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
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[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
[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
[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
[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
[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
[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
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