具有 Hellinger-Kantorovich 正则系数的非齐次连续性方程的叠加原理
A superposition principle for the inhomogeneous continuity equation with Hellinger-Kantorovich-regular coefficients
AI总结：
本文证明低正则系数下非齐次连续性方程的动态 Hellinger-Kantorovich 叠加原理，并用于建立最小总变差解唯一性及刻画相关正则化项极点。
摘要：
我们研究非齐次连续性方程 $\partial_t ρ_t + {\rm div}(vρ_t) = g ρ_t$ 的测度值解，其中系数 $v$ 和 $g$ 具有低正则性。针对正测度解以及最近引入的动态 Hellinger-Kantorovich 能量有限的系数，我们证明了一个新的叠加原理。该原理将解分解为曲线 $t \mapsto h(t)δ_{γ(t)}$，这些曲线在适当意义下满足特征系统 $\dot γ(t) = v(t, γ(t))$、$\dot h(t) = g(t, γ(t)) h(t)$。特别地，它将现有叠加原理推广到 $g$ 为低正则性、且关于 $h$ 的特征不唯一的情形。文中给出了该原理的两个应用。首先，如果特征在其可能的消失时刻之前是唯一的，则得到非齐次连续性方程的最小总变差解的唯一性。其次，刻画了动态 Hellinger-Kantorovich 型正则化项的极点。这类正则化项例如出现在动态反问题和动态最优传输的背景中。
英文摘要：
We study measure-valued solutions of the inhomogeneous continuity equation $\partial_t ρ_t + {\rm div}(vρ_t) = g ρ_t$ where the coefficients $v$ and $g$ are of low regularity. A new superposition principle is proven for positive measure solutions and coefficients for which the recently-introduced dynamic Hellinger-Kantorovich energy is finite. This principle gives a decomposition of the solution into curves $t \mapsto h(t)δ_{γ(t)}$ that satisfy the characteristic system $\dot γ(t) = v(t, γ(t))$, $\dot h(t) = g(t, γ(t)) h(t)$ in an appropriate sense. In particular, it provides a generalization of existing superposition principles to the low-regularity case of $g$ where characteristics are not unique with respect to $h$. Two applications of this principle are presented. First, uniqueness of minimal total-variation solutions for the inhomogeneous continuity equation is obtained if characteristics are unique up to their possible vanishing time. Second, the extremal points of dynamic Hellinger-Kantorovich-type regularizers are characterized. Such regularizers arise, e.g., in the context of dynamic inverse problems and dynamic optimal transport.
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