Scholar

[19]

S. Selberg and A. Tesfahun, Global well-posedness of the Chern-Simons-Higgs equations with finite energy, Discrete Cont. Dyn. Syst., 33 (2013), 2531-2546.  doi: 10.3934/dcds.2013.33.2531.  Google Scholar

[20]

A. Vargas and L. Vega, Global wellposedness for 1D non-linear Schrödinger equation for data with an infinite $L^2$-norm, J. Math. Pures Appl., 80 (2001), 1029-1044.  doi: 10.1016/S0021-7824(01)01224-7.  Google Scholar

show all references

References:
[1]

N. Bourneveas, T. Candy and S. Machihara, A note on the Chern-Simons-Dirac equations in the Coulomb gauge, Discr. Cont. Dyn. Syst., 34 (2014), 2693-2701.  doi: 10.3934/dcds.2014.34.2693.  Google Scholar

[2]

Y. M. Cho, J. W. Kim and D. H. Park, Fermionic vortex solutions in Chern-Simons electrodynamics, Phys. Rev. D, 45 (1992), 3802-3806.  doi: 10.1103/PhysRevD.45.3802.  Google Scholar

[3]

P. d'Ancona, D. Foschi and S. Selberg, Product estimates for wave-Sobolev spaces in 2+1 and 1+1 dimensions, Contemp. Math., 526 (2010), 125-150.  doi: 10.1090/conm/526/10379.  Google Scholar

[4]

P. d'Ancona, D. Foschi and S. Selberg, Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system, J. Eur. Math. Soc., 9 (2007), 877-899.  doi: 10.4171/JEMS/100.  Google Scholar

[5]

D. Foschi and S. Klainerman, Bilinear space-time estimates for homogeneous wave equations, Ann. Sc. ENS. 4. serie, 33 (2000), 211-274.  doi: 10.1016/S0012-9593(00)00109-9.  Google Scholar

[6]

V. Grigoryan and A. Nahmod, Almost critical wee-posedmess for nonlinear wave equation with $Q_{\mu \nu}$ null forms in 2D, Math. Res. Letters, 21 (2014), 313-332.  doi: 10.4310/MRL.2014.v21.n2.a9.  Google Scholar

[7]

V. Grigoryan and A. Tanguay, Improved well-poseness for the quadratic derivative nonlinear wave equation in 2D, Preprint, arXiv: 1308.1719. Google Scholar

[8]

A. Grünrock, An improved local well-posedness result for the modified KdV equation, Int. Math. Res. Not., (2004), 3287–3308. doi: 10.1155/S1073792804140981.  Google Scholar

[9]

A. Grünrock, On the wave equation with quadratic nonlinearities in three space dimensions, Hyperbolic Diff. Equ., 8 (2011), 1-8.  doi: 10.1142/S0219891611002305.  Google Scholar

[10]

A. Grünrock and L. Vega, Local well-posedness for the modified KdV equation in almost critical $\hat{H}^r_s$ -spaces, Trans. Amer. Mat. Soc., 361 (2009), 5681-5694.  doi: 10.1090/S0002-9947-09-04611-X.  Google Scholar

[11]

J. Hong, Y. Kim and P. Y. Pac, Multivortex solutions of the abelian Chern-Simons-Higgs theory, Phys. Rev. Letters, 64 (1990), 2230-2233.  doi: 10.1103/PhysRevLett.64.2230.  Google Scholar

[12]

H. Huh, Cauchy problem for the Fermion field equation coupled with the Chern-Simons gauge, Lett. Math. Phys., 79 (2007), 75-94.  doi: 10.1007/s11005-006-0118-y.  Google Scholar

[13]

H. Huh and S.-J. Oh, Low regularity solutions to the Chern-Simons-Dirac and the Chern-Simons-Higgs equations in the Lorenz gauge, Comm. PDE, 41 (2016), 375-397.  doi: 10.1080/03605302.2015.1132730.  Google Scholar

[14]

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

[15]

S. Li and R. K. Bhaduri, Planar solitons of the gauged Dirac equation, Phys. Rev. D, 43 (1991), 3573-3574.   Google Scholar

[16]

H. Pecher, Low regularity solutions for Chern-Simons-Dirac systems in the temporal and Coulomb gauge, Electron. J. Differential Equations, 2016 (2016), 1-16.   Google Scholar

[17]

H. Pecher, Global well-posedness in energy space for the Chern-Simons-Higgs system in temporal gauge, J. Hyperbolic Diff. Equ., 13 (2016), 331-351.  doi: 10.1142/S0219891616500107.  Google Scholar

[18]

S. Selberg, Bilinear Fourier restriction estimates related to the 2D wave equation, Adv. Diff. Equ., 16 (2011), 667-690.   Google Scholar

[19]

S. Selberg and A. Tesfahun, Global well-posedness of the Chern-Simons-Higgs equations with finite energy, Discrete Cont. Dyn. Syst., 33 (2013), 2531-2546.  doi: 10.3934/dcds.2013.33.2531.  Google Scholar

[20]

A. Vargas and L. Vega, Global wellposedness for 1D non-linear Schrödinger equation for data with an infinite $L^2$-norm, J. Math. Pures Appl., 80 (2001), 1029-1044.  doi: 10.1016/S0021-7824(01)01224-7.  Google Scholar

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