On three-dimensional flows of thermo-viscoelastic fluids of Giesekus type

1. Problem Statement

The paper addresses the mathematical analysis of the flow of an incompressible, heat-conducting, viscoelastic fluid of the Giesekus type in a bounded three-dimensional domain $\Omega \subset \mathbb{R}^3$ over a time interval $(0, T)$.

The system couples the conservation of mass, linear momentum, internal energy, and the evolution of the elastic stress (represented by the left Cauchy–Green tensor $B$ or the deformation gradient $F$). The governing equations are:

• Incompressibility: $\text{div } v = 0$.
• Momentum Balance: $\rho \partial_t v + \rho \text{div}(v \otimes v) - \text{div } T = \rho f$.
• Stress Evolution: An evolution equation for the tensor $B$ (or $F$) involving the upper convected Oldroyd derivative and a dissipative term specific to the Giesekus model.
• Internal Energy Balance: $\rho \partial_t e + \rho \text{div}(ev) + \text{div } q - T : D = 0$.

Key Features of the Model:

• Temperature Dependence: All material parameters (viscosity $\nu$, relaxation time $\tau$, thermal conductivity $\kappa$, and the elastic modulus function $g$) depend on the temperature $\theta$.
• Giesekus Dissipation: The model includes a nonlinear dissipative term in the stress evolution, distinguishing it from the Oldroyd-B model.
• Boundary Conditions: Navier slip boundary conditions for velocity ($v \cdot n = 0, (Tn)_\tau = -\gamma^* v_\tau$) and zero heat flux ($\nabla \theta \cdot n = 0$).
• Initial Data: Large data is considered, meaning no smallness assumptions are imposed on the initial velocity, temperature, or deformation gradient.

2. Methodology

The authors employ a sophisticated weak-strong framework combined with advanced compactness tools to handle the strong nonlinearities and the lack of regularity in the solutions.

A. Thermodynamic Consistency and Reformulation

• Entropy and Energy: The authors derive the system from a thermodynamically consistent framework using the Helmholtz free energy. They replace the internal energy equation with an entropy inequality and a total energy inequality. This approach is crucial because the pressure $p$ may not be well-defined in the weak sense for certain boundary conditions, but the energy inequality remains valid.
• Variable Transformation: To handle the evolution equation for $B$, which contains terms not under derivatives (making limit identification difficult), the authors introduce a variable $F$ such that $B = FF^T$. They reformulate the stress evolution equation in terms of $F$ (a transport-diffusion type equation), following the strategy of Masmoudi. This allows for better control of the nonlinear terms.

B. Approximation Scheme

The existence proof relies on a multi-step approximation procedure involving:

1. Galerkin Approximation: Projecting the velocity, deformation gradient, and internal energy onto finite-dimensional subspaces.
2. Regularization: Introducing artificial diffusion terms ($\varepsilon_4 \Delta F$, $\varepsilon_7 \Delta e$) and cut-off functions ($\Lambda_{\varepsilon_3}$, $g_{\varepsilon_1}$) to ensure the existence of smooth approximate solutions.
3. Iterative Limits: The authors pass to the limit in a specific sequence of parameters ($m, n, l \to \infty$ for Galerkin indices, followed by $\varepsilon_7, \varepsilon_4, \varepsilon_2, \varepsilon_1, \varepsilon_5, \varepsilon_6, \varepsilon_3 \to 0$).

C. Compactness and Convergence Tools

The core of the analysis lies in establishing the strong convergence of the temperature $\theta$ and the deformation gradient $F$, which is necessary to pass to the limit in the nonlinear constitutive relations.

• Temperature Convergence: The authors use the Div-Curl Lemma (Murat-Tartar) combined with the entropy equation. By constructing specific test functions involving the entropy $\eta$ and the temperature, they prove that a subsequence of temperatures converges almost everywhere.
• Deformation Gradient Convergence: They adapt the renormalization technique (DiPerna-Lions) and a specific Gronwall-type inequality for the quantity $|F|^2 - |F|^2$ (where $F$ is the limit). This involves splitting the velocity field into a "Stokes-like" part and a remainder to handle the convective terms, ultimately proving the strong convergence of $F$ in $L^2$.
• Biting Limits: Chacon's biting lemma is used to handle weak convergence in $L^1$ spaces for quantities like the dissipation terms.

3. Key Contributions

1. Global-in-Time Large-Data Existence: The paper establishes the existence of a global weak solution for the full 3D Giesekus model with temperature dependence, without requiring:
   • Smallness of initial data.
   • Artificial stress diffusion (a common regularization in previous literature).
   • Unphysical growth assumptions on material parameters (e.g., thermal conductivity is allowed to be bounded, unlike previous works requiring polynomial growth).
2. Thermodynamically Consistent Weak Solution: The solution is defined in a "dissipative weak" sense, satisfying the total energy inequality and the entropy inequality. This ensures the solution respects the second law of thermodynamics.
3. Handling the Giesekus Nonlinearity: The authors successfully treat the specific nonlinear dissipation term of the Giesekus model ($\alpha(B-I)$ and $\tau(B^2-B)$) in the presence of temperature coupling, a problem that remained open for large data in 3D.
4. General Material Parameters: The result holds for a broad class of continuous material functions (viscosity, relaxation time, etc.) satisfying only natural boundedness and monotonicity conditions, making the model applicable to a wide range of physical scenarios.

4. Main Results

Theorem: Under the assumption that the domain $\Omega$ is Lipschitz and the initial data $(v_0, F_0, \theta_0)$ belong to the natural energy-entropy function spaces (specifically $v_0 \in L^2$, $F_0 \in L^2$, $\theta_0 \in L^1$ with $\det F_0 > 0$), there exists a global-in-time dissipative weak solution $(v, \theta, F)$ to the system.

The solution satisfies:

• Weak Sequential Stability: The solution is stable under limits of approximating sequences.
• Energy Inequality: The total energy (kinetic + internal) satisfies a differential inequality accounting for boundary dissipation.
• Entropy Inequality: The entropy production is non-negative, satisfying the second law of thermodynamics.
• Regularity: The solution possesses specific regularity properties, including $v \in L^\infty(0,T; L^2) \cap L^2(0,T; W^{1,2})$, $\theta \in L^\infty(0,T; L^1) \cap L^{5/3}(Q_T)$, and $F \in L^\infty(0,T; L^2) \cap L^4(Q_T)$.

5. Significance

This work represents a significant breakthrough in the mathematical theory of complex fluids. Prior to this, global existence results for heat-conducting viscoelastic fluids often relied on:

• Simplified models (e.g., Oldroyd-B with stress diffusion).
• Small data assumptions.
• Two-dimensional settings.
• Unphysical assumptions on the growth of thermal conductivity.

By removing these restrictions, Bulíček, Los, and Woźnicki provide a rigorous foundation for modeling realistic, large-scale, three-dimensional flows of complex materials (such as polymers and biological fluids) where thermal effects and elastic stresses are strongly coupled. The methodology developed, particularly the combination of entropy-based compactness and the specific treatment of the Giesekus nonlinearity without artificial diffusion, sets a new standard for the analysis of rate-type viscoelastic fluids.