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Mv-strong uniqueness for density dependent, incompressible, non-Newtonian fluids Woznicki, Jakub

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We analyse the system of the form \begin{align*} {\partial}_t{\rho} +{\rm div \,}_x(\rho u) = 0\\ {\partial}_t(\rho u) +{\rm div \,}_x(\rho u\otimes u) + \nabla_x p = {\rm div \,}_x {\mathbb{S}}\label{secondequation}\\ {\rm div \,}_x(u) = 0 \end{align*} where $\rho$ is the mass density, $u$ denotes velocity field, ${\mathbb{S}}$ the stress tensor and $p$ is the pressure. We are interested in the measure-valued solutions to those equations and prove the mv-strong uniqueness property. This work bases its assumptions on the recent paper by Abbatiello and Feireisl [1], but differs from it in density dependency. Surprisingly the solutions are not defined by the Young measures, but by the similar tool to the so-called defect measure.

BIBLIOGRAPHY [1] A. Abbatiello and E. Feireisl. On a class of generalized solutions to equations describing incompressible viscous fluids. Ann. Mat. Pura Appl. (4), 199(3):1183â 1195, 2020.

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