American Institute of Mathematical Sciences

Advanced
March  2014, 13(2): 673-685. doi: 10.3934/cpaa.2014.13.673

Local well-posedness for the nonlinear Dirac equation in two space dimensions

Hartmut Pecher 1,

1. 

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

Received  March 2013 Revised  August 2013 Published  October 2013

The Cauchy problem for the cubic nonlinear Dirac equation in two space dimensions is locally well-posed for data in $H^s$ for $ s > 1/2$. The proof given in spaces of Bourgain-Klainerman-Machedon type relies on the null structure of the nonlinearity as used by d'Ancona-Foschi-Selberg for the Dirac-Klein-Gordon system before and bilinear Strichartz type estimates for the wave equation by Selberg and Foschi-Klainerman.
Keywords: Dirac equation, well-posedness, Fourier restriction norm method..
Mathematics Subject Classification: 35Q55, 35L70.
Citation: Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure & Applied Analysis, 2014, 13 (2) : 673-685. doi: 10.3934/cpaa.2014.13.673
References:
[1]

P. d'Ancona, D. Foschi and S. Selberg, Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system,, Journal of the EMS, 9 (2007), 877.   Google Scholar

[2]

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,, Journal Hyperbolic Diff. Equations, 4 (2007), 295.   Google Scholar

[3]

T. Candy, Global existence for an $L^2$ critical nonlinear Dirac equation in one dimension,, Adv. Differential Equations, 16 (2011), 643.   Google Scholar

[4]

V. Delgado, Global solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac and other nonlinear Dirac equations in one space dimension,, Proc. AMS, 69 (1978), 289.   Google Scholar

[5]

M. Escobedo and L. Vega, A semilinear Dirac equation in $H^s(R^3)$ for $s>1$,, SIAM J. Math. Anal., 28 (1997), 338.   Google Scholar

[6]

R. Finkelstein, C. Fronsdal and P. Kaus, Nonlinear spinor field,, Phys. Rev., 103 (1956), 1571.   Google Scholar

[7]

R. Finkelstein, R. LeLevier and M. Ruderman, Nonlinear spinor fields,, Phys. Rev., 83 (1951), 326.   Google Scholar

[8]

D. Foschi and S. Klainerman, Homogeneous $L^2$ bilinear estimates for wave equations,, Ann. Scient. ENS $4^e$ serie, 23 (2000), 211.   Google Scholar

[9]

J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation,, J. Funct. Analysis, 133 (1995), 50.   Google Scholar

[10]

D. Gross and A. Neveu, Dynamical symmetry breaking in asymptotically free field theories,, Phys. Rev. D, 10 (1974), 3235.   Google Scholar

[11]

A. Grünrock and H. Pecher, Global solutions for the Dirac-Klein-Gordon system in two space dimensions,, Comm. Partial Differential Equations, 35 (2010), 89.   Google Scholar

[12]

S. Machihara, M. Nakamura, K. Nakanishi and T. Ozawa, Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation,, J. Funct. Anal., 219 (2005), 1.   Google Scholar

[13]

S. Machihara, K. Nakanishi and T. Ozawa, Small global solutions and the relativistic limit for the nonlinear Dirac equation,, Rev. Math. Iberoamericana, 19 (2003), 179.   Google Scholar

[14]

S. Selberg, "Multilinear Spacetime Estimates and Applications to Local Existence Theory for Nonlinear Wave Equations,", Ph.D. thesis, (1999).   Google Scholar

[15]

S. Selberg and A. Tesfahun, Low regularity well-posedness for some nonlinear Dirac equations in one space dimension,, Diff. Int. Equ., 23 (2010), 265.   Google Scholar

[16]

M. Soler, Classical, stable, nonlinear spinor field with positive rest energy,, Phys. Rev. D, 1 (1970), 2766.   Google Scholar

[17]

W.E. Thirring, A soluble relativistic field theory,, Ann. Physics, 3 (1958), 91.   Google Scholar

show all references

References:
[1]

P. d'Ancona, D. Foschi and S. Selberg, Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system,, Journal of the EMS, 9 (2007), 877.   Google Scholar

[2]

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,, Journal Hyperbolic Diff. Equations, 4 (2007), 295.   Google Scholar

[3]

T. Candy, Global existence for an $L^2$ critical nonlinear Dirac equation in one dimension,, Adv. Differential Equations, 16 (2011), 643.   Google Scholar

[4]

V. Delgado, Global solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac and other nonlinear Dirac equations in one space dimension,, Proc. AMS, 69 (1978), 289.   Google Scholar

[5]

M. Escobedo and L. Vega, A semilinear Dirac equation in $H^s(R^3)$ for $s>1$,, SIAM J. Math. Anal., 28 (1997), 338.   Google Scholar

[6]

R. Finkelstein, C. Fronsdal and P. Kaus, Nonlinear spinor field,, Phys. Rev., 103 (1956), 1571.   Google Scholar

[7]

R. Finkelstein, R. LeLevier and M. Ruderman, Nonlinear spinor fields,, Phys. Rev., 83 (1951), 326.   Google Scholar

[8]

