A superposition principle for the inhomogeneous continuity equation with Hellinger-Kantorovich-regular coefficients

Abstract

We study measure-valued solutions of the inhomogeneous continuity equation ${\partial }_t{\rho }_t+\mathrm{div}(v{\rho }_t)=g{\rho }_t$ 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 $t\mapsto h(t){\delta }_{\gamma (t)}$ that satisfy the characteristic system $\dot{\gamma }(t)=v(t,\gamma (t))$, $\dot{h}(t)=g(t,\gamma (t))h(t)$ 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.

References in corpus (2)

• Young measures, superposition and transport
• A generalized conditional gradient method for dynamic inverse problems with optimal transport regularization

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• On extreme points and representer theorems for the Lipschitz unit ball on finite metric spaces