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

In this study, we investigate the incompressible generalised Navier-Stokes-Voigt equations within a bounded domain $\Omega \subset \mathbb{R}^d$, where $d \geq 2$. The governing momentum equation is expressed as: $$\partial_t(\boldsymbol{v}- \kappa \Delta \boldsymbol{v}) + \nabla \cdot (\boldsymbol{v} \otimes \boldsymbol{v}) + \nabla \pi - \nu \nabla \cdot \left( |\mathbf{D}(\boldsymbol{v})|^{p-2} \mathbf{D}(\boldsymbol{v}) \right) = \boldsymbol{f}.$$ Here, for $d \in \{2,3,4\}$, $\boldsymbol{v}$ represents the velocity field, $\pi$ denotes the pressure, and $\boldsymbol{f}$ is the external forcing term. The constants $\kappa$ and $\nu$ correspond to the relaxation time and kinematic viscosity, respectively. The parameter $p \in (1, \infty)$ characterizes the fluid's flow behavior, and $\mathbf{D}(\boldsymbol{v})$ denotes the symmetric part of the velocity gradient $\nabla \boldsymbol{v}$. For power-law exponents satisfying $p>1$ when $2\leq d\leq 3$, 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 $2 \leq d \leq 3$. Moreover, for $p>\frac{2d}{d+2}$ with $d>3$, the framework uses a Gelfand triple, allowing the Aubin--Dubinski\u{\i} 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 $d=4$.