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Cauchy problem for the non-Newtonian viscous incompressible fluid. (English) Zbl 0863.76003

The article is devoted to the problem of the existence, uniqueness and regularity of the weak solution of the Cauchy problem for non-Newtonian viscous incompressible fluid.
The Cauchy stress tensor has a form \[ \tau=-pI-\tau (e)-2\mu_1\Delta e, \tag{1} \] where \(e\) is the deformation velocity tensor and \(\tau(e)\) satisfies the following condition \[ \tau_{ij}(e) e_{ij}\geq c|e|^p \tag{2} \] or \[ \tau_{ij}(e) e_{ij}\geq c\big(|e|^2+|e|^p\big). \] Section 3 deals with the global existence and uniqueness of weak solution for bipolar problem (it means \(\mu_1\not= 0\)) \(p\geq 1\) for both models and also with the regularity.
In section 4 the author is interested in the case when \(\mu_1\to 0^+.\) It is proved the existence of weak solution for the first model when \(p>\tfrac{3n}{n+2}\) and its uniqueness and regularity for \(p\geq 1+\tfrac{2n}{n+2}\), \(n\) is dimension. For the second model the existence of weak solution is proved for \(p>1.\)

MSC:

76A05 Non-Newtonian fluids
35Q30 Navier-Stokes equations
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References:

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