doi: 10.3934/cpaa.2021091

Improved well-posedness results for the Maxwell-Klein-Gordon system in 2D

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

Received  December 2020 Revised  May 2021 Published  June 2021

The local well-posedness problem for the Maxwell-Klein-Gordon system in Coulomb gauge as well as Lorenz gauge is treated in two space dimensions for data with minimal regularity assumptions. In the classical case of data in $ L^2 $-based Sobolev spaces $ H^s $ and $ H^l $ for the electromagnetic field $ \phi $ and the potential $ A $, respectively, the minimal regularity assumptions are $ s > \frac{1}{2} $ and $ l > \frac{1}{4} $, which leaves a gap of $ \frac{1}{2} $ and $ \frac{1}{4} $ to the critical regularity with respect to scaling $ s_c = l_c = 0 $. This gap can be reduced for data in Fourier-Lebesgue spaces $ \widehat{H}^{s, r} $ and $ \widehat{H}^{l, r} $ to $ s> \frac{21}{16} $ and $ l > \frac{9}{8} $ for $ r $ close to $ 1 $, whereas the critical exponents with respect to scaling fulfill $ s_c \to 1 $, $ l_c \to 1 $ as $ r \to 1 $. Here $ \|f\|_{\widehat{H}^{s, r}} : = \| \langle \xi \rangle^s \tilde{f}\|_{L^{r'}_{\tau \xi}} \, , \, 1 < r \le 2 \, , \, \frac{1}{r}+\frac{1}{r'} = 1 \, . $ Thus the gap is reduced for $ \phi $ as well as $ A $ in both gauges.

Citation: Hartmut Pecher. Improved well-posedness results for the Maxwell-Klein-Gordon system in 2D. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021091
References:
[1]

P. d'AnconaD. 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

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P. d'AnconaD. Foschi and S. Selberg, Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system, J. EMS, 9 (2007), 877-898.  doi: 10.4171/JEMS/100.  Google Scholar

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M. Czubak and N. Pikula, Low regularity well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge., Commun. Pure Appl. Anal., 13 (2014), 1669-1683.  doi: 10.3934/cpaa.2014.13.1669.  Google Scholar

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

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

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V. Grigoryan and A. Tanguay, Improved well-posedness for the quadratic derivative nonlinear wave equation in 2D, J. Math. Anal. Appl., 475 (2019), 1578-1595.  doi: 10.1016/j.jmaa.2019.03.032.  Google Scholar

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A. Grünrock, An improved local well-posedness result for the modified KdV equation., Int. Math. Res. Not., 61 (2004), 3287-3308.  doi: 10.1155/S1073792804140981.  Google Scholar

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

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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

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M. KeelT. Roy and T. Tao, Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm, Discrete Cont. Dyn. Syst., 30 (2011), 573-621.  doi: 10.3934/dcds.2011.30.573.  Google Scholar

[12]

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.  Google Scholar

[13]

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

[14]

M. Machedon and J. Sterbenz, Almost optimal local well-posedness for the (3+1)- dimensional Maxwell-Klein-Gordon equations, J. AMS, 17 (2004), 297-359.  doi: 10.1090/S0894-0347-03-00445-4.  Google Scholar

[15]

H. Pecher, Low regularity local well-posedness for the Maxwell-Klein-Gordon equations in Lorenz gauge, Adv. Differ. Equ., 19 (2014), 359-386.   Google Scholar

[16]

H. Pecher, Almost optimal local well-posedness for the Maxwell-Klein-Gordon system in Fourier-Lebesgue spaces, Commun. Pure Appl. Anal., 19 (2020), 3303-3321.  doi: 10.3934/cpaa.2020146.  Google Scholar

[17]

S. Selberg, Almost optimal local well-posedness of the Maxwell-Klein-Gordon equations in 1+4 dimensions, Commun. Partial Differ. Equ., 27 (2002), 1183-1227.  doi: 10.1081/PDE-120004899.  Google Scholar

[18]

S. Selberg and A. Tesfahun, Finite-energy global well-posedness of the Maxwell-Klein-Gordon system in Lorenz gauge, Commun. Partial Differ. Equ., 35 (2010), 1029-1057.  doi: 10.1080/03605301003717100.  Google Scholar

[19]

T. Tao, Multilinear weighted convolutions of $L^2$-functions, and applications to non-linear dispersive equations, Amer. J. Math., 123 (2001), 839-908.   Google Scholar

show all references

References:
[1]

P. d'AnconaD. 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

[2]

P. d'AnconaD. Foschi and S. Selberg, Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system, J. EMS, 9 (2007), 877-898.  doi: 10.4171/JEMS/100.  Google Scholar

[3]

S. Cuccagna, On the local existence for the Maxwell-Klein-Gordon system in $\mathbb{R}^{3+1}$, Commun. Partial Differ. Equ., 24 (1999), 851-867.  doi: 10.1080/03605309908821449.  Google Scholar

[4]

M. Czubak and N. Pikula, Low regularity well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge., Commun. Pure Appl. Anal., 13 (2014), 1669-1683.  doi: 10.3934/cpaa.2014.13.1669.  Google Scholar

[5]

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

[6]

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

[7]

V. Grigoryan and A. Tanguay, Improved well-posedness for the quadratic derivative nonlinear wave equation in 2D, J. Math. Anal. Appl., 475 (2019), 1578-1595.  doi: 10.1016/j.jmaa.2019.03.032.  Google Scholar

[8]

A. Grünrock, An improved local well-posedness result for the modified KdV equation., Int. Math. Res. Not., 61 (2004), 3287-3308.  doi: 10.1155/S1073792804140981.  Google Scholar

[9]

A. Grünrock, On the wave equation with quadratic nonlinearities in three space dimensions., Hyperbolic Differ. 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]

M. KeelT. Roy and T. Tao, Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm, Discrete Cont. Dyn. Syst., 30 (2011), 573-621.  doi: 10.3934/dcds.2011.30.573.  Google Scholar

[12]

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.  Google Scholar

[13]

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

[14]

M. Machedon and J. Sterbenz, Almost optimal local well-posedness for the (3+1)- dimensional Maxwell-Klein-Gordon equations, J. AMS, 17 (2004), 297-359.  doi: 10.1090/S0894-0347-03-00445-4.  Google Scholar

[15]

H. Pecher, Low regularity local well-posedness for the Maxwell-Klein-Gordon equations in Lorenz gauge, Adv. Differ. Equ., 19 (2014), 359-386.   Google Scholar

[16]

H. Pecher, Almost optimal local well-posedness for the Maxwell-Klein-Gordon system in Fourier-Lebesgue spaces, Commun. Pure Appl. Anal., 19 (2020), 3303-3321.  doi: 10.3934/cpaa.2020146.  Google Scholar

[17]

S. Selberg, Almost optimal local well-posedness of the Maxwell-Klein-Gordon equations in 1+4 dimensions, Commun. Partial Differ. Equ., 27 (2002), 1183-1227.  doi: 10.1081/PDE-120004899.  Google Scholar

[18]

S. Selberg and A. Tesfahun, Finite-energy global well-posedness of the Maxwell-Klein-Gordon system in Lorenz gauge, Commun. Partial Differ. Equ., 35 (2010), 1029-1057.  doi: 10.1080/03605301003717100.  Google Scholar

[19]

T. Tao, Multilinear weighted convolutions of $L^2$-functions, and applications to non-linear dispersive equations, Amer. J. Math., 123 (2001), 839-908.   Google Scholar

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