Low-regularity well-posedness for a mixed-sign quadratic Dirac equation on $N$-star metric graphs

Abstract

We study the Cauchy problem for a mixed-sign quadratic Dirac equation on a noncompact $N$-star metric graph $G$, \[ \mathrm{i}\partial_t ψ= Dψ- \mathcal N(ψ), \qquad ψ(0)=ψ_0, \] where $\ddot{I}ˆ=(\ddot{I}{ˆ}_1,\ddot{I}{ˆ}_2{)}^{\mathsf{T}}:\mathbb{R}\times G\to {\mathbb{C}}^2$ and $D$ denotes the self-adjoint Dirac-Kirchhoff operator on $G$. The nonlinearity acts edgewise and is given by a bilinear interaction between the positive and negative spectral parts, \[ \mathcal N(ψ)=\mathcal B\bigl(Î _+ψ,Î _-ψ\bigr), \] where ${\hat{I}}_{\pm }$ are the spectral projections of $D$ and $\mathcal{B}$ is a fixed bilinear map on ${\mathbb{C}}^2$ applied componentwise on each edge. This is a model quadratic interaction tailored to the mixed-sign Bourgain-space mechanism, rather than a general nonlinear Dirac equation on graphs. Using Bourgain-type spaces associated with the spectral resolution of $D$ and a mixed-sign bilinear estimate on $N$-star graphs, we prove local well-posedness in the operator Sobolev space $H_D^s(G)$ for \(s>-\frac18\). We also establish a blow-up alternative in $H_D^s(G)$ for the maximal forward lifespan.

19 pages, comments are welcome