The zero capillarity limit for the Euler-Korteweg system with no-flux boundary conditions

1. Problem Statement

The paper investigates the vanishing capillarity limit ($\varepsilon \to 0$) of the Euler-Korteweg (EK) system in a domain $\Omega \subset \mathbb{R}^d$ with a smooth boundary. The EK system models compressible, inviscid fluids with internal capillary forces (diffuse interfaces).

The system is given by:
$$\begin{cases} \partial_t \rho + \text{div}(\rho u) = 0 \\ \partial_t (\rho u) + \text{div}(\rho u \otimes u) + \nabla p(\rho) = \varepsilon^2 \rho \nabla \left( \text{div}(k(\rho)\nabla\rho) - \frac{1}{2}k'(\rho)|\nabla\rho|^2 \right) \end{cases}$$
subject to no-flux boundary conditions:
$$u \cdot n|_{\partial\Omega} = 0, \quad \partial_n \rho|_{\partial\Omega} = 0.$$

The authors aim to prove the convergence of finite energy weak solutions of the EK system to strong (classical) solutions of the standard compressible Euler system (where $\varepsilon = 0$) as $\varepsilon \to 0$.

Key Challenge:
There is a mismatch between the boundary conditions of the EK system (which requires $\partial_n \rho = 0$) and the limiting Euler system (which only requires $u \cdot n = 0$). This mismatch typically leads to the formation of a boundary layer near $\partial\Omega$. In similar problems (like the vanishing viscosity limit), controlling this layer often requires strong assumptions on energy dissipation or specific smallness conditions on initial data. The authors aim to handle this without such restrictive conditions.

2. Methodology

The core methodology relies on the Relative Energy (or Relative Entropy) Method, a powerful tool for proving weak-strong uniqueness and singular limits in PDEs.

A. Standard Relative Energy

The authors first define a standard relative energy functional $\mathcal{E}(\rho, J | r, U)$ measuring the distance between the weak solution $(\rho, J)$ and a smooth test solution $(r, U)$:
$$\mathcal{E} = \int_\Omega \left( \frac{1}{2}|\Lambda - \sqrt{\rho}U|^2 + f(\rho|r) + \frac{\varepsilon^2}{2}|\nabla\beta(\rho)|^2 \right) dx$$
where $\Lambda = \sqrt{\rho}u$ is the pseudo-velocity, $f(\rho|r)$ is the relative internal energy, and $\beta(\rho)$ is an auxiliary state function related to the capillarity coefficient $k(\rho)$.

• Result: Using a Grönwall-type estimate, they prove that if the initial relative energy vanishes, the weak solution converges to the strong Euler solution in $L^1$ (density) and $L^2$ (momentum). However, this does not control the gradient of the density near the boundary.

B. Augmented Relative Energy & Boundary Layer Correction

To address the gradient convergence and the boundary layer, the authors introduce a higher-order relative energy functional $\mathcal{E}_h$.

1. Augmented Formulation: They interpret the capillary energy $\frac{\varepsilon^2}{2}k(\rho)|\nabla\rho|^2$ as the kinetic energy of an auxiliary velocity field $v(\rho) = \sqrt{k(\rho)/\rho}\nabla\rho$. This allows them to treat the capillary term as a kinetic term in the relative energy.
2. Boundary Layer Correction: Since the limiting Euler density $\rho_E$ generally does not satisfy the Neumann condition $\partial_n \rho_E = 0$, the corresponding auxiliary velocity $v(\rho_E)$ does not satisfy the no-flux condition.
   • The authors construct a boundary layer corrector $v_{bl}$ using a cut-off function near the boundary.
   • They define a corrected test function $v_{bl}^E = v(\rho_E) - v_{bl}$ which satisfies the required boundary conditions.
3. Error Analysis: They derive an energy inequality for $\mathcal{E}_h$ using the corrected test functions. The remainder terms are split into:
   • $R_{rel}$: Standard relative energy errors.
   • $R_{in}$: Errors because the Euler solution doesn't solve the full EK system.
   • $R_{bl}$: Errors generated by the boundary layer correction.

