On the stability of solutions to non-Newtonian Navier--Stokes--Fourier-like systems in the supercritical case

Technical Summary: On the Stability of Solutions to Non-Newtonian Navier–Stokes–Fourier-Like Systems in the Supercritical Case

Problem Statement
The paper investigates the mathematical analysis of a three-dimensional, homogeneous, incompressible, non-Newtonian, heat-conducting fluid. The system is governed by a modified Navier–Stokes–Fourier system comprising the momentum equation, the incompressibility constraint, and the heat equation with a source term representing the dissipation of kinetic energy. The constitutive relation for the Cauchy stress tensor is general, encompassing power-law and Ladyzhenskaya models with a power-law exponent $p$ in the range $p \in [8/5, 11/5)$. This range is identified as "supercritical," a regime where standard regularity, uniqueness, and energy equality results are not known to hold. The problem is posed on a bounded Lipschitz domain $\Omega \subset \mathbb{R}^3$ with no-slip boundary conditions for velocity and prescribed non-uniform temperature (Dirichlet) on the boundary. The primary objective is to establish the existence of global-in-time solutions for arbitrary finite-energy initial data and to prove the nonlinear stability of the unique steady state $(0, \hat{\vartheta})$ as time tends to infinity.

Methodology and Novel Concepts
The authors address the lack of regularity in the supercritical regime by introducing a novel concept of solution, termed the $b$-weak solution. This formulation departs from classical weak solutions by replacing the internal energy equality with an entropy inequality and, crucially, introducing a $b$-corrected total energy inequality.

1. Entropy Inequality: Instead of the heat equation, the solution satisfies an entropy inequality derived by dividing the heat equation by temperature $\vartheta$. This utilizes the entropy variable $\eta = \ln \vartheta$, which possesses better integrability properties than $\vartheta$ itself.
2. $\gamma$-Correction Function: To handle the non-homogeneous Dirichlet boundary conditions for temperature, which prevent the direct use of standard total energy inequalities (due to uncontrollable boundary flux terms), the authors introduce a $\gamma$-correction function $b(\vartheta, \hat{\vartheta})$. This function allows for the construction of a modified energy functional that accounts for heat flux through the boundary.
3. $b$-Corrected Total Energy Inequality: By multiplying the heat equation by the correction function and combining it with the momentum equation, the authors derive a modified energy balance. This inequality includes a correction term involving the primitive of $b$, denoted $B$, and ensures that the total energy (kinetic plus thermal, adjusted by $B$) satisfies a differential inequality suitable for stability analysis.
4. Existence Proof: The existence of $b$-weak solutions is established via Faedo–Galerkin approximations. The authors derive a priori estimates independent of the approximation parameter $n$, utilizing the kinetic energy inequality, the entropy inequality, and the $b$-corrected total energy inequality. The limit passage relies on the monotonicity of the stress tensor, the solenoidal Lipschitz truncation method to identify the stress limit, and compactness arguments (Aubin–Lions lemma).
5. Stability Analysis: The stability of the steady state is analyzed using a Lyapunov functional $L_{\alpha, \beta}$, which measures the "distance" between the solution and the steady state. This functional incorporates the auxiliary functions $H_\alpha$ and $G$ to handle non-constant heat conductivity. The proof involves renormalizing the entropy inequality and combining it with the $b$-corrected energy inequality to derive a differential inequality for the Lyapunov functional, leading to exponential decay estimates.

Key Results

• Theorem 1.3 (Global Existence): For any initial data with finite energy ($v_0 \in L^2_{0,div}$, $\vartheta_0 \in L^1$) and boundary temperature $\hat{\vartheta}$ satisfying standard regularity, there exists a global-in-time $b$-weak solution. The solution satisfies a uniform global estimate in $L^\infty(0, +\infty; L^1(\Omega))$ for the temperature. Notably, this result holds for power-law exponents $p > 6/5$, extending previous results that were restricted to $p > 9/5$ or required Neumann boundary conditions.
• Theorem 1.4 (Nonlinear Stability): The unique steady state $(0, \hat{\vartheta})$ is nonlinearly stable. Specifically:
  • The velocity field decays exponentially in time: $\|v(\tau)\|_2^2 \leq e^{-\mu \tau} \|v(0)\|_2^2$ (with specific modifications for large initial data when $p < 2$).
  • The full solution (velocity and temperature) converges to the steady state in the sense of the Lyapunov functional $L_{\alpha, \beta}$. For any bounded set of initial data, there exist constants $\mu, \beta > 0$ such that the Lyapunov functional decays exponentially: $\int_\Omega L_{\alpha, \beta}(v(\tau), \vartheta(\tau), \hat{\vartheta}) dx \leq e^{-\mu(\tau-\sigma)} \int_\Omega L_{\alpha, \beta}(v(\sigma), \vartheta(\sigma), \hat{\vartheta}) dx$.

Significance and Claims
The paper claims to provide the first result that simultaneously establishes the existence of global-in-time solutions and their long-time nonlinear stability for this specific class of non-Newtonian, heat-conducting fluids in the supercritical regime with non-homogeneous Dirichlet boundary conditions.

The authors emphasize that previous works on similar systems either assumed the existence of smooth solutions, restricted analysis to subcritical regimes where classical energy equalities hold, or relied on uniqueness results that are not available in the supercritical case for shear-thinning fluids. The introduction of the $b$-weak solution concept is presented as a necessary technical advancement to overcome the lack of energy equality and the challenges posed by the boundary conditions. The stability result is described as a significant step forward, demonstrating that despite the potential non-uniqueness of weak solutions in this regime, the physically relevant class of $b$-weak solutions exhibits a stable long-time behavior converging to the unique equilibrium. The analysis explicitly avoids the Oberbeck–Boussinesq approximation, addressing the full coupling where kinetic energy dissipation acts as a source in the heat equation, a feature that introduces substantial mathematical difficulties compared to simplified models.