Well Posedness of Variational Dissipative Conservative Partial Differential Equation Systems

This deposit contains the paper “Well-Posedness of Variational Dissipative–Conservative Partial Differential Equation Systems” by Kevin Ellis Anderson. The paper establishes rigorous well-posedness results for a class of coupled variational partial differential equation systems generated by a single free-energy functional. The framework combines gradient-flow evolution for a real scalar field with Hamiltonian (unitary) evolution for a complex-valued field, capturing the coexistence of irreversible dissipation and conservative dynamics within a unified variational structure. Under standard regularity and growth assumptions on the free-energy density, the paper proves local and global existence, uniqueness, and continuous dependence on initial data in appropriate Sobolev spaces. A strict Lyapunov functional provides uniform energy bounds and ensures global control of solutions. No physical interpretation or phenomenological assumptions are required for the analysis. The results supply the mathematical foundation needed to justify subsequent analytical, numerical, and regime-based studies of variational dissipative–conservative systems, including those exhibiting quantum-analogue, decoherent, and classical behaviour as different dynamical limits.