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

TL;DR

The paper addresses the existence and uniqueness of weak solutions to the incompressible generalised Navier–Stokes–Voigt equations with a power-law viscosity in a bounded domain. It introduces a two-regime approach, proving existence for large and small $p$ using Galerkin approximations, monotone operator theory, pressure decomposition, and compactness via a Gelfand triple to achieve strong convergence. A key novelty is handling the full range $p>\frac{2d}{d+2}$ without the Helmholtz–Hodge projection, facilitated by a pressure splitting into $\pi_1,\pi_2,$ and $\pi_h$ components and a regularization strategy for small $p$. The results contribute to the mathematical understanding of viscoelastic non-Newtonian flows, providing rigorous well-posedness and a foundation for future analysis of NSV-type models in practical settings.

Abstract

In this study, we investigate the incompressible generalised Navier-Stokes-Voigt equations within a bounded domain $Ω\subset \mathbb{R}^d$, where $d \geq 2$. The governing momentum equation is expressed as: \begin{align*} \partial_t(\boldsymbol{v} - κΔ\boldsymbol{v}) + \nabla \cdot (\boldsymbol{v} \otimes \boldsymbol{v}) + \nabla π- ν\nabla \cdot \big( |\mathbf{D}(\boldsymbol{v})|^{p-2} \mathbf{D}(\boldsymbol{v}) \big) = \boldsymbol{f}. \end{align*} Here, for $d \in \{2,3\}$, $\boldsymbol{v}$ represents the velocity field, $π$ denotes the pressure, and $\boldsymbol{f}$ is the external forcing term. The constants $κ$ and $ν$ 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 the power-law exponent $p \in \big( \frac{2d}{d+2}, \infty \big)$, we establish the existence of a weak solution to the generalised Navier-Stokes-Voigt equations. Furthermore, we demonstrate that the weak solution is unique for the same range of the exponent $p$. The optimality of our results lies in the framework's use of a Gelfand triple, which allows the Aubin-Dubinskii lemma to yield strong convergence of approximate solutions, essential for existence and valid precisely for $p > \frac{2d}{d+2}$.

Theorems & Definitions (24)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Definition 3.1
• Theorem 3.2
• Theorem 3.3
• Lemma 4.1