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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 \(0< T< T^*\leq \lim\inf_{c\to \infty} T^*_c\).

MSC:
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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