JournalsrlmVol. 30, No. 3pp. 511–531

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

  • Paolo Bonicatto

    Universität Basel, Switzerland
  • Nikolay A. Gusev

    Moscow Inst. of Physics and Technology, Moscow, RUDN University, Moscow and Russian Academy of Sciences, Moscow, Russia
Non-uniqueness of signed measure-valued solutions to the continuity equation in presence of a unique flow cover
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Abstract

We consider the continuity equation tμt+div(bμt)=0\partial_t\mu_t + \mathrm {div}(\mathbf b\mu_t) = 0, where {mu1}tR\{mu_1\}_{t \in \mathbb R} is a measurable family of (possibily signed) Borel measures on Rd\mathbb R^d and b:R×Rd toRd\mathbf b : \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). If the measure-valued solution μt\mu_t is non-negative, then the following superposition principle holds: μt\mu_t can be decomposed into a superposition of measures concentrated along the integral curves of b\mathbf b. For smooth b\mathbf b this result follows from the method of characteristics, and in the general case it was established by L. Ambrosio. A partial extension of this result for signed measure-valued solutions μt\mu_t was obtained in [AB08], where the following problem was proposed: does the superposition principle hold for signed measure-valued solutions in presence of unique flow of homeomorphisms solving the associated ordinary differential equation? We answer to this question in the negative, presenting two 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.

Cite this article

Paolo Bonicatto, Nikolay A. Gusev, Non-uniqueness of signed measure-valued solutions to the continuity equation in presence of a unique flow. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 30 (2019), no. 3, pp. 511–531

DOI 10.4171/RLM/858