magazinelogo

Journal of Applied Mathematics and Computation

ISSN Online: 2576-0653 Downloads: 177307 Total View: 1999022
Frequency: quarterly ISSN Print: 2576-0645 CODEN: JAMCEZ
Email: jamc@hillpublisher.com
Article Open Access http://dx.doi.org/10.26855/jamc.2023.03.005

Initial Boundary Value Problem for Maxwell-Dirac System in the Half Line

Fengxia Liu1, Yitong Pei2,*, Boling Guo3

1Institute of Artificial Intelligence, Beihang University, Beijing, China.

2Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing, China.

3Institute of Applied Physics and Computational Mathematics, Beijing, China.

*Corresponding author: Yitong Pei

Published: March 8,2023

Abstract

We consider the inhomogeneous Initial-Boundary value problem (IBVP) for the one dimensional Maxwell-Dirac system, which is a difficult and meaningful coupled equation to study. First, from Laplace transform and semigroup theory, we obtain the local well-posedness (LWP) of the system by contraction map, in addition, we give that the bound is independent of time T, and hence obtain the global well-posedness (GWP) with small solutions of the system. Finally, we obtain a sharp estimate in the long time asymptotics result.

References

[1] W. E. Thirring, A soluble relativistic fifield theory, Annals of Physics, (1), (1997), 91-112. 

[2] S. Selberg and A. Tesfahun, Low regularity well-posedness for some nonlinear Dirac equations in one space dimension, Differential Integral Equations, 23, 3-4(2010), 265-278. 

[3] V. Delgado, Global solutions of the Cauchy problem for the (classical) coupled Maxwell Dirac and other nonlinear Dirac equations in one space dimension [J].  Proc. Amer.  Math.  Soc. , 69(1978), 289-296. 

[4] M. Escobedo and L. Vega, A semilinear Dirac equation in Hs (R3) for s > 1, SIAM J. Math.  Anal. , 28(1997), 338- 362. 

[5] S. Machihara, Dirac equation with certain quadratic nonlinearities in one space dimension, Commun.  Contemp.  Math. , 9(3), (2007), 421-435. 

[6] H. Pecher, Local well-posedness for the nonlinear Dirac equation in two space dimensions, Commun.  Pure Appl.  Anal. , 13(2), (2014), 673-685. 

[7] L. Gross, The Cauchy problem for the coupled Maxwell and Dirac equations, Commun.  Pure Appl.  Math. 19(1966), 1-5. 

[8] T. Kato, Integration of the equation of evolution in a Banach space, J. AZ and z. Sot.  Japan, 5(1953), 208-234. 

[9] I. E. Segal, Non- linear semi-groups, Ann.  Math., 78(1963), 339-364. 

[10] J. M. Chadam, Global solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac equations in one space dimension, J. Funct.  Anal, 13(2), (1973), 173-184. 

[11] J. M. Chadam, On the Cauchy problem for the coupled Maxwell-Dirac equations, J. Math.  Phys., 13(1972), 597-604. 

[12] Y. Choquet-Bruhat, Solutions globales des equations de Maxwell-Dirac-Klein- Gordon (masses nuUes), C.R. Acad. Sci.  Paris 292, Ser.  I, (1981), 153-158. 

[13] M. Flato and G. Pinczon, J. Simon, Non- linear representations of Lie groups, Ann.  Sci.  Ec.  Norm.  Super. , 10(1977), 405-448. 

[14] J. L. Bona, S. M. Sun, B.-Y.  Zhang.  Forced Oscillations of A Damped Korteweg-de Vries Equations in A Quarter Plane.  Comm. Cont. Math. , 5(3): 369C400, 2003. 

[15] J. L. Bona, S. M. Sun, B- Y. Zhang.  Boundary Smoothing Properties of the Korteweg-de Vries Equation in A Quarter Plane and Applications, Dyn. PDE, 3: 1C69, 2006. 

[16] J. Holmer.  The Initial-Boundary Value Problem for the Korteweg-de Vries Equation.  Comm. Par. Differ.  Equa. , 31(8): 1151-1190, 2006. 

[17] I. P. Naumkin, Initial-boundary value problem for the one dimensional Thirring model, J. Differ.  Equa, 261(8), (2016), 4486-4523. 

[18] I.P. Naumkin, Cubic nonlinear Dirac equation in a quarter plane, J. Math.  Anal.  Appl. 434(2), (2016), 1633-1664. 

[19] Y. Cho and T. Ozawa, On the semirelativistic Hartree-type equation, SIAM J. Math.  Anal. , 38(4), (2006), 1060-1074. 

[20] T. Kato, On nonlinear Schr¨odinger equations II.  Hs-solutions and unconditional wellposedness, J. Anal.  Math. , 67(1995), 281-306.

How to cite this paper

Initial Boundary Value Problem for Maxwell-Dirac System in the Half Line

How to cite this paper:  Fengxia Liu, Yitong Pei, Boling Guo. (2023) Initial Boundary Value Problem for Maxwell-Dirac System in the Half Line. Journal of Applied Mathematics and Computation7(1), 28-52.

DOI: http://dx.doi.org/10.26855/jamc.2023.03.005