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Local well-posedness for nonlinear Dirac equation on N-star metric graphs
Abstract
We consider the Cauchy problem for the nonlinear Dirac equation on a noncompact N-star metric graph G, i∂t ψ= Dψ− |ψ|p−2ψ, ψ(0)=ψ0, where p≥3, ψ:ℝ× G→ℂ2 and D denotes the self-adjoint Dirac-Kirchhoff operator on G. Using Bourgain-type spaces defined through the spectral resolution of D, together with elementary L∞ bounds for the Dirac flow and fractional Nemytskii estimates below the trace threshold, we prove local well-posedness for initial data ψ0∈ HDs(G)∩ L∞(G;ℂ2), 0≤ s<1/2 . The corresponding solution belongs to C([0,T];HDs(G))∩ XTs,b∩ L∞([0,T]× G). Moreover, |ψ(t)|L2(G;ℂ2) is conserved along the solution on the existence interval. We also establish a blow-up alternative in the combined HDs and space-time L∞ control norm.
Versions
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BibTeX
@misc{xing2026local,
title = {{Local well-posedness for nonlinear Dirac equation on $N$-star metric graphs}},
author = {Huichao Xing and Zhipeng Yang},
howpublished = {arXiv},
year = {2026},
month = jul,
doi = {10.48550/arxiv.2607.09303},
eprint = {2607.09303},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.09303},
}
RIS
TY - UNPB
TI - Local well-posedness for nonlinear Dirac equation on N-star metric graphs
AU - Huichao Xing
AU - Zhipeng Yang
PY - 2026
DA - 2026/07/10
PB - arXiv
DO - 10.48550/arxiv.2607.09303
UR - https://arxiv.org/abs/2607.09303
AB - We consider the Cauchy problem for the nonlinear Dirac equation on a noncompact N-star metric graph G, i∂_t ψ= Dψ− |ψ|^p−2ψ, ψ(0)=ψ_0, where p≥3, ψ:ℝ× G→ℂ^2 and D denotes the self-adjoint Dirac-Kirchhoff operator on G. Using Bourgain-type spaces defined through the spectral resolution of D, together with elementary L^∞ bounds for the Dirac flow and fractional Nemytskii estimates below the trace threshold, we prove local well-posedness for initial data ψ_0∈ H_D^s(G)∩ L^∞(G;ℂ^2), 0≤ s<1/2 . The corresponding solution belongs to C([0,T];H_D^s(G))∩ X_T^s,b∩ L^∞([0,T]× G). Moreover, |ψ(t)|_L^2(G;ℂ^2) is conserved along the solution on the existence interval. We also establish a blow-up alternative in the combined H_D^s and space-time L^∞ control norm.
ER -