May  2012, 11(3): 1081-1096. doi: 10.3934/cpaa.2012.11.1081

Low regularity well-posedness for the 3D Klein - Gordon - Schrödinger system

1. 

Fachbereich Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, Gaußstr. 20, 42097 Wuppertal, Germany

Received  November 2010 Revised  April 2011 Published  December 2011

The Klein-Gordon-Schrödinger system in 3D is shown to be locally well-posed for Schrödinger data in $H^s$ and wave data in $H^{\sigma}\times H^{\sigma -1}$ , if $ s > - \frac{1}{4},$ $\sigma > - \frac{1}{2}$ , $\sigma -2s > \frac{3}{2} $ and $\sigma -2 < s < \sigma +1$ . This result is optimal up to the endpoints in the sense that the local flow map is not $C^2$ otherwise. It is also shown that (unconditional) uniqueness holds for $s = \sigma = 0$ in the natural solution space $C^0([0,T],L^2) \times C^0([0,T],L^2) \times C^0([0,T],H^{-\frac{1}{2}}).$ This solution exists even globally by Colliander, Holmer and Tzirakis [6]. The proofs are based on new well-posedness results for the Zakharov system by Bejenaru, Herr, Holmer and Tataru [3], and Bejenaru and Herr [4].
Citation: Hartmut Pecher. Low regularity well-posedness for the 3D Klein - Gordon - Schrödinger system. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1081-1096. doi: 10.3934/cpaa.2012.11.1081
References:
[1]

T. Akahori, Global solutions of the wave-Schrödinger system below $L^2$,, Hokkaido Math. J., 35 (2006), 779. Google Scholar

[2]

T. Akahori, Global solutions of the wave-Schrödinger system,, Doctoral thesis., (). Google Scholar

[3]

I. Bejenaru, S. Herr, J. Holmer and D. Tataru, On the 2D Zakharov system with $L^2$ Schrödinger data,, Nonlinearity, 22 (2009), 1063. doi: 10.1088/0951-7715/22/5/007. Google Scholar

[4]

I. Bejenaru and S. Herr, Convolutions of singular measures and applications to the Zakharov system,, J. Funct. Analysis, 261 (2011), 478. doi: 10.1016/j.jfa.2011.03.015. Google Scholar

[5]

J. Bourgain, Periodic Korteweg de Vries equation with measures as initial data,, Selecta Math. (N.S.), 3 (1997), 115. Google Scholar

[6]

J. Colliander, J. Holmer, and N. Tzirakis, Low regularity global well-posedness for the Zakharov and Klein - Gordon - Schrödinger systems,, Transact. AMS, 360 (2008), 4619. doi: 10.1090/S0002-9947-08-04295-5. Google Scholar

[7]

J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system,, J. Funct. Analysis, 151 (1997), 384. doi: 10.1006/jfan.1997.3148. Google Scholar

[8]

J. Holmer, Local ill-posedness of the 1D Zakharov system,, Electr. J. Differential Equations, 24 (2007), 1. Google Scholar

[9]

N. Masmoudi and K. Nakanishi, Uniqueness of finite energy solutions for Maxwell-Dirac and Maxwell-Klein-Gordon equations,, Commun. Math. Phys., 243 (2003), 123. doi: 10.1007/s00220-003-0951-0. Google Scholar

[10]

N. Masmoudi and K. Nakanishi, Uniqueness of solutions for Zakharov systems,, Funkcial. Ekvac., 52 (2009), 233. doi: 10.1619/fesi.52.233. Google Scholar

[11]

L. Molinet, J. C. Saut and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations,, SIAM J. Math. Anal., 33 (2001), 982. doi: 10.1137/S0036141001385307. Google Scholar

[12]

L. Molinet, J. C. Saut and N. Tzvetkov, Well-posedness and ill-posedness for the Kadomtsev-Petviashvili-I equation,, Duke Math. J., 115 (2002), 353. doi: 10.1215/S0012-7094-02-11525-7. Google Scholar

[13]

H. Pecher, Global solutions of the Klein - Gordon - Schrödinger system with rough data,, Diff. Int. Equations, 17 (2004), 179. Google Scholar

[14]

F. Planchon, On uniqueness for semilinear wave equations,, Math. Z., 244 (2003), 587. doi: 10.1007/s00209-003-0509-z. Google Scholar

