Gauge Transforms and Strichartz Estimates for Benjamin--Ono-Type Dirac Equations

Abstract

For $T>0$, we consider the Benjamin--Ono-type Dirac equation $$i{\partial }_t\psi (x,t)+D(-\Delta {)}^{\frac{1}{2}}\psi (x,t)=DG(\psi (x,t)),(x,t)\in {\mathbb{R}}^n\times [0,T],$$ where $(-\Delta {)}^{\frac{1}{2}}$ is the fractional Riesz operator, $D$ is the Dirac operator associated with the Clifford algebra of signature $(0,n)$, and $G(\cdot )$ a scalar-valued quadratic nonlinearity. Under a pointwise anti-commutation condition on $\psi$ and a uniform dyadic regularity assumption on its spatial Fourier profile, we construct a unit-modulus gauge map $u$ by $u=\exp (-iw),$ where $w$ is a solution of $Dw=\psi$. Assuming, in addition, the stated regularity and commutation conditions on $w$, we show that the Plemelj components $u_{\pm }=P_{\pm }u$ solve a coupled Schrödinger system with gauge nonlinearities $P_{\pm }N(\psi ,u)$, where $$N(\psi (x,t),u(x,t))=-2P_{-}\left[G(\psi (x,t))\right]u(x,t).$$ We also obtain a priori estimates for the homogeneous Sobolev seminorms ${\dot{H}}^{\sigma }$ of $\psi (\cdot ,t)$, with at most exponential growth on $[0,T]$, together with estimates for $P_{\pm }N(\psi (\cdot ,t),u(\cdot ,t))$. Combining these bounds with the homogeneous and inhomogeneous Strichartz estimates for the Schrödinger system, we obtain, for $n\ge 3$ and every sharp Schrödinger admissible pair $(q,p)$, an a priori Strichartz-type estimate for $\psi$ in $L_T^q{\dot{W}}_x^{\sigma ,p}$, expressed in terms of the initial datum $\psi (\cdot ,0)$ and the a priori controls of $\psi$ in $L_T^{\infty }{H}_x^{\sigma +1}$ and $L_T^{\infty }{L}_x^{\infty }$.

arXiv comment29 pages