Rigorous analytical theory and long-time convergence

Develop a complete rigorous analytical theory for the spontaneous aggregation Fokker–Planck equation with exponential response, including well-posedness, regularity, asymptotic compactness, and a rigorous proof of long-time convergence of solutions to stationary states.

Background

The paper derives the gradient-flow energy dissipation identity and formally outlines a convergence argument based on compactness, a LaSalle-type invariance principle, and a Wasserstein–Łojasiewicz inequality. However, the analysis assumes smooth positive global solutions and the uniform-in-time estimates needed for parabolic regularity and precompactness. A complete treatment would need to establish these properties and rigorously justify convergence, including the trapping and finite-length arguments near stationary omega-limit points.

References

The present work is intended as a short structural note rather than a complete analytical treatment. We therefore emphasize the main ideas and their consequences without developing a full theory of well-posedness, regularity and asymptotic compactness. In particular, the discussion of long-time convergence is formal and indicates the assumptions and arguments that would be required for a rigorous result. A detailed analysis of these questions is left for future work.

— Gradient Flow Structure of the Spontaneous Aggregation Model  (2608.19005 - Haskovec, 19 Aug 2026) in Section 1, Introduction; Section 6, Long-time behavior