[15]

N. Tzvetkov, Remark on the local ill-posedness for KdV equation,, C.R. Acad. Sci. Paris Ser. I Math., 329 (1999), 1043. doi: 10.1016/S0764-4442(00)88471-2. Google Scholar

[16]

Y. Zhou, Uniqueness of weak solutions of the KdV equation,, Int. Math. Res. Notices No., 6 (1997), 271. doi: 10.1155/S1073792897000202. Google Scholar

[17]

Y. Zhou, Uniqueness of generalized solutions to nonlinear wave equations,, Amer. J. of Math., 122 (2000), 939. doi: 10.1353/ajm.2000.0040. Google Scholar

show all references

References:
[1]

T. Akahori, Global solutions of the wave-Schrödinger system below $L^2$,, Hokkaido Math. J., 35 (2006), 779. Google Scholar

[2]

T. Akahori, Global solutions of the wave-Schrödinger system,, Doctoral thesis., (). Google Scholar

[3]

I. Bejenaru, S. Herr, J. Holmer and D. Tataru, On the 2D Zakharov system with $L^2$ Schrödinger data,, Nonlinearity, 22 (2009), 1063. doi: 10.1088/0951-7715/22/5/007. Google Scholar

[4]

I. Bejenaru and S. Herr, Convolutions of singular measures and applications to the Zakharov system,, J. Funct. Analysis, 261 (2011), 478. doi: 10.1016/j.jfa.2011.03.015. Google Scholar

[5]

J. Bourgain, Periodic Korteweg de Vries equation with measures as initial data,, Selecta Math. (N.S.), 3 (1997), 115. Google Scholar

[6]

J. Colliander, J. Holmer, and N. Tzirakis, Low regularity global well-posedness for the Zakharov and Klein - Gordon - Schrödinger systems,, Transact. AMS, 360 (2008), 4619. doi: 10.1090/S0002-9947-08-04295-5. Google Scholar

[7]

J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system,, J. Funct. Analysis, 151 (1997), 384. doi: 10.1006/jfan.1997.3148. Google Scholar

[8]

J. Holmer, Local ill-posedness of the 1D Zakharov system,, Electr. J. Differential Equations, 24 (2007), 1. Google Scholar

[9]

N. Masmoudi and K. Nakanishi, Uniqueness of finite energy solutions for Maxwell-Dirac and Maxwell-Klein-Gordon equations,, Commun. Math. Phys., 243 (2003), 123. doi: 10.1007/s00220-003-0951-0. Google Scholar

[10]

N. Masmoudi and K. Nakanishi, Uniqueness of solutions for Zakharov systems,, Funkcial. Ekvac., 52 (2009), 233. doi: 10.1619/fesi.52.233. Google Scholar

[11]

L. Molinet, J. C. Saut and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations,, SIAM J. Math. Anal., 33 (2001), 982. doi: 10.1137/S0036141001385307. Google Scholar

[12]

L. Molinet, J. C. Saut and N. Tzvetkov, Well-posedness and ill-posedness for the Kadomtsev-Petviashvili-I equation,, Duke Math. J., 115 (2002), 353. doi: 10.1215/S0012-7094-02-11525-7. Google Scholar

[13]

H. Pecher, Global solutions of the Klein - Gordon - Schrödinger system with rough data,, Diff. Int. Equations, 17 (2004), 179. Google Scholar

[14]

F. Planchon, On uniqueness for semilinear wave equations,, Math. Z., 244 (2003), 587. doi: 10.1007/s00209-003-0509-z. Google Scholar

[15]

N. Tzvetkov, Remark on the local ill-posedness for KdV equation,, C.R. Acad. Sci. Paris Ser. I Math., 329 (1999), 1043. doi: 10.1016/S0764-4442(00)88471-2. Google Scholar

[16]

Y. Zhou, Uniqueness of weak solutions of the KdV equation,, Int. Math. Res. Notices No., 6 (1997), 271. doi: 10.1155/S1073792897000202. Google Scholar

[17]

Y. Zhou, Uniqueness of generalized solutions to nonlinear wave equations,, Amer. J. of Math., 122 (2000), 939. doi: 10.1353/ajm.2000.0040. Google Scholar

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