Variational framework for morphogenesis from two coupled vector fields: OMP Model

The OMP model is a variational framework for morphogenesis defined on a bounded regular domain $\Omega \subset \mathbb{R}^3$. It couples two vector fields $\vec{A}, \vec{B}$ through a local energy $E_{\text{loc}}(\vec{C}) = (\|\vec{C}\|-1)^2$, $\vec{C} = \alpha\vec{A}+\beta\vec{B}$, and an order parameter $\rho = \sigma(\theta - E_{\text{loc}})$ (sigmoid). A Lyapunov functional $\mathcal{L} = \frac12\int_{\Omega}(\rho-1)^2\,d\Omega$ ensures global stability. For dimensions $\le 3$, the regularized direction field $\vec{R}_\varepsilon = \vec{C}/\sqrt{\|\vec{C}\|^2+\varepsilon}$ is locally Lipschitz in the Sobolev space $H^2(\Omega)$ with Neumann boundary conditions, guaranteeing well-posedness and numerical stability. The model drives the immersive platform $\Phi$Crea for interactive geometric visualization.