论文标题
在非凸电势和记录气体的应用下，颗粒状培养基方程平衡的趋势
Trend to equilibrium for granular media equations under non-convex potential and application to log gases
论文作者
论文摘要
我们为颗粒状媒体方程式提供了新的HWI不平等，外部潜在$ V $和互动潜在的$ W $仅在空间的互补部分中严格凸出。特别是，不假定凸电势。在解决了与对数$ W $的奇异性有关的技术之后，我们将结果应用于在非刻痕或四分之一的外部电势下获得日志气体的稳定率。我们证明，对数气体的分布相对于瓦斯汀距离以平方根速率收敛。最后，我们在双孔电势$ v（x）= \ frac {x^4} {4} {4} + c \ frac {x^2} {2} $，$ c <0 $和非固定$ v（x）= g \ frac {x^4} + frac {x^4} + 2 $ g <0 $ for $ | c | $和$ | g | $ small。
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 $V(x) = \frac{x^4}{4} + c\frac{x^2}{2}$, $c < 0$ and the non-confining potential $V(x) = g\frac{x^4}{4} + \frac{x^2}{2}$, $g<0$ for $|c|$ and $|g|$ small enough.
