An analytical study of the static state of multi-junctions in a multi-phase field model (Q629023)

From the mathematical point of view, the paper deals with the system of ordinary differential equations \[ \dot{\phi}_{\alpha}={{1}\over{\tilde{N}}}\left(-\sum_{\beta=1}^{\tilde{N}} \mu_{\alpha\beta}\dot{\psi}_{1\alpha} + \sum_{\beta=1}^{\tilde{N}}\mu_{\alpha\beta} \dot{\psi}_{1\beta}\right),\quad \alpha=1,\dots,\tilde{N} \] which has the first integral \[ \sum_{\alpha=1}^{\tilde{N}}\dot{\phi}_{\alpha}=0. \] Consequently, the order of the system is lowered by 1, obtaining the vector matrix form \[ \dot{\Phi} = {{1}\over{\tilde{N}}}\;A\dot{\Psi} \] based on \(\mu_{\alpha\alpha}=0\), \(\mu_{\alpha\beta}=\mu_{\beta\alpha}\), \(\dot{\psi}_{\alpha\alpha}=0\), \(\dot{\psi}_{\alpha\beta}=\dot{\psi}_{\beta\alpha}\). Using Gershgorin's theorem, it is proved that \(A\) is invertible. From there, rigorous conclusions that validate experimental results showing the existence of equilibria are obtained.