Crucial Technical Insight: Unlike the vanishing viscosity limit (where boundary layers are strong and require assumptions on anomalous energy dissipation), the authors demonstrate that the zero capillarity limit generates a weaker boundary layer. The relative energy approach naturally absorbs the boundary layer errors without requiring additional assumptions on the lack of capillary energy concentration.

3. Key Contributions

1. Rigorous Convergence with Boundary Layers: The paper provides the first rigorous proof of the zero capillarity limit for the Euler-Korteweg system on domains with non-trivial boundaries and no-flux conditions.
2. Gradient Convergence: Unlike previous results that only established convergence in $L^p$ spaces for density and momentum, this work proves the strong convergence of the density gradient (specifically $\varepsilon \nabla \beta(\rho)$) in $L^2$, accounting for the boundary layer.
3. Relaxed Assumptions: The analysis does not require additional conditions on the lack of anomalous concentration of capillary energy near the boundary, a requirement often found in vanishing viscosity studies. This is attributed to the weaker nature of the capillary boundary layer.
4. Generalization: The results apply to a general class of capillarity coefficients $k(\rho) = c_\alpha \rho^\alpha$ (including the Quantum Hydrodynamics case where $k(\rho) = 1/4\rho$) and general barotropic pressure laws $p(\rho) = \rho^\gamma$.
5. Extension of Madelung Transform Analogy: The work extends the analogy between the EK system and the Nonlinear Schrödinger (NLS) equation (via the Madelung transform) to general domains with boundaries, showing that the hydrodynamic limit holds even when the wave function satisfies Neumann conditions.

4. Main Results

The paper establishes two main theorems:

• Theorem 2.6 (Standard Convergence):
  If the initial data are well-prepared (relative energy $\to 0$), the finite energy weak solutions $(\rho_\varepsilon, J_\varepsilon)$ converge strongly to the strong Euler solution $(\rho_E, u_E)$ in the topology:
  $$\|\rho_\varepsilon - \rho_E\|_{L^\infty(L^1 \cap L^\gamma)} + \|\Lambda_\varepsilon - \sqrt{\rho_\varepsilon}u_E\|_{L^\infty(L^2)} \to 0.$$

• Theorem 2.12 (Augmented Convergence with Gradient Control):
  Under similar assumptions but using the augmented relative energy, the convergence is strengthened to include the gradient term:
  $$\|\rho_\varepsilon - \rho_E\| + \|\Lambda_\varepsilon - \sqrt{\rho_\varepsilon}u_E\| + \varepsilon \|\nabla\beta(\rho_\varepsilon) - \sqrt{\rho_\varepsilon}v(\rho_E)\|_{L^\infty(L^2)} \to 0.$$
  This result explicitly quantifies the convergence of the capillary term and handles the boundary layer correction $v_{bl}$.

5. Significance

• Mathematical Physics: The paper resolves a long-standing technical difficulty in fluid dynamics regarding singular limits in bounded domains with capillary effects. It clarifies the structural difference between vanishing viscosity and vanishing capillarity limits regarding boundary layers.
• Quantum Fluids: Since the EK system with specific coefficients models Quantum Hydrodynamics (QHD) and Bose-Einstein Condensates, these results provide a rigorous mathematical foundation for the semiclassical limit of quantum fluids in confined geometries (e.g., quantum dots or traps with hard walls).
• Methodological Advance: The introduction of the augmented relative energy with a boundary layer correction for dispersive systems offers a new template for analyzing other singular limits involving higher-order derivatives and non-trivial boundary conditions.

In summary, Antonelli and Cacchio successfully bridge the gap between the dispersive Euler-Korteweg model and the inviscid Euler model in bounded domains, proving convergence without restrictive energy concentration assumptions by exploiting the specific weak nature of the capillary boundary layer.