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

Bredies, Kristian; Carioni, Marcello; Fanzon, Silvio

Abstract

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.

Citation

Bredies, K., Carioni, M., & Fanzon, S. (2022). A superposition principle for the inhomogeneous continuity equation with Hellinger–Kantorovich-regular coefficients. Communications in Partial Differential Equations, 47(10), 2023-2069. https://doi.org/10.1080/03605302.2022.2109172

• Journal Article Type: Article
• Acceptance Date: Jul 31, 2022
• Online Publication Date: Sep 15, 2022
• Publication Date: 2022
• Deposit Date: May 9, 2023
• Publicly Available Date: May 12, 2023
• Journal: Communications in Partial Differential Equations
• Print ISSN: 0360-5302
• Electronic ISSN: 1532-4133
• Publisher: Taylor and Francis Group
• Peer Reviewed: Peer Reviewed
• Volume: 47
• Issue: 10
• Pages: 2023-2069
• DOI: https://doi.org/10.1080/03605302.2022.2109172
• Keywords: Continuity equation; Dynamic inverse problems; Hellinger–Kantorovich energy; Optimal transport regularization; Superposition principle; Uniqueness
• Public URL: https://hull-repository.worktribe.com/output/4270989