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September  2021, 20(9): 2965-2989. 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  September 2021 Early access  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 and Applied Analysis, 2021, 20 (9) : 2965-2989. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[19]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[19]

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

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Hartmut Pecher. Almost optimal local well-posedness for the Maxwell-Klein-Gordon system with data in Fourier-Lebesgue spaces. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3303-3321. doi: 10.3934/cpaa.2020146

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