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Dissipative Solutions to a Compressible Non-Newtonian Korteweg System with Density-Dependent Viscous Stress Tensor

This paper establishes the existence of dissipative solutions and proves weak-strong uniqueness for a compressible non-Newtonian Korteweg system with density-dependent viscosity in periodic domains, thereby extending previous results on Newtonian flows to the non-Newtonian regime.

Original authors: Didier Bresch, Christophe Lacave, Maja Szlenk

Published 2026-07-13

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Didier Bresch, Christophe Lacave, Maja Szlenk

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where fluids don't just flow like water or honey, but act like a chaotic crowd of people who change their behavior based on how packed the room is. In some places, they are thick and sluggish; in others, they are thin and fast. This is the world of compressible non-Newtonian fluids. Scientists have been trying to write the "rulebook" for how these tricky fluids move for a long time, but there's a big problem: the math gets so messy that we can't prove a solution exists, or that the solution is unique. It's like trying to predict the exact path of every single person in a mosh pit without a clear set of rules.

Enter the researchers Didier Bresch, Christophe Lacave, and Maja Szlenk. They decided to tackle this chaos by adding a special ingredient to the mix: capillarity.

Think of capillarity as the fluid's "memory" or its "surface tension." It's the force that makes water bead up on a leaf or allows a paper towel to soak up a spill. In this paper, the authors argue that if you include this force in the equations, the chaos suddenly becomes manageable. They proved that dissipative solutions exist for these fluids in a 2D or 3D space (specifically a periodic domain, which is like a video game world where if you walk off the right edge, you reappear on the left).

Here is the magic trick they used: Relative Entropy.

Imagine you have two versions of the same fluid system. One is the messy, real-world version (the "weak solution") that we are trying to understand. The other is a perfect, smooth, ideal version (the "strong solution") that we can calculate easily. The authors created a mathematical "distance meter" called Relative Entropy. This meter measures how far apart the messy version is from the perfect one.

Their main finding is that if you start with the same initial conditions (the same starting crowd density and speed), this distance meter never grows out of control. In fact, they proved a weak-strong uniqueness property. This means: if a perfect, smooth solution exists, then the messy, real-world solution must be exactly the same as it. The messy crowd and the perfect crowd are forced to march in lockstep.

To get this result, they had to build a bridge. They couldn't jump straight to the answer, so they created a "regularized" version of the problem. Think of this as training wheels. They added a few extra, slightly artificial terms to the equations (like a tiny bit of extra friction or diffusion) to make the math behave nicely. They proved that solutions exist for this "training wheels" version. Then, they carefully removed the training wheels (letting the extra parameters go to zero) and showed that the solution didn't fall apart; it settled into a valid dissipative solution.

The paper is very specific about what it doesn't do. It does not claim to solve the problem for all types of fluids or in all conditions. It specifically focuses on fluids where the viscosity (thickness) depends on the density (how crowded the fluid is). It also explicitly rules out the idea that we can easily find standard "weak solutions" (the usual type of solution mathematicians look for) for these systems without adding the capillarity term. Without that capillary force, the door to proving existence remains closed.

The authors are confident in their proof. They didn't just run a computer simulation and say, "It looks like it works." They used rigorous mathematical logic, including the Galerkin method (a technique that approximates complex shapes using simpler building blocks) and monotonicity methods (using the fact that certain forces always push in a specific direction) to prove that a solution must exist. They showed that for the specific system described by their equations (1.1) with the initial conditions (1.2) and (1.3), a global-in-time dissipative solution exists.

So, what's the takeaway? If you want to model these super-complex, density-changing fluids, you can't ignore the capillary forces. Once you include them, the math finally clicks into place, guaranteeing that a solution exists and that if a perfect solution is found, it's the only one that matters. It's a solid step forward in understanding the wild, shifting behavior of non-Newtonian fluids, turning a chaotic mosh pit into a predictable dance.

