Navier–Stokes Equations with Shear-Thickening Viscosity. Regularity up to the Boundary

In this article we prove some sharp regularity results for the stationary and the evolution Navier–Stokes equations with shear dependent viscosity, see (1.1), under the no-slip boundary condition(1.4). We are interested in regularity results for the second order derivatives of the velocity and for the first order derivatives of the pressure up to the boundary, in dimension n ≥ 3. In reference [4] we consider the stationary problem in the half space \({\mathbb{R}}_+^n\) under slip and no-slip boundary conditions. Here, by working in a simpler context, we concentrate on the basic ideas of proofs. We consider a cubic domain and impose our boundary condition (1.4) only on two opposite faces. On the other faces we assume periodicity, as a device to avoid unessential technical difficulties. This choice is made so that we work in a bounded domain Ω and, at the same time, with a flat boundary. In the last section we provide the extension of the results from the stationary to the evolution problem.

Authors and Affiliations

1. Department of Applied Mathematics “U. Dini”, University of Pisa, Via Buonarrotti, 1/C, 56100, Pisa, Italy

   H. Beirão da Veiga

Corresponding author

Correspondence to H. Beirão da Veiga.

Partly supported by CMAF/UL and Fundação Calouste Gulbenkian (Lisbon).

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