American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Asymptotic profiles of steady states for a diffusive SIS epidemic model with spontaneous infection and a logistic source
  • CPAA Home
  • This Issue
  • Next Article
    Disconjugacy conditions and spectrum structure of clamped beam equations with two parameters
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 and 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.

[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.

[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. Sci. Ec. Norm. Super., 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 equations with $Q_{\mu \nu}$ null forms in 2D, Math. Res. Lett., 21 (2014), 313-332.  doi: 10.4310/MRL.2014.v21.n2.a9.

[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.

[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.

[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.

[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.

[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.

[12]

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

[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.

[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.

[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.

[16]

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

[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), 1029-105.  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'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.

[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.

[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. Sci. Ec. Norm. Super., 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 equations with $Q_{\mu \nu}$ null forms in 2D, Math. Res. Lett., 21 (2014), 313-332.  doi: 10.4310/MRL.2014.v21.n2.a9.

[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.

[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.

[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.

[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.

[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.

[12]

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

[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.

[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.

[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.

[16]

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

[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), 1029-105.  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. 

[1]

M. Keel, Tristan Roy, Terence Tao. Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 573-621. doi: 10.3934/dcds.2011.30.573
[2]

Magdalena Czubak, Nina Pikula. Low regularity well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1669-1683. doi: 10.3934/cpaa.2014.13.1669
[3]

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
[4]

Hartmut Pecher. Local well-posedness for the Klein-Gordon-Zakharov system in 3D. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1707-1736. doi: 10.3934/dcds.2020338
[5]

Axel Grünrock, Sebastian Herr. The Fourier restriction norm method for the Zakharov-Kuznetsov equation. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2061-2068. doi: 10.3934/dcds.2014.34.2061
[6]

Hartmut Pecher. Local well-posedness for the Maxwell-Dirac system in temporal gauge. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 3065-3076. doi: 10.3934/dcds.2022008
[7]

Shinya Kinoshita. Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in 2D. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1479-1504. doi: 10.3934/dcds.2018061
[8]

Isao Kato. Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in four and more spatial dimensions. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2247-2280. doi: 10.3934/cpaa.2016036
[9]

Hartmut Pecher. Low regularity well-posedness for the 3D Klein - Gordon - Schrödinger system. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1081-1096. doi: 10.3934/cpaa.2012.11.1081
[10]

Radhia Ghanmi, Tarek Saanouni. Well-posedness issues for some critical coupled non-linear Klein-Gordon equations. Communications on Pure and Applied Analysis, 2019, 18 (2) : 603-623. doi: 10.3934/cpaa.2019030
[11]

Hartmut Pecher. Low regularity solutions for the (2+1)-dimensional Maxwell-Klein-Gordon equations in temporal gauge. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2203-2219. doi: 10.3934/cpaa.2016034
[12]

Jishan Fan, Yueling Jia. Local well-posedness of the full compressible Navier-Stokes-Maxwell system with vacuum. Kinetic and Related Models, 2018, 11 (1) : 97-106. doi: 10.3934/krm.2018005
[13]

Yuan Xu, Fujun Zhou, Weihua Gong. Global Well-posedness and Optimal Decay Rate of the Quasi-static Incompressible Navier–Stokes–Fourier–Maxwell–Poisson System. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1537-1565. doi: 10.3934/cpaa.2022028
[14]

E. Compaan, N. Tzirakis. Low-regularity global well-posedness for the Klein-Gordon-Schrödinger system on $ \mathbb{R}^+ $. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 3867-3895. doi: 10.3934/dcds.2019156
[15]

Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007
[16]

Pietro d’Avenia, Lorenzo Pisani, Gaetano Siciliano. Klein-Gordon-Maxwell systems in a bounded domain. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 135-149. doi: 10.3934/dcds.2010.26.135
[17]

Pierre-Damien Thizy. Klein-Gordon-Maxwell equations in high dimensions. Communications on Pure and Applied Analysis, 2015, 14 (3) : 1097-1125. doi: 10.3934/cpaa.2015.14.1097
[18]

Percy D. Makita. Nonradial solutions for the Klein-Gordon-Maxwell equations. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2271-2283. doi: 10.3934/dcds.2012.32.2271
[19]

Christopher Henderson, Stanley Snelson, Andrei Tarfulea. Local well-posedness of the Boltzmann equation with polynomially decaying initial data. Kinetic and Related Models, 2020, 13 (4) : 837-867. doi: 10.3934/krm.2020029
[20]

Yong Zhou, Jishan Fan. Local well-posedness for the ideal incompressible density dependent magnetohydrodynamic equations. Communications on Pure and Applied Analysis, 2010, 9 (3) : 813-818. doi: 10.3934/cpaa.2010.9.813

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (157)
  • HTML views (82)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy