## Kyoto Journal of Mathematics

### Well-posedness for nonlinear Dirac equations in one dimension

#### Abstract

We completely determine the range of Sobolev regularity for the Dirac- Klein-Gordon system, the quadratic nonlinear Dirac equations, and the wave-map equation to be well posed locally in time on the real line. For the Dirac-Klein-Gordon system, we can continue those local solutions in nonnegative Sobolev spaces by the charge conservation. In particular, we obtain global well-posedness in the space where both the spinor and scalar fields are only in $L^2({\mathbb R})$. Outside the range for well-posedness, we show either that some solutions exit the Sobolev space instantly or that the solution map is not twice differentiable at zero.

#### Article information

Source
Kyoto J. Math., Volume 50, Number 2 (2010), 403-451.

Dates
First available in Project Euclid: 7 May 2010

https://projecteuclid.org/euclid.kjm/1273236821

Digital Object Identifier
doi:10.1215/0023608X-2009-018

Mathematical Reviews number (MathSciNet)
MR2666663

Zentralblatt MATH identifier
1248.35170

#### Citation

Machihara, Shuji; Nakanishi, Kenji; Tsugawa, Kotaro. Well-posedness for nonlinear Dirac equations in one dimension. Kyoto J. Math. 50 (2010), no. 2, 403--451. doi:10.1215/0023608X-2009-018. https://projecteuclid.org/euclid.kjm/1273236821

#### References

• [1] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, II: The KdV-equation, Geom. Funct. Anal. 3 (1993), 209–262.
• [2] N. Bournaveas, A new proof of global existence for the Dirac-Klein-Gordon equations in one space dimension, J. Funct. Anal. 173 (2000), 203–213.
• [3] N. Bournaveas, Local well-posedness for a nonlinear Dirac equation in spaces of almost critical dimension, Discrete Contin. Dyn. Syst. 20 (2008), 605–616.
• [4] N. Bournaveas and D. Gibbeson, Global charge class solutions of the Dirac-Klein-Gordon equations in one space dimension, Differential Integral Equations 19 (2006), 1001–1018.
• [5] N. Bournaveas and D. Gibbeson, Low regularity global solutions of the Dirac-Klein-Gordon equations in one space dimension, Differential Integral Equations 19 (2006), 211–222.
• [6] J. M. Chadam, Global solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac equations in one space dimension, J. Funct. Anal. 13 (1973), 173–184.
• [7] P. D’Ancona, D. Foschi, and S. Selberg, Local well-posedness below the charge norm for the Dirac-Klein-Gordon system in two space dimensions, J. Hyperbolic Differ. Equ. 4 (2007), 295–330.
• [8] P. D’Ancona, D. Foschi, and S. Selberg, Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system, J. Eur. Math. Soc. (JEMS) 9 (2007), 877–899.
• [9] Y.-F. Fang, “A direct proof of global existence for the Dirac-Klein-Gordon equations in one space dimension” in Proceedings of the Third East Asia Partial Differential Equation Conference, Taiwanese J. Math. 8 Math. Soc. Repub. China (Taiwan), Kaohsiung, 2004, 33–41.
• [10] Y.-F. Fang, On the Dirac-Klein-Gordon equations in one space dimension, Differential Integral Equations 17 (2004), 1321–1346.
• [11] Y.-F. Fang and M. G. Grillakis, On the Dirac-Klein-Gordon equations in three space dimensions, Comm. Partial Differential Equations 30 (2005), 783–812.
• [12] Y.-F. Fang and M. G. Grillakis, On the Dirac-Klein-Gordon system in 2+1 dimensions, preprint.
• [13] Y.-F. Fang and H.-C. Huang, A critical case of the Dirac-Klein-Gordon equations in one space dimension, Taiwanese J. Math. 12 (2008), 1045–1059.
• [14] J. Ginibre, Y. Tsutsumi, and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal. 151 (1997), 384–436.
• [15] A. Gruenrock and H. Pecher, Global solutions for the Dirac-Klein-Gordon system in two space dimensions, preprint, arXiv:0903.3189v1 [math.AP]
• [16] J. Holmer, Local ill-posedness of the 1D Zakharov system, Electron. J. Differential Equations 2007, no. 24.
• [17] M. Keel and T. Tao, Local and global well-posedness of wave maps on R1+1 for rough data, Internat. Math. Res. Notices 1998, no. 21, 1117–1156.
• [18] S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math. 46 (1993), 1221–1268.
• [19] S. Machihara, One dimensional Dirac equation with quadratic nonlinearities, Discrete Contin. Dyn. Syst. 13 (2005), 277–290.
• [20] S. Machihara, Dirac equation with certain quadratic nonlinearities in one space dimension, Commun. Contemp. Math. 9 (2007), 421–435.
• [21] S. Machihara, The Cauchy problem for the 1-D Dirac-Klein-Gordon equation, NoDEA Nonlinear Differential Equations Appl. 14 (2007), 625–641.
• [22] F. Melnyk, Local Cauchy problem for the nonlinear Dirac and Dirac-Klein-Gordon equations on Kerr space-time, J. Math. Phys. 47 (2006), no. 052503.
• [23] H. Pecher, Low regularity well-posedness for the one-dimensional Dirac-Klein-Gordon system, Electron. J. Differential Equations 2006, no. 150.
• [24] T. Roy, Adapted linear-nonlinear decomposition and global well-posedness for solutions to the defocusing cubic wave equation on3, Discrete Contin. Dyn. Syst. 24 (2009), 1307–1323.
• [25] S. Selberg, Global well-posedness below the charge norm for the Dirac-Klein-Gordon system in one space dimension, Int. Math. Res. Not. IMRN 2007, no. 17, art. ID rnm058.
• [26] S. Selberg and A. Tesfahun, Low regularity well-posedness of the Dirac-Klein-Gordon equations in one space dimension, Commun. Contemp. Math. 10 (2008), 181–194.