A Priori Uniform Bounds as Measure-Theoretic Tools: Long-Term Analysis via Classical-Enhanced Synthesis

This work presents a systematic study of nonlinear differential equations within Sobolev spaces, focusing on mild solutions and their qualitative properties. An iterative reconstruction method is developed to obtain uniform a priori bounds, which ensure both the existence and tightness of invariant measures. Furthermore, uniqueness of these measures is established under appropriate structural conditions. The results provide a rigorous foundation for analyzing the asymptotic behavior of nonlinear dynamical systems.