The aim of this paper is to investigate the regime of high Mach number flows for compressible barotropic fluids of Korteweg type with density dependent viscosity. In particular we consider the models for isothermal capillary and quantum compressible fluids. For the capillary case we prove the existence of weak solutions and related properties for the system without pressure, and the convergence of the solution in the high Mach number limit. This latter is proved also in the quantum case for which a weak-strong uniqueness analysis is also discussed in the framework of the so-called "augmented" version of the system. Moreover, as byproduct of our results, in the case of a capillary fluid with a special choice of the initial velocity datum, we obtain an interesting property concerning the propagation of vacuum zones.
We study the inhomogeneous incompressible Navier-Stokes system endowed with a general capillary term. Thanks to recent methods based on Lagrangian change of variables, we obtain local well-posedness in critical Besov spaces (even if the integration index p is different from 2) and for variable viscosity and capillary terms. In the case of constant coefficients and for initial data that are perturbations of a constant state, we are able to prove that the lifespan goes to infinity as the capillary coefficient goes to zero, connecting our result to the global existence result obtained by Danchin …
Global existence and uniqueness of the density-dependent incompressible Navier-Stokes-Korteweg system with variable capillarity and viscosity coefficients
In the first main result of this paper we prove that one can approximate discontinious solutions of the 1d Navier Stokes system with solutions of the 1d Navier-Stokes-Korteweg system as the capilarity parameter tends to 0. Moreover, we allow the viscosity coefficients $μ$ = $μ$ ($ρ$) to degenerate near vaccum. In order to obtain this result, we propose two main technical novelties. First of all, we provide an upper bound for the density verifing NSK that does not degenerate when the capillarity coefficient tends to 0. Second of all, we are able to show that the positive part of the effective v…
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. Vasse…
In this paper we analyze the high friction regime for the Navier Stokes Korteweg equations for multicomponent systems. According to the shape of the mixing and friction terms, we shall perform two limits: the high friction limit toward an equilibrium system for the limit densities and the barycentric velocity, and, after an appropriate time scaling, the diffusive relaxation toward parabolic, gradient flow equations for the limit densities. The rigorous justification of these limits is done by means of relative entropy techniques in the framework of weak, finite energy solutions of the relaxati…
The aim of this paper is to investigate the singular relaxation limits for the Euler-Korteweg and the Navier-Stokes-Korteweg system in the high friction regime. We shall prove that the viscosity term is present only in higher orders in the proposed scaling and therefore it does not affect the limiting dynamics, and the two models share the same equilibrium equation. The analysis of the limit is carried out using the relative entropy techniques in the framework of weak, finite energy solutions of the relaxation models converging toward smooth solutions of the equilibrium. The results proved her…
Existence of global strong solution for the compressible Navier-Stokes system and the Korteweg system in two-dimension
本文研究了N=2维粘性可压缩正压流体。研究了可压缩Navier-Stokes方程组和Korteweg方程组在大初值条件下的整体强解的存在性问题。在第一种情况下，我们感兴趣的是稍微推广了V.A. Vaigant和A. V. Kazhikhov在\cite{VG}中关于在环面中适当选择粘性系数（$μ(ρ)=μ>0$和$λ(ρ)=λρ^β$ with $β>3$）的二维全局强解的存在性。我们将通过只假设$β>2$来削弱$β$的条件，实质上是通过利用Coifman等人在\cite{4M}中引入的交换子估计，并使用\cite{VG}中的\textit{effective velocity}的概念。在第二种情况下，我们研究了具有退化粘性系数和摩擦项的Korteweg方程组在大初值条件下整体强解的存在性。
In this paper, we consider global weak solutions to com-pressible Navier-Stokes-Korteweg equations with density dependent viscosities , in a periodic domain $Ω= \mathbb T^3$, with a linear drag term with respect to the velocity. The main result concerns the exponential decay to equilibrium of such solutions using log-sobolev type inequalities. In order to show such a result, the starting point is a global weak-entropy solutions definition introduced in D. Bresch, A. Vasseur and C. Yu [12]. Assuming extra assumptions on the shear viscosity when the density is close to vacuum and when the densit…
The global solutions with large initial data for the isothermal compressible fluid models of Korteweg type has been studied by many authors in recent years. However, little is known of global large solutions to the nonisothermal compressible fluid models of Korteweg type up to now. This paper is devoted to this problem, and we are concerned with the global existence of smooth and non-vacuum solutions with large initial data to the Cauchy problem of the one-dimensional nonisothermal compressible fluid models of Korteweg type. The case when the viscosity coefficient $μ(ρ)=ρ^α$, the capillarity c…
This paper is concerned with the large-time behavior of solutions to the Cauchy problem of the one-dimensional compressible fluid models of Korteweg type with density- and temperature-dependent viscosity, capillarity, and heat conductivity coefficients, which models the motions of compressible viscous fluids with internal capillarity. We show that the combination of the viscous contact wave with two rarefaction waves is asymptotically stable with a large initial perturbation if the strength of the composite wave and the heat conductivity coefficient satisfy some smallness conditions. The proof…
Low Mach number limit for the Navier--Stokes--Korteweg equations with a stationary force
School of Mathematics, Nanjing University（南京大学数学系）；The Institute of Mathematical Sciences, The Chinese University of Hong Kong（香港中文大学数学科学研究所）；School of Mathematical Sciences, Ocean University of China（中国海洋大学数学科学学院）
On the Cauchy problem for the multi-dimensional compressible Navier-Stokes-Korteweg system: Global strong solutions with arbitrarily large initial data
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学学院）
Vanishing viscosity limit for $n\times n$ hyperbolic system of conservation laws in 1-d with nonlinear viscosity: Part-I Uniform BV estimates
