On Navier–Stokes–Korteweg and Euler–Korteweg Systems: Application to Quantum Fluids Models

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.

Weak Solutions for the Compressible Navier-Stokes Equations with Density Dependent Viscosities

“…Remark. As mentioned in [14], the equation on µ(ρ) is important: By taking ψ = divϕ for all ϕ ∈ C ∞ 0 , we can write the equation satisfied by ∇µ(ρ) namely…”

Rankine–Hugoniot conditions for fluids whose energy depends on space and time derivatives of density

“…This is the case of the model of fluids endowed with capillarity (the Lagrangian depends on the density gradient) [21,22,23,24] and the model of fluids containing gas bubbles [25,26,39,27] (the Lagrangian depends on the material derivative of density). Mathematically equivalent models also appear in quantum mechanics where the nonlinear Schrödinger equation is reduced to the equations of capillary fluids via the Madelung transform [28,29,30], and in the long-wave theory of free surface flows [31,32,33] where the equations of motion (Serre-Green-Naghdi equations) have the form which is equivalent to the equations of bubbly fluids. We show that the Hamilton principle implies not only classical Rankine-Hugoniot conditions for the mass, momentum and energy, but also additional relations.…”

On the compactness of weak solutions to the Navier–Stokes–Korteweg equations for capillary fluids

“…then it is possible to overcome this difficulty by considering an auxiliary system written in terms of an effective velocity field where the Korteweg tensor vanishes. Note that this relation plays a crucial role in the theory, see for example [13] where the authors study the vanishing viscosity limit for the Quantum Navier-Stokes equations, or [5] where (1.7) is extensively exploited to construct the approximating system and [9] where numerical methods are performed. We stress that in (1.1)-(1.2) the viscosity and capillarity coefficients do not satisfy the relation (1.7) and hence in this paper we cannot rely on a similar analysis.…”

“…Non uniqueness results by using convex integration methods has been proved in [16]. Relative entropy methods to study singular limits for the equations (1.4)-(1.6) have been exploited in [13,16,23,18], in particular we mention the incompressible limit in [1] in the quantum case, the quasineutral limit [17] for the constant capillarity case and the vanishing viscosity limit in [13]. Finally, the analysis of the long time behaviour for the isothermal Quantum-Navier-Stokes equations has been performed in [14].…”