Stability and Localisation in Variational Dissipative Conservative PDE Systems

This deposit contains the paper “Stability and Localisation in Variational Dissipative–Conservative PDE Systems” by Kevin Ellis Anderson. The paper analyses stability, instability, and localisation phenomena in a class of coupled variational partial differential equation systems generated by a single free-energy functional. The framework combines gradient-flow evolution for a dissipative sector with Hamiltonian (unitary) evolution for a conservative sector, allowing irreversible dissipation and oscillatory dynamics to coexist within a unified variational formulation. Linear and nonlinear stability criteria are established for homogeneous and weakly inhomogeneous stationary states in appropriate Sobolev spaces. When the curvature of the free energy admits unstable directions, coupling between the dissipative and conservative sectors can trigger irreversible energy transfer and the formation of spatially localised absorbing regions. These results provide a rigorous mathematical basis for localisation and event-like behaviour in variational dissipative–conservative systems, showing how spatial trapping and stability selection arise dynamically from instability and dissipation rather than from additional postulates.