ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Uniqueness and Non-Uniqueness of Signed Measure-Valued Solutions to the Continuity Equation

Discontinuity, Nonlinearity, and Complexity 9(4) (2020) 489--497 | DOI:10.5890/DNC.2020.12.001

Paolo Bonicatto

Departement Mathematik und Informatik, Universit"at Basel, Spiegelgasse 1, CH-4051, Basel, Switzerland

Abstract

We consider the continuity equation $\partial_t \mu_t + \text{div}(b \mu_t) = 0$, where $\{\mu_t\}_{t \in \mathbb R}$ is a measurable family of (possibily signed) Borel measures on $\mathbb R^d$ and $b \colon \mathbb R \times \mathbb R^d \to \mathbb R^d$ is a bounded Borel vector field (and the equation is understood in the sense of distributions). We discuss some uniqueness and non-uniqueness results for this equation: in particular, we report on some counterexamples in which uniqueness of the flow of the vector field holds but one can construct non-trivial signed measure-valued solutions to the continuity equation with zero initial data. This is based on a joint work with N.A. Gusev \cite{lincei}.

Acknowledgments

The author was supported by the ERC Starting Grant 676675 FLIRT.

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