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

Abstract

UDC 517.51
We 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).