Technical Summary: Dissipative Solutions to a Compressible Non-Newtonian Korteweg System

Problem Statement
This paper addresses the mathematical well-posedness of compressible non-Newtonian fluid flows where the viscous stress tensor depends on the density. Specifically, the authors investigate the existence of solutions to the compressible Navier–Stokes–Korteweg system in a periodic domain $\mathbb{T}^d$ ( $d=2,3$ ). The system is characterized by a density-dependent viscosity tensor $S(Du)$ satisfying growth and monotonicity conditions (generalizing power-law and Bingham-like fluids) and includes a capillarity term (Korteweg stress) modeled via a logarithmic density gradient. While the existence of weak solutions for such systems remains an open problem in the standard Leray-Hopf framework, the authors aim to establish the existence of dissipative solutions and prove the weak-strong uniqueness property.

Methodology
The proof strategy relies on the construction of approximate solutions followed by a limit passage, utilizing the relative entropy method.

Approximation Scheme: The authors introduce a regularized system (Equation 3.1) containing two key artificial terms:
- A parabolic regularization $\varepsilon \Delta \varrho$ in the continuity equation to ensure better regularity and bounds on the density.
- A high-order nonlinear viscosity term $\nu \text{div}(|Du|^{q-2}Du)$ with $q > \max(3, 3p/2)$ . This term provides higher integrability for the velocity gradient $Du$, allowing the use of $u$ as a test function and facilitating the application of monotonicity methods for the nonlinear stress tensor $S$ .
Existence of Approximate Solutions: Using the Galerkin method, the authors construct global-in-time weak solutions to the regularized system. They derive uniform energy estimates (Equation 3.4) that control the kinetic energy, internal energy, capillarity energy, and the dissipation terms associated with both the physical stress $S$ and the artificial viscosity.
Limit Passage ( $N \to \infty$ ): The authors pass to the limit in the Galerkin approximation. Key steps include:
- Utilizing the Aubin–Lions–Simon lemma to obtain strong convergence for $\sqrt{\varrho_N}$ and $\varrho_N u_N$ .
- Applying the monotonicity method (Minty-Browder technique) to handle the nonlinear stress tensor $S(Du)$, proving that $S(Du_N)$ converges weakly to $S(Du)$.
Relative Entropy Inequality: A central component of the analysis is the derivation of a relative entropy inequality. The authors define a relative entropy functional $E(t)$ measuring the distance between a weak solution $(\varrho, u)$ and a smooth test pair $(r, v)$ . They prove that for the approximate solutions, this functional satisfies an inequality involving error terms dependent on the regularization parameters $\varepsilon$ and $\nu$ .
Limit Passage ( $\nu, \varepsilon \to 0$ ): The authors demonstrate that the error terms in the relative entropy inequality vanish as the regularization parameters tend to zero. This relies on uniform bounds derived from the energy estimates and the specific structure of the capillarity term.

Key Contributions and Results

Existence of Dissipative Solutions: The main result (Theorem 1.2) establishes the existence of a global-in-time dissipative solution to the compressible non-Newtonian Navier–Stokes–Korteweg system. The solution satisfies the relative entropy inequality (Definition 2.4), which serves as a generalized weak solution concept.
Weak-Strong Uniqueness: As a direct consequence of the relative entropy inequality (Corollary 2.5), the paper proves that if a dissipative solution and a sufficiently regular strong solution share the same initial data, they must coincide. This ensures that the dissipative solution is unique within the class of strong solutions.
Extension to Non-Newtonian Flows: The work extends previous results concerning the link between Euler–Korteweg and Navier–Stokes–Korteweg systems (previously established for Newtonian flows with density-dependent viscosity) to the non-Newtonian case. It accommodates stress tensors $S$ that satisfy general growth conditions (e.g., $|S(A)| \sim |A|^{p-1}$ ) and monotonicity, covering models relevant to granular media and avalanches.

Significance
The paper claims significance by addressing a gap in the theory of compressible non-Newtonian fluids. Since the standard existence theory for weak solutions (à la Leray) remains open for these systems, the introduction of dissipative solutions provides a viable framework for analyzing their behavior. By incorporating capillarity effects, the authors show that these physical factors are sufficient to guarantee the existence of solutions that satisfy weak-strong uniqueness. This result generalizes the work of Bresch, Gisclon, and Lacroix-Violet (2019) from Newtonian to non-Newtonian fluids, offering a robust mathematical tool for studying complex viscoplastic compressible flows.

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