[Paper Review] On Navier-Stokes-Korteweg and Euler-Korteweg Systems: Application to Quantum Fluids Models

Didier Bresch, Marguerite Gisclon+1 more|arXiv (Cornell University)|Mar 28, 2017

Navier-Stokes equation solutions28 references4 citations

TL;DR

This paper extends relative entropy methods to the Navier-Stokes-Korteweg and Euler-Korteweg systems with density-dependent viscosities satisfying the BD relation, enabling weak-strong uniqueness and proving that global weak solutions of the quantum Navier-Stokes system converge to dissipative solutions of the quantum Euler system as viscosity vanishes. The key contribution is a simplified relative entropy formulation via an augmented system, reducing assumptions on the capillary coefficient compared to prior work.

ABSTRACT

In this paper, the main objective is to generalize to the Navier-Stokes-Korteweg (with density dependent viscosities satisfying the BD relation) and Euler-Korteweg systems a recent relative entropy [proposed by D. Bresch, P. Noble and J.--P. Vila, (2016)] introduced for the compressible Navier-Stokes equations with a linear density dependent shear viscosity and a zero bulk viscosity. As a concrete application, this helps to justify mathematically the convergence between global weak solutions of the quantum Navier-Stokes system [recently obtained simultaneously by I. Lacroix-Violet and A. Vasseur (2017)] and dissipative solutions of the quantum Euler system when the viscosity coefficient tends to zero: This selects a dissipative solution as the limit of a viscous system. We also get weak-strong uniqueness for the Quantum-Euler and for the Quantum-Navier-Stokes equations. Our results are based on the fact that Euler-Korteweg systems and corresponding Navier--Stokes-Korteweg systems can be reformulated through an augmented system such as the compressible Navier-Stokes system with density dependent viscosities satisfying the BD algebraic relation. This was also observed recently [by D. Bresch, F. Couderc, P. Noble and J.--P. Vila, (2016)] for the Euler-Korteweg system for numerical purposes. As a by-product of our analysis, we show that this augmented formulation helps to define relative entropy estimates for the Euler Korteweg systems in a simplest way compared to recent works [See D. Donatelli, E. Feireisl, P. Marcati (2015) and J. Giesselmann, C. Lattanzio, A.-E. Tzavaras (2017)] with less hypothesis required on the capillary coefficient.

Motivation & Objective

• To generalize a relative entropy framework for compressible Navier-Stokes equations with density-dependent viscosity to the Navier-Stokes-Korteweg and Euler-Korteweg systems.
• To justify the convergence of global weak solutions of the quantum Navier-Stokes system to dissipative solutions of the quantum Euler system as viscosity tends to zero.
• To establish weak-strong uniqueness for both quantum Euler and quantum Navier-Stokes equations using the relative entropy approach.
• To simplify relative entropy estimates for Euler-Korteweg systems by introducing an augmented system formulation, requiring fewer hypotheses on the capillary coefficient than previous works.

Proposed method

• The authors reformulate the Euler-Korteweg and Navier-Stokes-Korteweg systems as augmented compressible Navier-Stokes systems with density-dependent viscosities satisfying the BD algebraic relation.
• They apply a recently developed relative entropy framework—originally for linear density-dependent shear viscosity and zero bulk viscosity—to the augmented system.
• The relative entropy functional is constructed to measure the distance between a weak solution of the viscous system and a strong solution of the inviscid system.
• The method relies on deriving energy-type inequalities using the augmented system structure, enabling control of the relative entropy evolution.
• Key estimates involve the capillary term $ \varepsilon^2 \rho \nabla \left( K(\rho)\Delta\rho + \frac{1}{2}K'(\rho)|\nabla\rho|^2 \right) $, with $ K(\rho) = \rho^s $, $ -1 \leq s \leq 0 $, to ensure integrability and convergence.
• The analysis proves that vanishing viscosity selects a unique dissipative solution, linking weak solutions of the viscous system to dissipative solutions of the inviscid system.

Experimental results

Research questions

• RQ1 Can the relative entropy method be extended to the Euler-Korteweg and Navier-Stokes-Korteweg systems with density-dependent viscosities satisfying the BD relation?
• RQ2 Does the global weak solution of the quantum Navier-Stokes system converge to a dissipative solution of the quantum Euler system as viscosity tends to zero?
• RQ3 Can weak-strong uniqueness be established for the quantum Euler and quantum Navier-Stokes equations using this relative entropy framework?
• RQ4 How does the augmented system formulation simplify relative entropy estimates for Euler-Korteweg systems compared to prior approaches?
• RQ5 What are the minimal assumptions on the capillary coefficient $ K(\rho) $ required for the relative entropy estimates to hold?

Key findings

• The relative entropy framework is successfully extended to the Navier-Stokes-Korteweg and Euler-Korteweg systems with density-dependent viscosities satisfying the BD relation.
• The global weak solution of the quantum Navier-Stokes system converges to a dissipative solution of the quantum Euler system as the viscosity coefficient tends to zero.
• Weak-strong uniqueness is established for both the quantum Euler and quantum Navier-Stokes equations using the relative entropy method.
• The augmented system formulation allows for relative entropy estimates with significantly reduced assumptions on the capillary coefficient $ K(\rho) $, particularly for $ K(\rho) = \rho^s $ with $ -1 \leq s \leq 0 $.
• The condition $ \mathcal{E}_{\text{EuK}}^{\text{GLT}} = 0 $ implies $ \mathcal{E}_{\text{EuK}} = 0 $ under the specified $ K(\rho) $, ensuring equivalence of the relative entropy and its generalized form.
• The analysis confirms that the dissipative solution is selected as the limit of the viscous system, providing a mathematical justification for the selection of physically relevant solutions in the inviscid limit.