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June  2020, 19(6): 3303-3321. doi: 10.3934/cpaa.2020146

Almost optimal local well-posedness for the Maxwell-Klein-Gordon system with data in Fourier-Lebesgue spaces

Hartmut Pecher

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

Received  August 2019 Revised  November 2019 Published  March 2020

We prove a low regularity local well-posedness result for the Maxwell-Klein-Gordon system in three space dimensions for data in Fourier - Lebesgue spaces $ \widehat{H}^{s,r} $, where $ \|f\|_{\widehat{H}^{s,r}} = \|\langle \xi \rangle^s \widehat{f}(\xi)\|_{\widehat{L}^{r'}} $, $ \frac{1}{r}+\frac{1}{r'} = 1 $. The assumed regularity for the data is almost optimal with respect to scaling as $ r \to 1 $. This closes the gap between what is known in the case $ r = 2 $, namely $ s > \frac{3}{4} $, and the critical value $ s_c = \frac{1}{2} $ with respect to scaling.

Keywords: Maxwell-Klein-Gordon, local well-posedness, Fourier restriction norm method.
Mathematics Subject Classification: 35Q61, 35L70.
Citation: Hartmut Pecher. Almost optimal local well-posedness for the Maxwell-Klein-Gordon system with data in Fourier-Lebesgue spaces. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3303-3321. doi: 10.3934/cpaa.2020146
References:
[1]

P. d'Ancona, D. Foschi and S. Selberg, Null structure and almost optimal local regularity of the Maxwell-Dirac system, Amer. J. Math., 132 (2010), 771-839.  doi: 10.1353/ajm.0.0118.  Google Scholar

[2]

P. d'Ancona, D. Foschi and S. Selberg, Atlas of products for wave-Sobolev spaces in $\mathbb {R}^{1+3}$, Trans. Amer. Math. Soc., 364 (2012), 31-63.  doi: 10.1090/S0002-9947-2011-05250-5.  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. Sci. Ec. Norm. Super., 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 equations 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]

A. Grünrock, An improved local well-posedness result for the modified KdV equation, Int. Math. Res. Notices, 61 (2004), 3287-3308.  doi: 10.1155/S1073792804140981.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

H. Pecher, Local well-posedness for low regularity data for the higher-dimensional Maxwell-Klein-Gordon system in Lorenz gauge, J. Math. Phys., 59 (2018), 101503. doi: 10.1063/1.5035408.  Google Scholar

[15]

H. Pecher, Low regularity local well-posedness for the (N+1)-dimensional Maxwell-Klein-Gordon equation in Lorenz gauge, preprint, arXiv: 1705.00599. doi: 10.3934/cpaa.2016034.  Google Scholar

[16]

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

[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), 1029-105.  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'Ancona, D. Foschi and S. Selberg, Null structure and almost optimal local regularity of the Maxwell-Dirac system, Amer. J. Math., 132 (2010), 771-839.  doi: 10.1353/ajm.0.0118.  Google Scholar

[2]

P. d'Ancona, D. Foschi and S. Selberg, Atlas of products for wave-Sobolev spaces in $\mathbb {R}^{1+3}$, Trans. Amer. Math. Soc., 364 (2012), 31-63.  doi: 10.1090/S0002-9947-2011-05250-5.  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. Sci. Ec. Norm. Super., 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 equations 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]

A. Grünrock, An improved local well-posedness result for the modified KdV equation, Int. Math. Res. Notices, 61 (2004), 3287-3308.  doi: 10.1155/S1073792804140981.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

H. Pecher, Local well-posedness for low regularity data for the higher-dimensional Maxwell-Klein-Gordon system in Lorenz gauge, J. Math. Phys., 59 (2018), 101503. doi: 10.1063/1.5035408.  Google Scholar

[15]

H. Pecher, Low regularity local well-posedness for the (N+1)-dimensional Maxwell-Klein-Gordon equation in Lorenz gauge, preprint, arXiv: 1705.00599. doi: 10.3934/cpaa.2016034.  Google Scholar

[16]

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

[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), 1029-105.  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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Magdalena Czubak, Nina Pikula. Low regularity well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1669-1683. doi: 10.3934/cpaa.2014.13.1669
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