Solenoidal Lipschitz truncation for parabolic PDE's

TL;DR

This paper develops a solenoidal Lipschitz truncation method for functions in specific Sobolev spaces, enabling better approximation of solutions to nonlinear parabolic PDEs, especially in fluid mechanics, and simplifies existing proofs.

Contribution

It introduces a divergence-free Lipschitz approximation technique for parabolic PDE solutions, improving analysis in fluid mechanics and simplifying prior proofs.

Findings

Constructed a solenoidal Lipschitz approximation for functions in L^(L^2)  L^p(W^{1,p})

Revised the existence proof for non-stationary generalized Newtonian fluids using the new approximation

Provided a simplified approach to stationary solenoidal Lipschitz truncation

Abstract

We consider functions $u\in L^\infty(L^2)\cap L^p(W^{1,p})$ with $1<p<\infty$ on a time space domain. Solutions to non-linear evolutionary PDE's 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 (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 ${\rm div} u_\lambda=0$, 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 [BDF12].