论文标题
与hellinger-kantorovich-gromard系数的不均匀连续性方程的叠加原理
A superposition principle for the inhomogeneous continuity equation with Hellinger-Kantorovich-regular coefficients
论文作者
论文摘要
我们研究了不均匀的连续性方程的测量值解决方案$ \ partial_tρ_t + {\ rm div}（vρ_t）= gρ_t$，其中系数$ v $和$ g $是低规律性的。一个新的叠加原理被证明是用于阳性措施解决方案和系数，而最近引入的动态Hellinger-Kantorovich能量是有限的。该原理将溶液分解为曲线$ t \ mapsto h（t）δ_{γ（t）} $满足特征系统$ \dotγ（t）= v（t，γ（t）$，$ \ dot h（t，t）$，$ \ dot h（t）= g（t）= g（t，γ（t，γ（t））h（t，γ（t））$。特别是，它提供了现有的叠加原理对$ g $的低规范案例的概括，而在$ h $方面并非独特的特征。提出了该原则的两个应用。首先，如果特征在可能消失的时间之前是唯一的，则获得了不均匀连续性方程的最小总变化溶液的唯一性。其次，充满活力的Hellinger-Kantorovich型正规化器的极端点是特征的。例如，在动态逆问题和动态最佳传输的背景下，出现了这样的正规化器。
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.
