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

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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 - A, 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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