Solenoidal Lipschitz truncation for parabolic PDEs

Solenoidal Lipschitz truncation for parabolic PDEs

Tue, 05/06/2012
12:30 		Dominic Breit (Universität München) OxPDE Special Seminar Add to calendar Gibson 1st Floor SR
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.