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Divergence-Free Positive Symmetric Tensors. Applications to Fluid Dynamics

Lecturer

Denis Serre, ENS Lyon

Location

601 Pao Yue-Kong Library

Time

Abstract

Most of the models of fluid dynamics share a common structure, described by a non-negative symmetric tensor whose lines are divergence free. Although this feature has been overlooked so far, it contains a far-reaching information in terms of integrability. This is related to a new functional inequality, which we prove with sharp constant. The equality case is especially interesting. As a matter of fact, this structure also arises in periodic homogenisation, when the effective tensor equals the theoretical upper bound. Then the relevant inequality can be viewed as a non-commutative version of Gagliardo’s inequality. A third aspect of the same theory is a property of weakly upper semi-continuity of some functional involving the determinant of this tensor.