-
Previous Article
Discrete gradient fields on infinite complexes
- DCDS Home
- This Issue
- Next Article
Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm
1. | Department of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church St. S.E., Minneapolis, MN 55455 |
2. | Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, United States |
3. | Department of Mathematics, UCLA, 405 Hilgard Ave, Los Angeles, CA 90095 |
References:
[1] |
J. Bourgain, "Global Solutions of Nonlinear Schrödinger Equations," AMS Publications, 1999. |
[2] |
H. Bahouri and J-Y. Chemin, On global well-posedness for defocusing cubic wave equation, Int. Math. Res. Not., 2006, Art. ID 54873, 12 pp. |
[3] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness result for KdV in Sobolev spaces of negative index, Elec. J. Diff. Eq., 2001 (2001), 1-7 (electronic). |
[4] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for the Schrodinger equations with derivative, SIAM J. Math., 33 (2001), 649-669.
doi: 10.1137/S0036141001384387. |
[5] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Almost conservation laws and global rough solutions to a nonlinear Schrodinger equation, Math. Res. Letters, 9 (2002), 659-682. |
[6] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, A refined global well-posedness for the Schrodinger equations with derivative, SIAM J. Math., 34 (2002), 64-86.
doi: 10.1137/S0036141001394541. |
[7] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\R$ and $\T$, J. Amer. Math. Soc., 16 (2003), 705-749.
doi: 10.1090/S0894-0347-03-00421-1. |
[8] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Multilinear estimates for periodic KdV equations and applications, J. Funct. Anal., 211 (2004), 173-218.
doi: 10.1016/S0022-1236(03)00218-0. |
[9] |
J. Colliander, M. Keel, G. Staffilani, H. Takoka and T. Tao, Resonant decompositions and the I-method for cubic nonlinear Schrodinger on $\R^2$, Disc. Cont. Dynam. Systems A, 21 (2008), 665-686.
doi: 10.3934/dcds.2008.21.665. |
[10] |
S. Cuccagna, On the local existence for the Maxwell Klein Gordon system in $\R^{3+1}$, Comm. Partial Differential Equations, 24 (1999), 851-867.
doi: 10.1080/03605309908821449. |
[11] |
D. Eardley and V. Moncrief, The global existence of Yang-Mills-Higgs fields in $\R^{3+1}$, Comm. Math. Phys., 83 (1982), 171-212.
doi: 10.1007/BF01976040. |
[12] |
D. Foschi and S. Klainerman, Bilinear space-time estimates for homogeneous wave equations, Les Annales Scientifiques de l'Ecole Normale Superieure, 33 (2000), 211-274.
doi: 10.1016/S0012-9593(00)00109-9. |
[13] |
I. Gallagher and F. Planchon, On global solutions to a defocusing semi-linear wave equation, Revista Mat. Iberoamericana, 19 (2003), 161-177. |
[14] |
L. Kapitanski, Weak and yet weaker solutions of semilinear wave equations, Comm. Partial Differential Equations, 19 (1994), 1629-1676.
doi: 10.1080/03605309408821067. |
[15] |
M. Keel and T. Tao, Endpoint strichartz estimates, Amer. Math. J., 120 (1998), 955-980.
doi: 10.1353/ajm.1998.0039. |
[16] |
M. Keel and T. Tao, Local and global well-posedness of wave maps on $\R^{1+1}$ for rough data, Internat. Math. Res. Not., 21 (1998), 1117-1156.
doi: 10.1155/S107379289800066X. |
[17] |
C. Kenig, G. Ponce and L. Vega, Global well-posedness for semi-linear wave equations, Comm. Partial Differential Equations, 25 (2000), 1741-1752.
doi: 10.1080/03605300008821565. |
[18] |
S. Klainerman, On the regularity of classical field theories in Minkowski space-time $\R^{3+1}$, Prog. in Nonlin. Diff. Eq. and their Applic., Birkhäuser, 29 (1997), 113-150. |
[19] |
S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math., 46 (1993), 1221-1268.
doi: 10.1002/cpa.3160460902. |
[20] |
S. Klainerman and M. Machedon, Finite energy solutions of the Yang-Mills equations in $\R^{3+1}$, Ann. of Math., 142 (1995), 39-119.
