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Boundary Regularity of Shear Thickening Flows

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Abstract

This article is concerned with the global regularity of weak solutions to systems describing the flow of shear thickening fluids under the homogeneous Dirichlet boundary condition. The extra stress tensor is given by a power law ansatz with shear exponent p≥ 2. We show that, if the data of the problem are smooth enough, the solution u of the steady generalized Stokes problem belongs to \({W^{1,(np+2-p)/(n-2)}(\Omega)}\) . We use the method of tangential translations and reconstruct the regularity in the normal direction from the system, together with anisotropic embedding theorem. Corresponding results for the steady and unsteady generalized Navier–Stokes problem are also formulated.

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Authors and Affiliations

  1. Department of Applied Mathematics “U. Dini”, Pisa University, Via F. Buonarroti 1, 56127, Pisa, Italy

    Hugo Beirão da Veiga

  2. Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 18675, Praha 8, Czech Republic

    Petr Kaplický

  3. Mathematisches Institut, Universität Freiburg, Eskerstr 1, 79104, Freiburg, Germany

    Michael Růžička

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  1. Hugo Beirão da Veiga
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  2. Petr Kaplický
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  3. Michael Růžička
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Correspondence to Hugo Beirão da Veiga.

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da Veiga, H.B., Kaplický, P. & Růžička, M. Boundary Regularity of Shear Thickening Flows. J. Math. Fluid Mech. 13, 387–404 (2011). https://doi.org/10.1007/s00021-010-0025-y

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