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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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