LIPSCHITZOVSKÉ FUNKCE V ANALÝZE SYTÉMŮ PARCIÁLNÍCH DIFERENCIÁLNÍCH ROVNIC

Abstract

We consider a steady ow of a homogeneous incompressible nonNewtonian uid. We suppose that the viscosity of the uid depends on the mean normal stress (the pressure) and on the shear rate as this dependence is motivated by many technologically important experiments and studies. We study a system of partial dierential equations that govern such ows of uids subject to the homogeneous Dirichlet (no-slip) boundary condition and establish a global existence of a weak solution under certain specied assumptions on the structure of the viscosity. This is carried out by passing to the limit in the weak solution of a previously introduced approximate system, the existence of which is also shown. The fact that the viscosity is monotone in some sense plays an important role. A decomposition of the pressure and Lipschitz test functions as Lipschitz approximations of Sobolev functions are incorporated in order to obtain almost everywhere convergence of the pressure and the symmetric part of the velocity gradient.