Harish-Chandra Research Institute, A CI of Homi Bhabha National Institute（哈里什-昌德拉研究所，霍米·巴巴国立学院成员机构）；Istituto di Matematica Applicata e Tecnologie Informatiche “Enrico Magenes” - Consiglio Nazionale delle Ricerche（恩里科·马格内斯应用数学与信息技术研究所-意大利国家研究委员会）
This work is devoted to prove the existence of global strong solution in dimension $N\geq 2$ for a isothermal model of capillary fluids derived by J.E Dunn and J.Serrin (1985) (see \cite{fDS}), which can be used as a phase transition model. We will restrict us to the case of the so called compressible Navier-Stokes system with quantum pressure which corresponds to consider the capillary coefficient $κ(ρ)=\frac{κ_1}ρ$ with $κ_1>0$. In a first part we prove the existence of strong solution in finite time for large initial data with a precise bound by below on the life span $T^*$. This one depend…
On the large-time behavior of strong solutions to the generalized compressible Navier-Stokes-Korteweg system in 2D and 3D for arbitrarily large initial data
We present a new proof for the equivalence of potential theoretic weak solutions and viscosity solutions to the $\texttt{p}(\cdot)$-Laplace equation in $\mathbb{R}^n$. The proof of the equivalence in the variable exponent case in Euclidean space was first given by Juutinen, Lukkari, and Parviainen (2010) and extended the equivalence of potential theoretic weak solutions and viscosity solutions to the $\texttt{p}$-Laplace equation in $\mathbb{R}^n$, given by Juutinen, Lindqvist, and Manfredi (2001). In both the fixed exponent case and the variable exponent case, the main argument is based on th…
We consider the compressible Navier-Stokes system describing the motion of a viscous fluid confined to a straight layer $Ω_δ=(0,δ)\times\mathbb{R}^2$. We show that the weak solutions in the 3D domain converge strongly to the solution of the 2D incompressible Navier-Stokes equations (Euler equations) when the Mach number $ε$ tends to zero as well as $δ\rightarrow 0$ (and the viscosity goes to zero).
In this paper, we investigate the low Mach and low Froude numbers limit for the compressible Navier-Stokes equations with degenerate, density-dependent, viscosity coefficient, in the strong stratification regime. We consider the case of a general pressure law with singular component close to vacuum, and general ill-prepared initial data. We perform our study in the three-dimensional periodic domain. We rigorously justify the convergence to the generalised anelastic approximation, which is used extensively to model atmospheric flows.
The low Mach number limit for one-dimensional non-isentropic compressible Navier-Stokes system without viscosity is investigated, where the density and temperature have different asymptotic states at far fields. It is proved that the solution of the system converges to a nonlinear diffusion wave globally in time as Mach number goes to zero. It is remarked that the velocity of diffusion wave is proportional with the variation of temperature. Furthermore, it is shown that the solution of compressible Navier-Stokes system also has the same phenomenon when Mach number is suitably small.
In this paper we study the incompressible inviscid limit for a compressible micro-polar model. We prove that the weak solution of the compressible micro-polar system converges to the solution of the Navier-Stokes equations (Euler equations) in the limit of small Mach number (and vanishing viscosity).
As a framework for handling low Mach number limits we consider a notion of measure-valued solution of the incompressible Euler system which explicitly formulates the usually suppressed dependence on the pressure variable. We clarify in which sense such a measure-valued solution is generated as a low Mach limit and state sufficient conditions for this convergence. For the special case of pressure-free solutions we are able to give such sufficient conditions only on the incompressible solution itself. As a necessary condition for low Mach limits we obtain a Jensen-type inequality. The same neces…
In the present paper we study the incompressible and fast rotation limit for the barotropic Navier-Stokes equations with Coriolis force, in the case when the Mach number $\rm Ma$ is large with respect to the Rossby number $\rm Ro$: namely, we focus on the regime ${\rm Ro}\ll{\rm Ma}$. For this, we follow a recent approach by Danchin and Mucha in \cite{D-M} and take also a large bulk viscosity coefficient. We prove that the limit dynamics is described by an incompressible Navier-Stokes type equation, recasted in the vorticy formulation, where however an additional unknown, linked to density osc…
In this paper we prove global in time existence of weak solutions to zero Mach number systems arising in fluid mechanics. Relaxing a certain algebraic constraint between the viscosity and the conductivity introduced in [D. Bresch, E.H. Essoufi, and M. Sy, J. Math. Fluid Mech. 2007] gives a more complete answer to an open question formulated in [P.-L. Lions, Oxford 1998]. A new mathematical entropy shows clearly the existence of two-velocity hydrodynamics with a fixed mixture ratio. As an application of our result we first discuss a model of gaseous mixture extending the results of [P. Embid, C…
In this paper we study a singular limit problem for a Navier-Stokes-Korteweg system with Coriolis force, in the domain $\R^2\times\,]0,1[\,$ and for general ill-prepared initial data. Taking the Mach and the Rossby numbers to be proportional to a small parameter $\veps$ going to $0$, we perform the incompressible and high rotation limits simultaneously. Moreover, we consider both the constant capillarity and vanishing capillarity regimes. In this last case, the limit problem is identified as a $2$-D incompressible Navier-Stokes equation in the variables orthogonal to the rotation axis. If the …
Hydrodynamic systems for quantum fluids are systems for compressible fluid flows for which quantum effects are macroscopically relevant. We discuss how the presence of the dispersive tensor describing the quantum effects alters the acoustic dispersion at the example of the Quantum Hydrodynamic system (QHD). For the QHD system the dispersion relation is given by the Bogoliubov dispersion relation for weakly interacting Bose gases. We provide refined Strichartz estimates allowing for an accurate control of acoustic oscillations. Applications to the low Mach number limit for the quantum Navier-St…