doi: 10.2307/2118611. |
[21] |
S. Klainerman and M. Machedon, Smoothing estimates for null forms and applications, Duke Math J., 81 (1995), 99-103.
doi: 10.1215/S0012-7094-95-08109-5. |
[22] |
S. Klainerman and M. Machedon, Remark on Strichartz-type inequalities, With appendices by Jean Bourgain and Daniel Tataru. Internat. Math. Res. Notices, 5 (1996), 201-220.
doi: 10.1155/S1073792896000153. |
[23] |
S. Klainerman and M. Machedon, Estimates for null forms and the spaces $H_{s,\delta}$, Internat. Math. Res. Notices, 17 (1996), 853-865.
doi: 10.1155/S1073792896000529. |
[24] |
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. |
[25] |
S. Klainerman and M. Machedon, On the optimal local regularity for gauge field theories, Diff. and Integral Eq., 10 (1997), 1019-1030. |
[26] |
S. Klainerman and S. Selberg, Bilinear estimates and applications to nonlinear wave equations, Commun. Contemp. Math., 4 (2002), 223-295.
doi: 10.1142/S0219199702000634. |
[27] |
S. Klainerman and D. Tataru, On the optimal regularity for Yang-Mills equations in $\R^{4+1}$, J. Amer. Math. Soc., 12 (1999), 93-116.
doi: 10.1090/S0894-0347-99-00282-9. |
[28] |
S. Klainerman, I. Rodnianski and T. Tao, A physical approach to wave equation bilinear estimate, Dedicated to the memory of Thomas H. Wolff, J. Anal. Math., 87 (2002), 299-336
doi: 10.1007/BF02868479. |
[29] |
H. Lindblad and C.D. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal., 130 (1995), 357-426.
doi: 10.1006/jfan.1995.1075. |
[30] |
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. |
[31] |
T. Roy, Adapted linear-nonlinear decomposition and global well-posedness for solutions to the defocusing cubic wave equation On $\R^{3}$, Discrete Contin. Dyn. Syst., 24 (2009), 1307-1323.
doi: 10.3934/dcds.2009.24.1307. |
[32] |
T. Roy, Global well-posedness for the radial defocusing cubic wave equation and for rough data, Elec. J. Diff. Eq., 166 (2007), 1-22. |
[33] |
S. Selberg, "Multilinear Space-Time Estimates and Applications to Local Existence Theory for Nonlinear Wave Equations," Princeton University Thesis, 1999. |
[34] |
S. Selberg, Almost optimal local well-posedness of the Klein-Gordon-Maxwell system in 1+4 dimensions, Communications in PDE, 27 (2002), 1183-1227.
doi: 10.1081/PDE-120004899. |
[35] |
E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton University Press, 1970. |
[36] |
C. D. Sogge, "Lectures on Nonlinear Wave Equations," Monographs in Analysis II, International Press, 1995. |
[37] |
T. Tao, Multilinear weighted convolution of $L^2_x$ functions, and applications to nonlinear dispersive equations, Amer. J. Math., 123 (2001), 839-908.
doi: 10.1353/ajm.2001.0035. |
[38] |
T. Tao, "Nonlinear Dispersive Equations: Local and Global Analysis," CBMS regional conference series in mathematics, 2006. |
[39] |
K. Uhlenbeck, Connections with $L^p$ bounds on curvature, Comm. Math. Phys., 83 (1982), 31-42.
doi: 10.1007/BF01947069. |
show all references
References:
[1] |
J. Bourgain, "Global Solutions of Nonlinear Schrödinger Equations," AMS Publications, 1999. |
[2] |
H. Bahouri and J-Y. Chemin, On global well-posedness for defocusing cubic wave equation, Int. Math. Res. Not., 2006, Art. ID 54873, 12 pp. |
[3] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness result for KdV in Sobolev spaces of negative index, Elec. J. Diff. Eq., 2001 (2001), 1-7 (electronic). |
[4] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for the Schrodinger equations with derivative, SIAM J. Math., 33 (2001), 649-669.
doi: 10.1137/S0036141001384387. |
[5] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Almost conservation laws and global rough solutions to a nonlinear Schrodinger equation, Math. Res. Letters, 9 (2002), 659-682. |
[6] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, A refined global well-posedness for the Schrodinger equations with derivative, SIAM J. Math., 34 (2002), 64-86.
