Solenoidal Lipschitz truncation and applications in fluid mechanics

Abstract

We extend the Lipschitz truncation method to the setting of solenoidal functions. In particular, we approximate a solenoidal Sobolev function by a solenoidal Lipschitz function which differs from the original function only on a small set.Our main application is the existence of weak solutions to the two-dimensional Prandtl–Eyring fluid model which has almost linear growth. In this situation a correction via Bogovskiĭ operators does not work.Furthermore, we extend the concept of almost A-harmonicity to the fluid context in the pressure free formulation.