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 $G$, \[ \mathrm{i}\partial_t ψ= Dψ- |ψ|^{p-2}ψ, \qquad ψ(0)=ψ_0, \] where $p\ge 3$, $\psi :\mathbb{R}\times G\to {\mathbb{C}}^2$ 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 $L^{\infty }$ bounds for the Dirac flow and fractional Nemytskii estimates below the trace threshold, we prove local well-posedness for initial data \[ ψ_0\in H_D^s(G)\cap L^\infty(G;\mathbb C^2), \qquad 0\le s<\frac12 . \] The corresponding solution belongs to \[ C([0,T];H_D^s(G))\cap X_T^{s,b}\cap L^\infty([0,T]\times G). \] Moreover, $\Vert \psi (t){\Vert }_{L^2(G;{\mathbb{C}}^2)}$ 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 $L^{\infty }$ control norm.

20 pages, comments are welcome