Solenoidal lipschitz truncation for parabolic PDEs

We consider functions u is an element of L-infinity(L-2) boolean AND L-p(W-1,W- p) with 1 <p <infinity on a time-space domain. Solutions to nonlinear evolutionary PDEs typically belong to these spaces. Many applications require a Lipschitz approximation u(lambda) of u which coincides with u on a large set. For problems arising in fluid mechanics one needs to work with solenoidal (divergencefree) functions. Thus, we construct a Lipschitz approximation, which is also solenoidal. As an application we revise the existence proof for non-stationary generalized Newtonian fluids of Diening, R u. zi. cka and Wolf, Existence of weak solutions for unsteady motions of generalized Newtonian fluids, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9 (2010) 1-46. Since div u(lambda) = 0, we are able to work in the pressure free formulation, which heavily simplifies the proof. We also provide a simplified approach to the stationary solenoidal Lipschitz truncation of Breit, Diening and Fuchs, Solenoidal Lipschitz truncation and applications in fluid mechanics.