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

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  • Jianchao Bai

    (School of Mathematics and Physics, Yibin University, Yibin 644000, China)

  • Jinxing Liu

    (School of Mathematics and Physics, Yibin University, Yibin 644000, China)

Abstract

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.

Suggested Citation

  • Jianchao Bai & Jinxing Liu, 2025. "A Priori Uniform Bounds as Measure-Theoretic Tools: Long-Term Analysis via Classical-Enhanced Synthesis," Mathematics, MDPI, vol. 13(19), pages 1-22, September.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:19:p:3095-:d:1759030
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