Discontinuity, Nonlinearity, and Complexity
Uniqueness and NonUniqueness of Signed MeasureValued Solutions to the Continuity Equation
Discontinuity, Nonlinearity, and Complexity 9(4) (2020) 489497  DOI:10.5890/DNC.2020.12.001
Paolo Bonicatto
Departement Mathematik und Informatik, Universit"at Basel,
Spiegelgasse 1, CH4051, Basel, Switzerland
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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 nonuniqueness 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 nontrivial signed measurevalued 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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