Decay estimates for massive Dirac equation in a constant magnetic field

Abstract

We study the deacy and Strichartz estimates for the massive Dirac Hamiltonian in a constant magnetic fields in ${\mathbb{R}}_t\times {\mathbb{R}}_x^2$: \begin{equation*} \begin{cases} i\partial_tu(t,x)-\mathcal{D}_Au(t,x)=0, u(0,x)=f, \end{cases} \end{equation*} where ${\mathcal{D}}_A=-i\sigma \cdot (\nabla -i\mathbf{A}(x))+{\sigma }_{3}m$ with $m\ge 0$ being the mass and ${\sigma }_i$ being the Dirac matrices and the potential $\mathbf{A}(x)=\frac{B_0}{2}(-x_2,x_1),B_0>0$. In particular, we show the $L^1({\mathbb{R}}^2)\to L^{\infty }({\mathbb{R}}^2)$ type micro-localized decay estimates, for any finite time $T>0$, there exists a constant $C_T$ such that \begin{equation*} \|e^{it\mathcal{D}_{A}}φ(2^{-j}|\mathcal{D}_{A}|)f(x)\|_{[L^{\infty}(\mathbb{R}^2)]^2} \leq C_T 2^{2j}(1+2^{j}|t|)^{-\frac12} \|φ(2^{-j}|\mathcal{D}_{A}|)f\|_{[L^1{(\mathbb{R}^2)]^2}}, \quad |t|\leq T, \end{equation*} and we further prove the local-in-time Strichartz estimates for the Dirac equations with this unbounded potential.

18 pages, 0 figures