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

1. Problem Statement

The paper investigates the incompressible Generalised Navier-Stokes-Voigt (NSV) equations in a bounded domain $\Omega \subset \mathbb{R}^d$ ($d \geq 2$) over a time interval $[0, T]$. The system models the flow of viscoelastic fluids with non-Newtonian (power-law) behavior.

The governing momentum equation is:
$$\partial_t(v - \kappa\Delta v) + \nabla \cdot (v \otimes v) + \nabla \pi - \nu \nabla \cdot (|D(v)|^{p-2}D(v)) = f$$
subject to:

• Incompressibility: $\nabla \cdot v = 0$.
• Initial Condition: $v(0) = v_0$.
• Boundary Condition: $v = 0$ on $\partial \Omega$.

Key Parameters:

• $v$: Velocity field, $\pi$: Pressure, $f$: External force.
• $\kappa > 0$: Relaxation time (Voigt term).
• $\nu > 0$: Kinematic viscosity.
• $p \in (1, \infty)$: Power-law exponent characterizing the fluid.
  • $1 < p < 2$: Shear-thinning (pseudo-plastic).
  • $p > 2$: Shear-thickening (dilatant).
  • $p = 2$: Newtonian fluid (classical NSV).
• $D(v) = \frac{1}{2}(\nabla v + (\nabla v)^\top)$: Symmetric part of the velocity gradient.

The primary mathematical challenge is establishing the existence and uniqueness of weak solutions for the full range of $p$, particularly in the "critical" or "sub-critical" regimes where standard compactness arguments fail due to the lack of sufficient integrability of the convective term $v \otimes v$.

2. Methodology

The authors employ a combination of Galerkin approximations, monotone operator theory, compactness arguments, and a refined pressure decomposition technique.

A. Galerkin Approximation

The solution is constructed as the limit of finite-dimensional approximations $v_n$ in a subspace spanned by eigenfunctions of the Laplacian. The approximate system reduces to a system of ODEs, for which local existence is guaranteed by Carathéodory's theorem.

B. Uniform Estimates and Compactness

• Energy Estimates: The authors derive uniform bounds for $v_n$ in spaces like $L^\infty(0, T; V)$ and $L^p(0, T; V_p)$.
• Time Derivative Estimates: Bounds on $\partial_t v_n$ are established in dual spaces (e.g., $L^2(0, T; V)$ or negative Sobolev spaces) to apply compactness lemmas.
• Aubin-Dubinskiĭ Lemma: Used to extract a strongly convergent subsequence. The paper highlights that for $d \leq 3$, standard embeddings suffice. However, for $d=4$ and specific ranges of $p$, a Gelfand triple framework is essential to secure strong convergence.

C. Handling the Convective Term and Pressure

A major novelty is the avoidance of the Helmholtz-Hodge projection, which typically restricts analysis to $p \geq 2$. Instead, the authors:

1. Decompose the Pressure: Following the techniques of Wolf [51] and Diening et al. [21], they decompose the pressure $\pi$ into distinct components ($\pi_h, \pi_1, \pi_2$) corresponding to the inertial, viscous, and convective terms.
   • $\pi_h$: Harmonic part associated with the Voigt term.
   • $\pi_1, \pi_2$: Parts associated with the viscous stress and convective/forcing terms.
2. Regularization (for $d=4$): For the case $d=4$ and $p \in (\frac{2d}{d+2}, \frac{d}{2})$, a regularized problem is introduced with an additional term $\frac{1}{n} \text{div}(|D(v_n)|^{\beta-2}D(v_n))$ where $\beta \geq d/2$. This ensures the existence of approximate solutions via Theorem 3.2 before passing to the limit.

D. Minty Trick

To identify the limit of the nonlinear viscous term $|D(v_n)|^{p-2}D(v_n)$, the authors utilize the Minty trick (monotonicity method). By testing the weak formulation with specific test functions and utilizing the strict monotonicity of the operator $A(v) = |D(v)|^{p-2}D(v)$, they prove that the weak limit of the nonlinear term is indeed the nonlinear term of the weak limit.

3. Key Contributions and Novelties

1. Optimal Existence Ranges:

   • For $2 \leq d \leq 3$: Existence and uniqueness are proven for $p > 1$. This is optimal, as $p=1$ corresponds to the limit case where standard Sobolev embeddings fail.
   • For $d = 4$: Existence and uniqueness are proven for $p > \frac{2d}{d+2} = \frac{4}{3}$. This is the critical lower bound required to apply the Aubin-Dubinskiĭ lemma within the specific Gelfand triple framework used.
   • Significance: These results extend previous literature (which often required $p \geq 2$ or $p \geq \frac{3d+2}{d+2}$) to the minimal possible ranges for these dimensions.

2. Avoidance of Helmholtz-Hodge Projection:
   The authors explicitly avoid using the Helmholtz-Hodge projection, which is ill-behaved for $p < 2$ in the context of controlling the convective term. By working directly with the pressure decomposition, they handle the full range $p > 1$ for $d \leq 3$ without technical obstructions.

3. Pressure Decomposition in 4D:
   The paper provides a rigorous treatment of the 4-dimensional case, a borderline dimension for nonlinear PDEs. The decomposition of the pressure allows the authors to isolate the singular parts of the nonlinearity and recover strong convergence where standard methods fail.

4. Main Results

• Theorem 3.2 (Existence for $d \leq 3$ and $d > 3, p \geq d/2$):
  Under assumptions $v_0 \in V \cap V_p$ and $f \in L^{p'} \cap L^\infty(W^{-1,2})$, there exists a weak solution $v \in C([0, T]; V) \cap L^p(0, T; V_p)$ for:

  • $p > 1$ if $2 \leq d \leq 3$.
  • $p \geq d/2$ if $d > 3$.

• Theorem 3.3 (Existence for $d=4$ and critical $p$):
  For $d=4$ and $p \in (\frac{4}{3}, 2)$, assuming a smooth boundary and $f \in L^{p'}$, there exists a weak solution. This result relies on the regularization technique and the specific pressure decomposition.

• Theorem 3.4 (Uniqueness):
  The weak solution is unique in the class defined by Definition 3.1 for the ranges established in Theorems 3.2 and 3.3. The proof utilizes energy estimates on the difference of two solutions and Grönwall's inequality.

5. Significance and Impact

• Mathematical Rigor: The paper closes a gap in the mathematical theory of viscoelastic fluids by establishing existence for the optimal range of power-law exponents. It demonstrates that the Voigt regularization term ($-\kappa \partial_t \Delta v$) is powerful enough to handle the nonlinearities even in regimes where the classical Navier-Stokes equations are ill-posed or difficult to analyze.
• Physical Relevance: While physical applications are typically $d=2,3$, the analysis of $d=4$ provides insight into the structural stability of the equations and the mechanisms controlling solutions in higher dimensions. The results cover shear-thinning fluids ($1 < p < 2$), which are common in polymer processing and biological flows.
• Methodological Advancement: The successful application of pressure decomposition without Helmholtz-Hodge projection offers a robust template for analyzing other non-Newtonian fluid models where standard projection methods fail for $p < 2$.

In conclusion, this work represents a significant advancement in the analysis of generalized fluid models, providing the first proof of optimal existence and uniqueness for the Generalised NSV equations across the full spectrum of physically relevant power-law exponents in dimensions 2 and 3, and the critical range in dimension 4.