A superposition principle for the inhomogeneous continuity equation with Hellinger–Kantorovich-regular coefficients - Bredies, Kristian; Carioni, Marcello; Fanzon, Silvio (Communications in Partial Differential Equations, 2025)
We study measure-valued solutions of the inhomogeneous continuity equation (Formula presented.) 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 (Formula presented.) that satisfy the characteristic system (Formula presented.) 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, for example, in the context of dynamic inverse problems and dynamic optimal transport.
Keywords: Continuity equation; Dynamic inverse problems; Hellinger–Kantorovich energy; Optimal transport regularization; Superposition principle; Uniqueness