We study the behaviour of various Lyapunov functionals (relative entropies) along the solutions of a family of nonlinear drift-diffusion-reaction equations coming from statistical mechanics and population dynamics. These equations can be viewed as gradient flows over the space of Radon measures equipped with the Hellinger-Kantorovich distance. The driving functionals of the gradient flows are not assumed to be geodesically convex or semi-convex. We prove new isoperimetric-type functional inequalities, allowing us to control the relative entropies by their productions, which yields the exponential decay of the relative entropies.

中文翻译：

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与不平衡最优运输相关的凸 Sobolev 不等式

我们沿着来自统计力学和种群动力学的非线性漂移-扩散-反应方程族的解来研究各种李雅普诺夫函数（相对熵）的行为。这些方程可以看作是梯度流过配备 Hellinger-Kantorovich 距离的氡测量空间。梯度流的驱动函数不被假定为测地凸或半凸。我们证明了新的等周型函数不等式，允许我们通过它们的产生来控制相对熵，从而产生相对熵的指数衰减。