zbMATH — the first resource for mathematics

Small global solutions and the nonrelativistic limit for the nonlinear Dirac equation. (English) Zbl 1041.35061
The authors study the Cauchy problem for the Dirac equation (NLD): \[ i\partial_t\psi+ ic\alpha\nabla \psi- c^2\beta\psi+ 2\lambda\;(\beta\psi\mid\psi)\;\beta\psi= 0,\quad \psi(0)= \psi_0. \] Because of the cubic nonlinearity, they construct the \(L^2_t\) type Strichartz estimate for Dirac and Klein-Gordon equations. From estimates related to \[ K_\pm(t)= \exp(\pm it(1- \Delta)^{1/2}) \] derive Theorem 1. Let \(\psi_0\in H^s\), \(s> 1\), and let \(\| \psi_0\|_{H^s}\) be sufficiently small.
There exists a unique solution \(\psi: \mathbb R^4\to \mathbb C^4\) such that \(\psi\in C(\mathbb R; H^s)\cap L^2(\mathbb R; B^{s-\sigma}_r)\cap L^\infty(\mathbb R; H^s)\). Moreover, there exists a unique \(\psi_\pm\in H^s\) such that \[ \lim_{t\to\pm\infty} \|\psi(t)- U(t)\psi_\pm \|_{H^s}= 0, \] \(U(t)\psi_\pm\): the solution of the free Dirac equation. Substitution by \(u_c= 2\exp(it \beta c^2)\beta\psi\) yields the equation mNLD: \[ \partial_t u_c- c\cdot\exp(2ic^2 t\beta) \alpha\nabla u_c= -(i\lambda/2)(\beta u_c\mid u_c)\beta u_c, \] \(u_c(0)= \phi_c\). The authors also consider the NLS: \[ \partial_t v-(i\beta/2) \Delta v= -(i\lambda/2)(\beta v\mid v) \beta v, \] \(v(0)= \phi_\infty\), as \(c\to \infty\). They give estimates related to \[ V_\pm(t)= \exp(\pm it(c^4- c^2\Delta)^{1/2}), \] and prove Theorem 2. Let \(\phi_c\), \(\phi_\infty\in H^s\), \(s> 1\), and \(T^*_c\), \(T^*\): the existence times of \(u_c\), \(v\). If \(\lim_{c\to\infty} \|\phi_c- \phi_\infty\|= 0\) then \(\lim_{c\to \infty}\| u_c- v\|_{L^\infty(0,T; H_s)}= 0\) holds for \(\forall T\) with \(0< T< T^*\leq \lim\inf_{c\to \infty} T^*_c\).

35Q40 PDEs in connection with quantum mechanics
35Q55 NLS equations (nonlinear Schrödinger equations)
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Full Text: DOI EuDML
[1] Bergh, J. and Löfström, J.: Interpolation spaces. Springer, Berlin- Heiderberg-New York, 1976. · Zbl 0344.46071
[2] Brenner, P.: On scattering and everywhere defined scattering operators for nonlinear Klein-Gordon equations. J. Differential Equations 56 (1985), 310-344. · Zbl 0513.35066 · doi:10.1016/0022-0396(85)90083-X
[3] Cazenave, T. and Weissler, F. B.: The Cauchy problem for the critical nonlinear Schrödinger equations in Hs. Nonlinear Anal. 14 (1990), 807-836. · Zbl 0706.35127 · doi:10.1016/0362-546X(90)90023-A
[4] Dias, J. P. and Figueira, M.: Global existence of solutions with small initial data in Hs for the massive nonlinear Dirac equations in three space dimensions. Boll. Un. Mat. Ital. B(7) 1 (1987), 861-874. 193 · Zbl 0637.35014
[5] Escobedo, M. and Vega, L.: A semilinear Dirac equation in Hs(R3) for s &gt; 1. SIAM J. Math. Anal. 2 (1997), 338-362. · Zbl 0877.35028 · doi:10.1137/S0036141095283017
[6] Finkelstein, R., Lelevier, R. and Ruderman, M.: Nonlinear spinor fields. Phys. Rev. 83 (1951), 326-332. · Zbl 0043.21603 · doi:10.1103/PhysRev.83.326