doi: 10.1137/S0036141001394541. |
[7] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\R$ and $\T$, J. Amer. Math. Soc., 16 (2003), 705-749.
doi: 10.1090/S0894-0347-03-00421-1. |
[8] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Multilinear estimates for periodic KdV equations and applications, J. Funct. Anal., 211 (2004), 173-218.
doi: 10.1016/S0022-1236(03)00218-0. |
[9] |
J. Colliander, M. Keel, G. Staffilani, H. Takoka and T. Tao, Resonant decompositions and the I-method for cubic nonlinear Schrodinger on $\R^2$, Disc. Cont. Dynam. Systems A, 21 (2008), 665-686.
doi: 10.3934/dcds.2008.21.665. |
[10] |
S. Cuccagna, On the local existence for the Maxwell Klein Gordon system in $\R^{3+1}$, Comm. Partial Differential Equations, 24 (1999), 851-867.
doi: 10.1080/03605309908821449. |
[11] |
D. Eardley and V. Moncrief, The global existence of Yang-Mills-Higgs fields in $\R^{3+1}$, Comm. Math. Phys., 83 (1982), 171-212.
doi: 10.1007/BF01976040. |
[12] |
D. Foschi and S. Klainerman, Bilinear space-time estimates for homogeneous wave equations, Les Annales Scientifiques de l'Ecole Normale Superieure, 33 (2000), 211-274.
doi: 10.1016/S0012-9593(00)00109-9. |
[13] |
I. Gallagher and F. Planchon, On global solutions to a defocusing semi-linear wave equation, Revista Mat. Iberoamericana, 19 (2003), 161-177. |
[14] |
L. Kapitanski, Weak and yet weaker solutions of semilinear wave equations, Comm. Partial Differential Equations, 19 (1994), 1629-1676.
doi: 10.1080/03605309408821067. |
[15] |
M. Keel and T. Tao, Endpoint strichartz estimates, Amer. Math. J., 120 (1998), 955-980.
doi: 10.1353/ajm.1998.0039. |
[16] |
M. Keel and T. Tao, Local and global well-posedness of wave maps on $\R^{1+1}$ for rough data, Internat. Math. Res. Not., 21 (1998), 1117-1156.
doi: 10.1155/S107379289800066X. |
[17] |
C. Kenig, G. Ponce and L. Vega, Global well-posedness for semi-linear wave equations, Comm. Partial Differential Equations, 25 (2000), 1741-1752.
doi: 10.1080/03605300008821565. |
[18] |
S. Klainerman, On the regularity of classical field theories in Minkowski space-time $\R^{3+1}$, Prog. in Nonlin. Diff. Eq. and their Applic., Birkhäuser, 29 (1997), 113-150. |
[19] |
S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math., 46 (1993), 1221-1268.
doi: 10.1002/cpa.3160460902. |
[20] |
S. Klainerman and M. Machedon, Finite energy solutions of the Yang-Mills equations in $\R^{3+1}$, Ann. of Math., 142 (1995), 39-119.
doi: 10.2307/2118611. |
[21] |
S. Klainerman and M. Machedon, Smoothing estimates for null forms and applications, Duke Math J., 81 (1995), 99-103.
doi: 10.1215/S0012-7094-95-08109-5. |
[22] |
S. Klainerman and M. Machedon, Remark on Strichartz-type inequalities, With appendices by Jean Bourgain and Daniel Tataru. Internat. Math. Res. Notices, 5 (1996), 201-220.
doi: 10.1155/S1073792896000153. |
[23] |
S. Klainerman and M. Machedon, Estimates for null forms and the spaces $H_{s,\delta}$, Internat. Math. Res. Notices, 17 (1996), 853-865.
doi: 10.1155/S1073792896000529. |
[24] |
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. |
[25] |
S. Klainerman and M. Machedon, On the optimal local regularity for gauge field theories, Diff. and Integral Eq., 10 (1997), 1019-1030. |
[26] |
S. Klainerman and S. Selberg, Bilinear estimates and applications to nonlinear wave equations, Commun. Contemp. Math., 4 (2002), 223-295.
