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Large data existence theory for unsteady flows of fluids with pressure- and shear-dependent viscosities. (English) Zbl 1325.35160

Summary: A generalization of Navier-Stokes’ model is considered, where the Cauchy stress tensor depends on the pressure as well as on the shear rate in a power-law-like fashion, for values of the power-law index \(r \in(\frac{2 d}{d + 2}, 2]\). We develop existence of generalized (weak) solutions for the resultant system of partial differential equations, including also the so far uncovered cases \(r \in(\frac{2 d}{d + 2}, \frac{2 d + 2}{d + 2}]\) and \(r = 2\). By considering a maximal sensible range of the power-law index \(r\), the obtained theory is in effect identical to the situation of dependence on the shear rate only.

MSC:

35Q35 PDEs in connection with fluid mechanics
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35D30 Weak solutions to PDEs
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