Trend to equilibrium for granular media equations under non-convex potential and application to log gases

2020/11/17 by Scander Mustapha, Mustapha, Scander
Environmental Science · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Groundwater flow and contamination studies #Nonlinear Partial Differential Equations #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2011.08547

openalex publication_date 2020/11/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We derive new HWI inequalities for the granular media equation, which external potential V and interaction potential W are only strictly convex on complementary parts of the space. Particularly, potentials are not assumed convex. After solving technicalities related to the singularity of a logarithmic W, we apply our result to obtain stability rates of log gases under non-strictly convex or quartic external potentials. We prove that the distribution of a log gas converges towards an equilibrium with respect to the Wasserstein distance at a square root rate. Finally, we establish exponential stability of log gases under the double-well potential V(x) = (x⁴)/(4) + c(x²)/(2), c < 0 and the non-confining potential V(x) = g(x⁴)/(4) + (x²)/(2), g<0 for |c| and |g| small enough.

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