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

Abstract

In this study, we investigate the incompressible generalised Navier-Stokes-Voigt equations within a bounded domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^d$, where $d\ge 2$. The governing momentum equation is expressed as: $${\partial }_t(\mathbf{v}-\hat{I}º\hat{I}”\mathbf{v})+\nabla \cdot (\mathbf{v}\otimes \mathbf{v})+\nabla \ddot{I}€-\hat{I}½\nabla \cdot \left(|\mathbf{D}(\mathbf{v}){|}^{p-2}\mathbf{D}(\mathbf{v})\right)=\mathbf{f}.$$ Here, for $d\in \{2,3,4\}$, $\mathbf{v}$ represents the velocity field, $\ddot{I}€$ denotes the pressure, and $\mathbf{f}$ is the external forcing term. The constants $\hat{I}º$ and $\hat{I}½$ correspond to the relaxation time and kinematic viscosity, respectively. The parameter $p\in (1,\infty )$ characterizes the fluid's flow behavior, and $\mathbf{D}(\mathbf{v})$ denotes the symmetric part of the velocity gradient $\nabla \mathbf{v}$. For power-law exponents satisfying $p>1$ when $2\le d\le 3$, and $p>\frac{2d}{d+2}$ for $d=4$, we establish the existence of weak solutions to the generalised Navier-Stokes-Voigt system. Moreover, we prove uniqueness of the weak solution for the same ranges of $p$. The results are optimal in the sense that $p>1$ is minimal for $2\le d\le 3$. Moreover, for $p>\frac{2d}{d+2}$ with $d>3$, the framework uses a Gelfand triple, allowing the Aubin--Dubinski\uı lemma to yield strong convergence of approximate solutions. This convergence is essential for the existence proof and holds precisely for $p>\frac{2d}{d+2}$ when $d=4$.