Bresch–Desjardins Entropy in Fluid Models

• Bresch–Desjardins Entropy is a higher-order energy/entropy framework that provides crucial a priori estimates for compressible fluid models with degenerate viscosity and singular transport phenomena.
• It employs an augmented velocity formulation to cancel out dangerous cross-derivative terms, enabling control over higher-order density and velocity derivatives even in vacuum regions.
• The methodology extends to various applications including quantum fluids, thin-film flows, and nonlocal alignment systems, thereby enhancing compactness and convergence in singular limits.

The Bresch–Desjardins entropy (BD entropy) is a higher-order energy/entropy structure arising in the analysis of compressible fluid models with degenerate viscosity, capillarity, drag, or other singularities in mass and momentum transport. First identified by D. Bresch and B. Desjardins in the context of the viscous shallow-water and compressible Navier–Stokes equations, the BD entropy provides supplementary a priori estimates that are essential for proving existence, compactness, and regularity of solutions—especially in regimes where standard energy methods fail due to vanishing density or degenerate dissipation. The BD entropy has become a critical tool in the analysis of degenerate parabolic-hyperbolic PDEs, quantum fluid models, thin-film flows, and nonlocal alignment systems.

1. BD Entropy: Definition and Functional Structure

The distinguishing feature of the BD entropy is the use of an "augmented velocity," typically of the form

$v = u + \alpha\frac{\nabla\rho}{\rho}$

where $u$ is the standard velocity, $\rho$ is density, and the coefficient $\alpha$ depends on the viscosity structure of the model. In the one-dimensional viscous shallow-water system with drag, the fundamental variables are height $h(x, t) > 0$ and horizontal velocity $u(x, t)$, and the BD-entropy functional is given by

$$E_{BD}(h, u)(t) := \int_\Omega \left[ \tfrac12 h v^2 + \tfrac{1}{2Fr^2} h^2 + \tfrac{1}{2W_e} (\partial_x h)^2 \right] dx + \alpha\int_0^t\int_\Omega \frac{h^2}{F(h)} dx\,ds$$

where $v = u + \tfrac{4}{R_e}\frac{\partial_x h}{h}$, and $F(h)$ is a positive drag-weight function dictating the nature of the dissipation (Bresch et al., 2018).

The prototype in higher dimensions for compressible Navier–Stokes with degenerate viscosities is

$$J_{BD}(t) = \int_\Omega \left[ \tfrac12 \rho |u + \nabla\ln\rho|^2 + \frac{\rho^\gamma}{\gamma-1} \right] dx.$$

This functional directly controls higher-order derivatives of the density and velocity, yielding bounds not accessible through the standard kinetic-plus-potential energy alone (Vasseur et al., 2015).

2. Algebraic Mechanisms and Structural Requirements

The existence of a BD-type estimate hinges on crucial algebraic relationships between the viscosity coefficients. In compressible Navier–Stokes, the structure

$\lambda(\rho) = 2[\rho\mu'(\rho) - \mu(\rho)]$

is essential. For instance, $\mu(\rho) = \rho$, $\lambda(\rho) = 0$ is a canonical case. This choice yields remarkable cancellation of dangerous cross-derivative terms when the momentum equations are tested by the augmented velocity $u + \nabla\ln\rho$, enabling the derivation of gradient and second-derivative controls on $\rho$, even in vacuum regions (Vasseur et al., 2015, Bresch et al., 2017).

For quantum or capillarity models, the BD entropy naturally incorporates capillarity or quantum Bohm terms: $$B_\varepsilon(t) = \int_{\mathbb{R}^3} \tfrac12 \left|\sqrt\rho\,u + 2c\nabla\sqrt\rho\right|^2 + |\nabla\sqrt\rho|^2 + \pi_\varepsilon(\rho)\, dx,$$ with time-differentiation producing nonnegative dissipative terms corresponding to quantum regularity (Antonelli et al., 2019, Donatelli et al., 2024).

3. Dissipation and Entropy Inequalities

The BD entropy functional enjoys a precise dissipation identity. Formally,

$$\frac{d}{dt} E_{BD}(h, u)(t) + D_{BD}(h, u)(t) = 0,$$

where the dissipation term $D_{BD}$ typically includes:

• Drag-weighted dissipation (e.g., $\alpha h^2 u^2 / F(h)$),
• Viscous dissipation ($h |\partial_x u|^2$ or $\rho |D(u)|^2$),
• Capillarity or higher-order derivative dissipation (e.g., $|\partial_x^2 h|^2$, $|\nabla^2 \sqrt\rho|^2$) (Bresch et al., 2018, Vasseur et al., 2015, Antonelli et al., 2019).

