We prove inequalities connecting a flow through the (n−1)-dimensional surface S of a smooth solenoidal vector field with its Lp(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 ∈ Lp(U) (or, more precisely, of the class of vector fields that are equal almost everywhere with respect to the Lebesgue measure).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, No. 8, pp. 1141–1149, August, 2019.
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Noarov, A.I. On the Correct Determination of Flow of a Discontinuous Solenoidal Vector Field. Ukr Math J 71, 1303–1311 (2020) doi:10.1007/s11253-019-01715-7