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

Abstract

For $T>0$, we consider the Benjamin--Ono-type Dirac equation $$ i\partial_tψ(x,t) + D(-Δ)^{\frac{1}{2}}ψ(x,t) = D\,G(ψ(x,t)), \qquad (x,t)\in\mathbb{R}^n\times[0,T], $$ where $(-Δ)^{\frac{1}{2}}$ 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 $ψ$ and a uniform dyadic regularity assumption on its spatial Fourier profile, we construct a unit-modulus gauge map $u$ by $ u=\exp\bigl(-iw\bigr),$ where $w$ is a solution of $Dw=ψ$. 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 $P_{\pm}N(ψ,u)$, where $$ N(ψ(x,t),u(x,t)) = -2P_{-}\!\left[ G(ψ(x,t)) \right]u(x,t). $$ We also obtain a priori estimates for the homogeneous Sobolev seminorms $\dot H^σ$ of $ψ(\cdot,t)$, with at most exponential growth on $[0,T]$, together with estimates for $P_{\pm}N(ψ(\cdot,t),u(\cdot,t))$. Combining these bounds with the homogeneous and inhomogeneous Strichartz estimates for the Schrödinger system, we obtain, for $n\geq 3$ and every sharp Schrödinger admissible pair $(q,p)$, an a priori Strichartz-type estimate for $ψ$ in $L_T^q\dot W_x^{σ,p}$, expressed in terms of the initial datum $ψ(\cdot,0)$ and the a priori controls of $ψ$ in $L_T^\infty H_x^{σ+1}$ and $L_T^\infty L_x^\infty$.