Careful analysis reveals that the evolution equations for the augmented velocity and mass density interlock, with structural cancellations arising due to the BD algebraic relation. Such structure is robust enough to extend to various nonlinear, nonlocal, and stochastic systems (Donatelli et al., 2024, Mucha et al., 2024).

4. Applications to Existence, Regularity, and Compactness

The BD entropy method serves as the backbone for existence theories in multiple degenerate settings:

• Viscous shallow-water and thin-film equations: Uniform BD-entropy bounds allow for the passage to the lubrication limit, wherein the BD entropy degenerates into the Bernis–Friedman entropy for the thin-film equation, providing global weak solutions and sharp compactness (Bresch et al., 2018).
• Compressible Navier–Stokes with degenerate viscosities: BD entropy ensures control in vacuum and is critical for proving the existence of global weak solutions, even with degenerate diffusion (Vasseur et al., 2015).
• Quantum and capillarity fluids: Incorporating the BD structure accommodates capillarity/quantum terms, extending compactness to higher-order derivatives (Antonelli et al., 2019, Bresch et al., 2017, Donatelli et al., 2024).
• Primitive equations and multidimensional stratified flows: The BD estimate extends to multidimensional systems, yielding fine compactness of the vertical velocities and enabling existence of global weak solutions to atmospheric models (Liu et al., 2018).
• Nonlocal and swarming models: By choosing the nonlocal viscosity and interaction terms to fit the BD structure, global compactness and long-time asymptotics can be addressed in models with alignment, drag, and singular interaction kernels (Mucha et al., 2024, Chaudhuri et al., 2024).

5. Families of BD Entropies and Drag Structures

The framework admits a variety of entropy functionals indexed by the choice of drag-weight $F(h)$ or more generally the structure of viscosity and capillarity:

• $F(h) = h^2$ gives the standard BD entropy with linear drag.
• $F(h) = h^2 + h^3$ or other powers produces families of entropies with nonlinear drag-dissipation structure (Bresch et al., 2018).
• In general, the BD approach accommodates multi-parameter families by modulating the weight function, always contingent on retaining the key algebraic cancellation in the system.

This flexibility underpins its relevance across models as disparate as thin films, degenerate Navier–Stokes, and nonlocal PDEs.

6. Limit Theories and Bridging with Other Entropies

The BD entropy naturally connects to other entropy estimates under scaling limits. In particular, in the vanishing inertia and viscosity limit, the BD entropy for the viscous shallow-water system converges to the Bernis–Friedman (BF) entropy for the lubrication (thin-film) equation: $$\int_\Omega G_0(h(T)) dx + \int_0^T \int_\Omega \left(\frac{1}{\alpha W_e}|\partial_x^2 h|^2 + \frac{1}{\alpha Fr^2}\frac{F'(h)}{F(h)}|\partial_x h|^2\right) dx\,dt = \int_\Omega G_0(h_0) dx$$ where $G_0$ is related to the drag structure. The rigorous bridge between BD and BF entropies provides a robust pathway for transferring existence, regularity, and compactness results from viscous models to their degenerate, singular, or non-viscous limits (Bresch et al., 2018).

7. Broader Impact and Modern Extensions

In contemporary research, the BD entropy and its variants are central in analyzing the global dynamics, regularity, vacuum avoidance, and singular limits of degenerate dissipative PDEs in fluid mechanics, quantum hydrodynamics, and collective behavior. Its capability to replace pressure-based estimates with regularity in the density and augmented velocities is indispensable for passing to the limit in approximation schemes, ensuring compactness, and enabling convergence—often in situations where standard energy methods are ineffective. The versatility of the entropy extends to stochastic PDEs, swarming hydrodynamics with alignment, pressureless flows, and problems with strong singularities or degenerate structure (Donatelli et al., 2024, Mucha et al., 2024, Chaudhuri et al., 2024).

The continued development and adaptation of the BD entropy methodology, including its manifestations as modulated energies, effective velocities, or two-velocity reformulations, underpin important advances in the mathematical theory of nonlinear PDEs and their singular limits.