Weak solutions to the Navier-Stokes equations for steady compressible non-Newtonian fluids
Cosmin Burtea, Maja Szlenk
TL;DR
The paper establishes the existence of weak solutions for steady compressible non-Newtonian Navier–Stokes systems on bounded domains, under a monotone viscous stress with power-law growth $r$ and a $\gamma$-power law pressure $p(\varrho)=\varrho^{\gamma}$, with $r>\frac{3d}{d+2}$ and sufficiently large $\gamma$. The authors develop a robust regularization framework (introducing $\alpha$, $\delta$, $\varepsilon$, $\eta$) and derive uniform energy bounds, enabling a limit passage that identifies the nonlinear pressure and stress terms via monotonicity and renormalized continuity equations. A key novelty is the use of a measure-theoretic defect control and Egorov-type arguments to overcome density oscillations and to prove strong convergence of $\nabla u$ and $\varrho^{\gamma}$, thus obtaining a weak solution without near-isotropy assumptions. Additionally, the paper proves existence for a time-discretized Herschel–Bulkley model with a singular viscosity component, highlighting the approach’s flexibility to handle singular rheologies. The results advance the mathematical understanding of non-Newtonian compressible flows and provide a foundation for numerical and stability analyses in anisotropic settings.
Abstract
We prove the existence of weak solutions to steady, compressible non-Newtonian Navier-Stokes system on a bounded, two- or three-dimensional domain. Assuming the viscous stress tensor is monotone satisfying a power-law growth with power $r$ and the pressure is given by $\varrho^γ$, we construct a solution provided that $r>\frac{3d}{d+2}$ and $γ$ is sufficiently large, depending on the values of $r$. Additionally, we also show the existence for time-discretized model for Herschel-Bulkley fluids, where the viscosity has a singular part.