# American Institute of Mathematical Sciences

March  2009, 8(2): 769-783. doi: 10.3934/cpaa.2009.8.769

## Turbulence models, $p-$fluid flows, and $W^{2, L}$ regularity of solutions

 1 Department of Applied Mathematics "U.Dini", Via F. Buonarroti 1/C, 56127-Pisa, Italy

Received  January 2008 Revised  June 2008 Published  December 2008

In this article we prove some sharp regularity results for the stationary and the evolution Navier-Stokes equations with shear dependent viscosity, under the no-slip boundary condition. This is a classical turbulence model, considered by von Neumann and Richtmeyer in the 50's, and by Smagorinski in the beginning of the 60's (for $p=3$). The model was extended to other physical situations, and deeply studied from a mathematical point of view, by Ladyzhenskaya in the second half of the 60's. In the sequel we consider the case $p> 2$. We are interested in regularity results in Sobolev spaces, up to the boundary, in dimension $n= 3$, for the second order derivatives of the velocity and the first order derivatives of the pressure. In spite of the very rich literature on this subject, sharp regularity results up to the boundary are quite new. In the sequel we improve in a very substantial way all the known results in the literature. In order to emphasize the very new ideas, we consider a flat boundary (the so called "cubic-domain" case). However, all the regularity results stated here hold in the presence of smooth boundaries, by following [3].
Citation: Hugo Beirão da Veiga. Turbulence models, $p-$fluid flows, and $W^{2, L}$ regularity of solutions. Communications on Pure and Applied Analysis, 2009, 8 (2) : 769-783. doi: 10.3934/cpaa.2009.8.769
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