Lyapunov Structure and Long Time Behaviour of Variational Dissipative Conservative PDE Systems

This deposit contains the paper “Lyapunov Structure and Long-Time Behaviour of Variational Dissipative–Conservative PDE Systems” by Kevin Ellis Anderson. The paper analyses the Lyapunov structure and asymptotic dynamics of 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 conservative motion to coexist within a unified variational formulation. Exploiting the strict monotonic decay of free energy, the paper establishes uniform boundedness of trajectories, characterises invariant sets via LaSalle-type arguments, and describes convergence properties in the long-time limit. Dissipation is shown to act as a dynamical selector, constraining asymptotic behaviour without modifying the conservative evolution. These results provide a rigorous description of relaxation, stability, and long-time behaviour in variational dissipative–conservative systems, completing the mathematical characterisation required to support further analytical, numerical, and regime-based investigations.