On the continuity in time of solutions to a generalized Navier--Stokes--Fourier system

1. Problem Statement

The paper investigates the mathematical analysis of a generalized Navier–Stokes–Fourier (NSF) system describing the flow of an incompressible, heat-conducting, non-Newtonian fluid in a bounded two-dimensional domain $\Omega \subset \mathbb{R}^2$.

Governing Equations:
The system consists of:

1. Momentum Balance: $\partial_t u + \text{div}(u \otimes u) - \text{div} S + \nabla \pi = f$
2. Incompressibility: $\text{div} u = 0$
3. Internal Energy Balance: $\partial_t \vartheta + \text{div}(\vartheta u) + \text{div} q = S : Du$
4. Entropy Balance: $\partial_t \eta + \text{div}(\eta u) + \text{div}(q/\vartheta) = S : Du/\vartheta - q \cdot \nabla \vartheta / \vartheta^2$

Key Features and Assumptions:

• Constitutive Relations:
  • Stress Tensor ($S$): The fluid exhibits non-Newtonian behavior modeled by a power-law relation $S = S^*(\vartheta, Du)$ with exponent $p \ge 2$. The stress satisfies monotonicity and coercivity conditions (e.g., $S:D \ge \nu |D|^p - \nu$).
  • Heat Flux ($q$): Follows Fourier's law $q = -\kappa(\vartheta)\nabla \vartheta$, where thermal conductivity $\kappa$ is bounded and continuous.
• Boundary Conditions: Homogeneous Dirichlet for velocity ($u=0$) and inhomogeneous Dirichlet for temperature ($\vartheta = \vartheta_b$).
• Initial Data: Finite energy initial data for velocity ($u_0 \in L^2$) and integrable initial temperature ($\vartheta_0 \in L^1$).
• Dimension: The primary existence result is established for 2D ($d=2$), while the regularity/continuity analysis is generalized to arbitrary dimensions $d$.

The Core Challenge:
In previous works on heat-conducting fluids with $L^1$ initial data and non-Newtonian stress, the existence of weak solutions was often established, but the time continuity of the temperature in the $L^1(\Omega)$ norm was not guaranteed. Typically, solutions were only known to satisfy the initial condition in a weak sense or as an essential limit. This paper aims to rigorously prove that the temperature is continuous in time, i.e., $\vartheta \in C([0, T]; L^1(\Omega))$.

2. Methodology

The authors employ a multi-stage approach combining approximation schemes, energy estimates, and entropy-based techniques.

A. Galerkin Approximation

• Construction: They construct approximate solutions $(u_{N,M}, \vartheta_{N,M})$ using a Galerkin method with finite-dimensional bases for velocity and temperature.
• Uniform Estimates: They derive a priori estimates uniform with respect to the approximation parameters ($N, M$).
  • Velocity: Standard energy estimates yield bounds in $L^\infty(0,T; L^2)$ and $L^p(0,T; W^{1,p})$.
  • Temperature: A crucial step involves testing the energy equation with specific nonlinear functions (e.g., $1 - (K/(K+\vartheta-\hat{\vartheta}))^\alpha$) to derive bounds in $L^\infty(0,T; L^1)$ and control the gradient $\nabla \vartheta$ in weighted spaces.
• Limit Passage: Using the Aubin–Lions lemma and compactness arguments, they pass to the limit to obtain a weak solution satisfying the momentum and internal energy equations.

B. Entropy Equality and Renormalization

• Entropy Formulation: Instead of relying solely on the internal energy equation, the authors utilize the entropy equation (derived formally from the energy equation via $\eta = \ln \vartheta$).
• Key Insight: Proving that the solution satisfies the entropy equality (rather than just an inequality) is the pivotal step. This allows them to bypass classical renormalization procedures (like DiPerna–Lions) which are often insufficient for establishing strong time continuity in this context.
• Convergence: They prove that the limit of the approximations satisfies the entropy equality in the distributional sense.