[7] Ginibre, J. and Velo, G.: Time decay of finite energy solutions of the non linear Klein-Gordon and Schrödinger equations. Ann. Inst. H. Poincaré Phys. Théor. 43 (1985), 399-442. · Zbl 0595.35089 · numdam:AIHPA_1985__43_4_399_0 · eudml:76307
[8] Ginibre, J. and Velo, G.: Smoothing properties and retarded estimates for some dispersive evolution equations. Commun. Math. Phys. 144 (1992), 163-188. · Zbl 0762.35008 · doi:10.1007/BF02099195
[9] Ginibre, J. and Velo, G.: Generalized Strichartz inequalities for the wave equation. J. Funct. Anal. 133 (1995), 50-68. · Zbl 0849.35064 · doi:10.1006/jfan.1995.1119
[10] Keel, M. and Tao, T.: Endpoint Strichartz estimates. Amer. J. Math. 120 (1998), 955-980. · Zbl 0922.35028 · doi:10.1353/ajm.1998.0039 · muse.jhu.edu
[11] Lindblad, H.: A sharp counterexample to local existence of low regularity solutions to nonlinear wave equations. Duke Math. J. 72 (1993), 503-539. · Zbl 0797.35123 · doi:10.1215/S0012-7094-93-07219-5
[12] Lindblad, H.: Counterexamples to local existence for semi-linear wave equations. Amer. J. Math. 118 (1996), 1-16. · Zbl 0855.35080 · doi:10.1353/ajm.1996.0002 · muse.jhu.edu
[13] Lindblad, H. and Sogge, C. D.: On existence and scattering with min- imal regularity for semilinear wave equations. J. Funct. Anal. 130 (1995), 357-426. · Zbl 0846.35085 · doi:10.1006/jfan.1995.1075
[14] Machihara, S., Nakanishi, K. and Ozawa, T.: Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations. Math. Ann. 322 (2002), 603-621. · Zbl 0991.35080 · doi:10.1007/s002080200008
[15] Matsuyama, T.: A remark on the nonrelativistic limit for semilinear Dirac equations. Nonlinear Anal. 25 (1995), 1139-1146. · Zbl 0846.35113 · doi:10.1016/0362-546X(94)00235-A
[16] Matsuyama, T.: Rapidly decreasing solutions and nonrelativistic limit of semilinear Dirac equations. Rev. Math. Phys. 7 (1995), 243-267. · Zbl 0841.35098 · doi:10.1142/S0129055X95000128
[17] Najman, B.: The nonrelativistic limit of the nonlinear Klein-Gordon equa- tion. Nonlinear Anal. 15 (1990), 217-228. · Zbl 0727.35122 · doi:10.1016/0362-546X(90)90158-D
[18] Najman, B.: The nonrelativistic limit of the Klein-Gordon and Dirac equations. In Differential Equations with Applications in Biology, Physics and Engineering, J. Goldstein, F. Kappel, W. Schappacher (Eds.), Lect. Notes Pure Appl. Math. 133 (1991), 291-299, Marcel Dekker. · Zbl 0744.35038
[19] Najman, B.: The nonrelativistic limit of the nonlinear Dirac equation. Ann. Inst. H. Poincaré, Anal. Non linéaire 9 (1992), 3-12. · Zbl 0746.35036 · numdam:AIHPC_1992__9_1_3_0 · eudml:78271
[20] Pecher, H.: Nonlinear small data scattering for the wave and Klein- Gordon equation. Math. Z. 185 (1984), 261-270. · Zbl 0538.35063 · doi:10.1007/BF01181697 · eudml:173400
[21] Ponce, G. and Sideris, T. C.: Local regularity of nonlinear wave equa- tions in three space dimensions. Comm. Partial Differential Equations 18 (1993), 169-177. · Zbl 0803.35096 · doi:10.1080/03605309308820925
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.