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

Jianchao Bai and Jinxing Liu ()
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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

Mathematics, 2025, vol. 13, issue 19, 1-22

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.

Keywords: nonlinear systems of differential equations; mild solution; solution; existence; uniqueness (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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