Gauge Transforms and Strichartz Estimates for Benjamin--Ono-Type Dirac Equations
Abstract
For T>0, we consider the Benjamin--Ono-type Dirac equation i∂tψ(x,t) + D(−Δ)1/2ψ(x,t) = D G(ψ(x,t)), (x,t)∈ℝn×[0,T], where (−Δ)1/2 is the fractional Riesz operator, D is the Dirac operator associated with the Clifford algebra of signature (0,n), and G(·) 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±=P±u solve a coupled Schrödinger system with gauge nonlinearities P±N(ψ,u), where N(ψ(x,t),u(x,t)) = −2P− [ G(ψ(x,t)) ]u(x,t). We also obtain a priori estimates for the homogeneous Sobolev seminorms Ḣσ of ψ(·,t), with at most exponential growth on [0,T], together with estimates for P±N(ψ(·,t),u(·,t)). Combining these bounds with the homogeneous and inhomogeneous Strichartz estimates for the Schrödinger system, we obtain, for n≥ 3 and every sharp Schrödinger admissible pair (q,p), an a priori Strichartz-type estimate for ψ in LTqẆxσ,p, expressed in terms of the initial datum ψ(·,0) and the a priori controls of ψ in LT∞ Hxσ+1 and LT∞ Lx∞.
Related
- Strichartz estimates to fractional Schrödinger equations
- Strichartz estimates for Klein-Gordon equations on asymptotically hyperbolic manifolds with applications
- Orthonormal Strichartz estimates for the Schrödinger equations with Hamiltonian on Wiener amalgam spaces
- Strichartz estimates for Schrödinger equations with nonlinear boundary interactions
- Bilinear Strichartz Estimates and Linear Morawetz Estimates for Schrödinger Operators with Potentials
BibTeX
@misc{faustino2026gauge,
title = {{Gauge Transforms and Strichartz Estimates for Benjamin--Ono-Type Dirac Equations}},
author = {Nelson Faustino},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16842},
eprint = {2610.16842},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16842},
}
RIS
TY - UNPB
TI - Gauge Transforms and Strichartz Estimates for Benjamin--Ono-Type Dirac Equations
AU - Nelson Faustino
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16842
UR - https://arxiv.org/abs/2610.16842
AB - For T>0, we consider the Benjamin--Ono-type Dirac equation i∂_tψ(x,t) + D(−Δ)^1/2ψ(x,t) = D G(ψ(x,t)), (x,t)∈ℝ^n×[0,T], where (−Δ)^1/2 is the fractional Riesz operator, D is the Dirac operator associated with the Clifford algebra of signature (0,n), and G(·) 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_±=P_±u solve a coupled Schrödinger system with gauge nonlinearities P_±N(ψ,u), where N(ψ(x,t),u(x,t)) = −2P_− [ G(ψ(x,t)) ]u(x,t). We also obtain a priori estimates for the homogeneous Sobolev seminorms Ḣ^σ of ψ(·,t), with at most exponential growth on [0,T], together with estimates for P_±N(ψ(·,t),u(·,t)). Combining these bounds with the homogeneous and inhomogeneous Strichartz estimates for the Schrödinger system, we obtain, for n≥ 3 and every sharp Schrödinger admissible pair (q,p), an a priori Strichartz-type estimate for ψ in L_T^qẆ_x^σ,p, expressed in terms of the initial datum ψ(·,0) and the a priori controls of ψ in L_T^∞ H_x^σ+1 and L_T^∞ L_x^∞.
ER -