doi: 10.1142/S0219199702000634. |
[27] |
S. Klainerman and D. Tataru, On the optimal regularity for Yang-Mills equations in $\R^{4+1}$, J. Amer. Math. Soc., 12 (1999), 93-116.
doi: 10.1090/S0894-0347-99-00282-9. |
[28] |
S. Klainerman, I. Rodnianski and T. Tao, A physical approach to wave equation bilinear estimate, Dedicated to the memory of Thomas H. Wolff, J. Anal. Math., 87 (2002), 299-336
doi: 10.1007/BF02868479. |
[29] |
H. Lindblad and C.D. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal., 130 (1995), 357-426.
doi: 10.1006/jfan.1995.1075. |
[30] |
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. |
[31] |
T. Roy, Adapted linear-nonlinear decomposition and global well-posedness for solutions to the defocusing cubic wave equation On $\R^{3}$, Discrete Contin. Dyn. Syst., 24 (2009), 1307-1323.
doi: 10.3934/dcds.2009.24.1307. |
[32] |
T. Roy, Global well-posedness for the radial defocusing cubic wave equation and for rough data, Elec. J. Diff. Eq., 166 (2007), 1-22. |
[33] |
S. Selberg, "Multilinear Space-Time Estimates and Applications to Local Existence Theory for Nonlinear Wave Equations," Princeton University Thesis, 1999. |
[34] |
S. Selberg, Almost optimal local well-posedness of the Klein-Gordon-Maxwell system in 1+4 dimensions, Communications in PDE, 27 (2002), 1183-1227.
doi: 10.1081/PDE-120004899. |
[35] |
E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton University Press, 1970. |
[36] |
C. D. Sogge, "Lectures on Nonlinear Wave Equations," Monographs in Analysis II, International Press, 1995. |
[37] |
T. Tao, Multilinear weighted convolution of $L^2_x$ functions, and applications to nonlinear dispersive equations, Amer. J. Math., 123 (2001), 839-908.
doi: 10.1353/ajm.2001.0035. |
[38] |
T. Tao, "Nonlinear Dispersive Equations: Local and Global Analysis," CBMS regional conference series in mathematics, 2006. |
[39] |
K. Uhlenbeck, Connections with $L^p$ bounds on curvature, Comm. Math. Phys., 83 (1982), 31-42.
doi: 10.1007/BF01947069. |
[1] |
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 |
[2] |
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 |
[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. 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 |
[5] |
Bassam Kojok. Global existence for a forced dispersive dissipative equation via the I-method. Communications on Pure and Applied Analysis, 2009, 8 (4) : 1401-1419. doi: 10.3934/cpaa.2009.8.1401 |
[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] |
Stefan Meyer, Mathias Wilke. Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces. Evolution Equations and Control Theory, 2013, 2 (2) : 365-378. doi: 10.3934/eect.2013.2.365 |
[8] |
Tadahiro Oh, Yuzhao Wang. On global well-posedness of the modified KdV equation in modulation spaces. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2971-2992. doi: 10.3934/dcds.2020393 |
[9] |
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 |
[10] |
Jianjun Yuan. On the well-posedness of Maxwell-Chern-Simons-Higgs system in the Lorenz gauge. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2389-2403. doi: 10.3934/dcds.2014.34.2389 |
[11] |
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 |
[12] |
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 |
[13] |
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 |
[14] |
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 |
[15] |
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 |
[16] |
Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843 |
[17] |
Sergey Zelik, Jon Pennant. Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$. Communications on Pure and Applied Analysis, 2013, 12 (1) : 461-480. doi: 10.3934/cpaa.2013.12.461 |
[18] |
Ming Wang. Sharp global well-posedness of the BBM equation in $L^p$ type Sobolev spaces. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5763-5788. doi: 10.3934/dcds.2016053 |
[19] |
Baoyan Sun, Kung-Chien Wu. Global well-posedness and exponential stability for the fermion equation in weighted Sobolev spaces. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2537-2562. doi: 10.3934/dcdsb.2021147 |
[20] |
Goro Akagi, Kei Matsuura. Well-posedness and large-time behaviors of solutions for a parabolic equation involving $p(x)$-Laplacian. Conference Publications, 2011, 2011 (Special) : 22-31. doi: 10.3934/proc.2011.2011.22 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]