D. Foschi and S. Klainerman, Homogeneous $L^2$ bilinear estimates for wave equations,, Ann. Scient. ENS $4^e$ serie, 23 (2000), 211.   Google Scholar

[9]

J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation,, J. Funct. Analysis, 133 (1995), 50.   Google Scholar

[10]

D. Gross and A. Neveu, Dynamical symmetry breaking in asymptotically free field theories,, Phys. Rev. D, 10 (1974), 3235.   Google Scholar

[11]

A. Grünrock and H. Pecher, Global solutions for the Dirac-Klein-Gordon system in two space dimensions,, Comm. Partial Differential Equations, 35 (2010), 89.   Google Scholar

[12]

S. Machihara, M. Nakamura, K. Nakanishi and T. Ozawa, Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation,, J. Funct. Anal., 219 (2005), 1.   Google Scholar

[13]

S. Machihara, K. Nakanishi and T. Ozawa, Small global solutions and the relativistic limit for the nonlinear Dirac equation,, Rev. Math. Iberoamericana, 19 (2003), 179.   Google Scholar

[14]

S. Selberg, "Multilinear Spacetime Estimates and Applications to Local Existence Theory for Nonlinear Wave Equations,", Ph.D. thesis, (1999).   Google Scholar

[15]

S. Selberg and A. Tesfahun, Low regularity well-posedness for some nonlinear Dirac equations in one space dimension,, Diff. Int. Equ., 23 (2010), 265.   Google Scholar

[16]

M. Soler, Classical, stable, nonlinear spinor field with positive rest energy,, Phys. Rev. D, 1 (1970), 2766.   Google Scholar

[17]

W.E. Thirring, A soluble relativistic field theory,, Ann. Physics, 3 (1958), 91.   Google Scholar

[1]

Axel Grünrock, Sebastian Herr. The Fourier restriction norm method for the Zakharov-Kuznetsov equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2061-2068. doi: 10.3934/dcds.2014.34.2061
[2]

Nikolaos Bournaveas. Local well-posedness for a nonlinear dirac equation in spaces of almost critical dimension. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 605-616. doi: 10.3934/dcds.2008.20.605
[3]

Hartmut Pecher. Corrigendum of "Local well-posedness for the nonlinear Dirac equation in two space dimensions". Communications on Pure & Applied Analysis, 2015, 14 (2) : 737-742. doi: 10.3934/cpaa.2015.14.737
[4]

M. Keel, Tristan Roy, Terence Tao. Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 573-621. doi: 10.3934/dcds.2011.30.573
[5]

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
[6]

Jerry Bona, Nikolay Tzvetkov. Sharp well-posedness results for the BBM equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (4) : 1241-1252. doi: 10.3934/dcds.2009.23.1241
[7]

Nils Strunk. Well-posedness for the supercritical gKdV equation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 527-542. doi: 10.3934/cpaa.2014.13.527
[8]

A. Alexandrou Himonas, Curtis Holliman. On well-posedness of the Degasperis-Procesi equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 469-488. doi: 10.3934/dcds.2011.31.469
[9]

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
[10]

Barbara Kaltenbacher, Irena Lasiecka. Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions. Conference Publications, 2011, 2011 (Special) : 763-773. doi: 10.3934/proc.2011.2011.763
[11]

Takamori Kato. Global well-posedness for the Kawahara equation with low regularity. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1321-1339. doi: 10.3934/cpaa.2013.12.1321
[12]

Ricardo A. Pastrán, Oscar G. Riaño. Sharp well-posedness for the Chen-Lee equation. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2179-2202. doi: 10.3934/cpaa.2016033
[13]

Zhaohui Huo, Boling Guo. The well-posedness of Cauchy problem for the generalized nonlinear dispersive equation. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 387-402. doi: 10.3934/dcds.2005.12.387
[14]

Zhaoyang Yin. Well-posedness, blowup, and global existence for an integrable shallow water equation. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 393-411. doi: 10.3934/dcds.2004.11.393
[15]

Jae Min Lee, Stephen C. Preston. Local well-posedness of the Camassa-Holm equation on the real line. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3285-3299. doi: 10.3934/dcds.2017139
[16]

Hideo Takaoka. Global well-posedness for the Kadomtsev-Petviashvili II equation. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 483-499. doi: 10.3934/dcds.2000.6.483
[17]

Jerry L. Bona, Hongqiu Chen, Chun-Hsiung Hsia. Well-posedness for the BBM-equation in a quarter plane. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1149-1163. doi: 10.3934/dcdss.2014.7.1149
[18]

Didier Pilod. Sharp well-posedness results for the Kuramoto-Velarde equation. Communications on Pure & Applied Analysis, 2008, 7 (4) : 867-881. doi: 10.3934/cpaa.2008.7.867
[19]

Yongye Zhao, Yongsheng Li, Wei Yan. Local Well-posedness and Persistence Property for the Generalized Novikov Equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 803-820. doi: 10.3934/dcds.2014.34.803
[20]

Hongmei Cao, Hao-Guang Li, Chao-Jiang Xu, Jiang Xu. Well-posedness of Cauchy problem for Landau equation in critical Besov space. Kinetic & Related Models, 2019, 12 (4) : 829-884. doi: 10.3934/krm.2019032

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (22)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences