Solenoidal Lipschitz truncation for parabolic PDEs

We consider functions $u\in L^\infty(0,T;L^2({B}))\cap L^p(0,T;W^{1,p}({B}))$ where $p\in(1,\infty)$, $T$ is positive and ${B}\subset\mathbb R^d$ bounded. Solutions to non-linear evolutionary PDE's typically belong to these spaces. Many applications require an approximation $u_\lambda$ of $u$ which is Lipschitz-continous and coincides with $u$ on a large set. For problems arising in fluid mechanics one needs to work with functions which are divergence-free thus we construct a function $u_\lambda\in L^\infty(0,T;W^{1,\mathrm{BMO}}({B}))$ which is in addition to the properties from the known truncation methods solenoidal. As an application we revisit the existence proof for non-stationary generalized Newtonian fluids. Since $\mathrm{div}\,u_\lambda=0$ we can completely avoid the appearance of the pressure term and the proof can be heavily simplified.