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

Abstract

We study measure-valued solutions of the inhomogeneous continuity equation $\partial_t \rho_t + {\rm div}(v\rho_t) = g \rho_t$ 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.