Abstract
We consider functions uELEMENT OFLoo(L2)INTERSECTIONLp(W1,p) with 1<p<oo on a time space domain. Solutions to non-linear evolutionary PDE's typically belong to these spaces.
Many applications require a Lipschitz approximation u位 of u which coincides with u on a large set. For problems arising in fluid mechanics one needs to work with solenoidal (divergence-free) 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 in [DRW10].
Since divu位=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 [BDF12].