C. Proof of Time Continuity

• Cauchy Sequence Argument: To prove $\vartheta \in C([0, T]; L^1(\Omega))$, the authors show that the sequence of approximate temperatures is a Cauchy sequence in the space $C([0, T]; L^1(\Omega))$.
• Truncation Technique: They utilize truncation functions $T_m$ and their mollifications $T_{m,\delta}$.
  • First, they prove that the truncated temperatures $T_{m,\delta}(\vartheta_N)$ form a Cauchy sequence in $C([0, T]; L^2)$.
  • Second, they establish uniform equiintegrability of the temperature sequence. This is achieved by analyzing the dissipation on high-level sets (where $\vartheta$ is large).
• Equivalence of Conditions: They demonstrate that time continuity is equivalent to the vanishing of dissipation on high-level sets of the temperature.

3. Key Contributions

1. First Proof of $L^1$ Time Continuity: The primary novelty is the rigorous proof that the temperature of a generalized NSF system with power-law stress ($p \ge 2$) and $L^1$ initial data is continuous in time with values in $L^1(\Omega)$. This resolves a long-standing gap where initial conditions were typically only attained in a weak or essential limit sense.
2. Entropy Equality as a Tool: The paper establishes that satisfying the entropy equality is a sufficient condition to guarantee time continuity. This provides a new pathway for analyzing thermodynamic systems without relying on the classical renormalized solution framework.
3. Generalized Regularity Theorem (Theorem 1.2): The authors prove a general result for the entropy equation with a convective term and $L^1$ source in arbitrary spatial dimensions. They show that time continuity is equivalent to several distinct conditions:
   • Vanishing dissipation on high-level sets (asymptotic control of gradients).
   • Validity of a truncated variational inequality (or equality).
   • Validity of a relative energy inequality/equality.
4. Handling Inhomogeneous Boundary Conditions: The analysis successfully accommodates spatially inhomogeneous Dirichlet boundary conditions for temperature, a technically difficult scenario often simplified in literature.

4. Main Results

Theorem 1.1 (Existence in 2D):
For a 2D bounded domain, $p \ge 2$, and finite-energy data, there exists a global weak solution $(u, S, \vartheta, \eta)$ such that:

• $u \in C([0, T]; L^2_{0,\text{div}}) \cap L^p(0, T; V)$.
• $\vartheta \in C([0, T]; L^1(\Omega))$.
• The solution satisfies the entropy equality (Eq. 1.24) and the initial condition $\vartheta(0) = \vartheta_0$ in the strong $L^1$ topology.

Theorem 1.2 (Equivalence of Continuity Conditions):
For arbitrary dimensions $d$, if a solution satisfies the entropy equality, then the following statements are equivalent:

• (A) The gradient dissipation vanishes on high-level sets: $\lim_{k \to \infty} \int_0^T \int_\Omega \frac{|\nabla \vartheta|^2}{\vartheta} \chi_{\{k \le \vartheta \le 2k\}} dx dt = 0$.
• (B/C) The solution satisfies a truncated variational inequality (or equality) for admissible test functions.
• (D/E) The solution satisfies a relative energy inequality (or equality).
• Conclusion: Any of these conditions implies $\vartheta \in C([0, T]; L^1(\Omega))$.

5. Significance

• Theoretical Advancement: This work bridges the gap between the existence of weak solutions and the strong topological properties required for physical interpretation. Time continuity is essential for defining the solution at $t=0$ rigorously and for studying long-time behavior (stability).
• Methodological Innovation: By leveraging the entropy equality to prove continuity, the authors offer a robust alternative to renormalization techniques. This approach is particularly powerful for systems with low regularity data (like $L^1$ temperature).
• Physical Relevance: The results validate the mathematical model for non-Newtonian heat-conducting fluids (e.g., polymers, blood, slurries) under realistic boundary conditions, ensuring that the temperature evolution is physically consistent (no "jumps" in the $L^1$ norm).
• Foundation for Stability: The establishment of time continuity and the relative energy inequality (Theorem 1.2, Statement E) lays the groundwork for future studies on the nonlinear stability and exponential decay of such systems to steady states, a topic hinted at in the introduction and related literature.

In summary, this paper provides a definitive answer to the regularity of temperature in generalized NSF systems, proving that under standard constitutive assumptions, the temperature evolves continuously in time, thereby strengthening the mathematical foundation of non-Newtonian fluid dynamics.