# On the correct definition of the flow of a discontinuous solenoidal vector field

### Abstract

UDC 517.51We prove inequalities connecting a flow through the $(n- 1)$-dimensional surface $S$ of a smooth solenoidal vector field with its $L^{p}(U)$-norm ($U$ is an $n$-dimensional domain that contains $S$). On the basis of these inequalities, we propose a correct definition of the flow through the surface $S$ of a discontinuous solenoidal vector field $f \in L^{p}(U)$ (or, more precisely, of the class of vector fields that are equal almost everywhere with respect to the Lebesque measure).

Published

25.08.2019

How to Cite

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 71, no. 8, Aug. 2019, pp. 1141-9, https://umj.imath.kiev.ua/index.php/umj/article/view/1505.

Issue

Section

Short communications