Optimal existence of weak solutions for the power-law Navier-Stokes-Voigt equations

Abstract

We study the power-law Navier--Stokes--Voigt system in a bounded domain, subject to appropriate initial conditions and no-slip boundary conditions. The analysis is governed by the power-law exponent $p$, ranging from shear-thinning ($1<p<2$) through the Newtonian case ($p=2$) to shear-thickening ($p>2$). We prove the global existence of weak solutions to the associated initial-boundary value problem. These results are optimal in the sense that they hold for the full admissible range of the exponent $p>1$ and for any space dimension $d\ge 2$. For $p>1$ when $2\le d\le 3$, and for $p\ge \frac{d}{2}$ when $d>3$, existence follows from a classical approach combining compactness arguments with the theory of monotone operators. The remaining case, $1<p<\frac{d}{2}$ with $d>3$, requires in addition a pressure decomposition technique and an ${\mathrm{L}}^{\infty }$--truncation argument. {Finally, we establish the uniqueness of weak solution for $p>1$ when $2\le d\le 4$, and for $p\ge \frac{d}{2}$ when $d>4$.}