Decay estimates for massive Dirac equation in a constant magnetic field

Abstract

We study the decay and Strichartz estimates for the massive Dirac Hamiltonian in a constant magnetic fields in $R_t\times R_x^2$:$\{\begin{array}{c}i{\partial }_{t}u(t,x)-D_{A}u(t,x)=0,\\ u(0,x)=f,\end{array}$ where $D_A=-i\sigma \cdot (\nabla -iA(x\left)\right)+{\sigma }_{3}m$ with $m\ge 0$ being the mass and ${\sigma }_i$ being the Dirac matrices and the potential $A\left(x\right)=\frac{B_0}{2}(-x_2,x_1),B_0>0$. In particular, we show the $L^1\left(R^2\right)\to L^{\infty }\left(R^2\right)$ type micro-localized decay estimates, for any finite time $T>0$, there exists a constant $C_T$ such that${\Vert e^{it{D}_A}\phi \left(2^{-j}\right|D_A\left|\right)f\left(x\right)\Vert }_{{\left[L^{\infty }\right(R^2\left)\right]}^2}\le C_T{2}^{2j}{(1+2^j|t\left|\right)}^{-\frac{1}{2}}{\Vert \phi \left(2^{-j}\right|D_A\left|\right)f\Vert }_{{\left[L^1\right(R^2\left)\right]}^2},\left|t\right|\le T,$ and we further prove the local-in-time Strichartz estimates for the Dirac equations with this unbounded potential.