Analysis of the flows of incompressible fluids with pressure dependent viscosity fulfilling \(\nu (p, \cdot ) \to + \infty \) as \(p \to +\infty \).

*(English)*Zbl 1224.35311Summary: Over a large range of the pressure, one cannot ignore the fact that the viscosity grows significantly (even exponentially) with increasing pressure. This paper concerns long-time and large-data existence results for a generalization of the Navier-Stokes fluid whose viscosity depends on the shear rate and the pressure. The novelty of this result stems from the fact that we allow the viscosity to be an unbounded function of pressure as it becomes infinite. In order to include a large class of viscosities and in order to explain the main idea in as simple a manner as possible, we restrict ourselves to a discussion of the spatially periodic problem.

##### MSC:

35Q30 | Navier-Stokes equations |

76A05 | Non-Newtonian fluids |

76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |

76D05 | Navier-Stokes equations for incompressible viscous fluids |

##### Keywords:

existence; weak solution; incompressible fluid; pressure-dependent viscosity; shear-dependent viscosity; spatially periodic problem
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\textit{M. Bulíček} et al., Czech. Math. J. 59, No. 2, 503--528 (2009; Zbl 1224.35311)

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