Non-uniqueness of signed measure-valued solutions to the continuity equation in presence of a unique flow

DOI10.4171/RLM/858zbMath1429.35046arXiv1809.10216OpenAlexW2972052036MaRDI QIDQ2335820

Yanyan Li

Publication date: 15 November 2019

Published in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1809.10216

zbMATH Keywords

superposition principlemethod of characteristicstwo counterexamples

Mathematics Subject Classification ID

Variational problems in a geometric measure-theoretic setting (49Q20) Geometric theory, characteristics, transformations in context of PDEs (35A30) Initial value problems for linear first-order PDEs (35F10) Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness (35A02)

Related Items (8)

Uniqueness and Non-Uniqueness of Signed Measure-Valued Solutions to the Continuity Equation ⋮ On the Extreme Points of the Unit Ball in the Space of Solenoidal Vector Measures on the Plane ⋮ Well-posedness of the deterministic transport equation with singular velocity field perturbed along fractional Brownian paths ⋮ Existence and uniqueness for the transport of currents by Lipschitz vector fields ⋮ Indecomposable sets of finite perimeter in doubling metric measure spaces ⋮ A directional Lipschitz extension lemma, with applications to uniqueness and Lagrangianity for the continuity equation ⋮ On the Ambrosio-Figalli-Trevisan superposition principle for probability solutions to Fokker-Planck-Kolmogorov equations ⋮ Inverse potential problems in divergence form for measures in the plane

Cites Work

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• Transport equations due to the non-Lipschitzian vector fields and fluid mechanics
• Well-posedness for the continuity equation for vector fields with suitable modulus of continuity
• On the one-dimensional continuity equation with a nearly incompressible vector field
• A uniqueness result for the decomposition of vector fields in \(\mathbb{R}^d\)
• On the uniqueness of solutions to continuity equations
• Some remarks on the continuity equation
• One-dimensional transport equations with discontinuous coefficients

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