This is the first of a series of papers devoted to the initial value problem for the Euler system of compressible fluids and augmented versions containing higher-order terms. We encompass solutions that have finite total energy and enjoy a certain symmetry (for instance, plane symmetry); these solutions may have unbounded amplitude and contain cavitation regions in which the mass density vanishes. In the present paper, we are interested in dispersive shock waves and analyze the zero viscosity-capillarity limit associated with the Navier-Stokes-Korteweg system. Specifically, we establish the existence of finite energy solutions as well as their convergence toward entropy solutions to the Euler system. We encompass a broad class of nonlinear Navier-Stokes-Korteweg constitutive laws, which is determined by two main conditions relating the viscosity and capillarity coefficients, that is, on one hand the strong coercivity condition (as we call it) which provides a favorable sign for the integrated dissipation associated with an effective energy, and on the other hand the tame capillarity condition (as we call it), which restricts pointwise the strength of the capillarity relatively to the viscosity. Rather mild conditions on the growth of the constitutive functions are aso imposed, which are required in order to define finite energy weak solutions to the Navier-Stokes-Korteweg system, even in the presence of cavitation. Our method of proof relies on fine algebraic properties of the Euler system and combines together energy and effective energy estimates, dissipation and effective dissipation estimates, a nonlinear Sobolev inequality, high-integrability properties for the mass density and for the velocity, and compactness properties based on entropies.
Vanishing capillary limit for compressible Navier-Stokes-Korteweg system in a bounded interval with large initial data
School of Mathematics, South China University of Technology（华南理工大学数学学院）；School of Mathematics and Systems Science, Guangdong Polytechnic Normal University（广东技术师范大学数学与系统科学学院）
On Navier-Stokes-Korteweg and Euler-Korteweg Systems: Application to Quantum Fluids Models
本文的主要目的是将最近由D. Bresch，P. Noble和J. P. Vila，（2016）]，其针对具有线性密度相关剪切粘度和零体积粘度的可压缩Navier-斯托克斯方程引入。作为一个具体的应用，这有助于在数学上证明量子Navier-Stokes系统[最近由I. Lacroix-Violet和A. Vasseur（2017）]和粘性系数趋于零时量子欧拉系统的耗散解：这选择耗散解作为粘性系统的极限。我们还得到了量子Euler方程和量子Navier-斯托克斯方程的弱-强唯一性。我们的结果是基于Euler-Korteweg系统和相应的Navier-Stokes-Korteweg系统可以通过一个增广系统，如可压缩的Navier-Stokes系统的密度依赖的粘度满足BD代数关系的重新制定。这一点最近也被D. Bresch，F. Couderc，P. Noble and J. P. Vila，（2016）]用于数值目的的Euler-Korteweg系统。作为我们分析的一个副产品，我们表明，这种增广的公式有助于定义相对熵估计的欧拉Korteweg系统在一个最简单的方式相比，最近的作品[见D。Donatelli，E. Feireisl，P. Marcati（2015）和J. Giesselmann，C. Lattanzio，A. E. Tzavaras（2017）]，对毛细管系数的假设较少。
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。
Existence of weak solution for compressible fluid models of Korteweg type
本文证明了由J. - E. Dunn和J. Serrin（1985）[6]，它可以用作相变模型。我们改进了文献[5]的结果，证明了在能量空间中，对于接近稳定平衡的初始数据，在特定的毛细系数选择下，存在二维整体弱解。特别是，我们对接近恒定毛细系数的毛细系数感兴趣。最后，我们证明了在能量空间中，对于一种特殊类型的毛细管系数，在大的初始数据下，在一维上整体弱解的存在性。
Convergence of a low order non-local Navier-Stokes-Korteweg system: the order-parameter model
本文考虑C. Rohde在Bandon，Lin和Rogers的作品之后，称为顺序参数模型，其目的是减少在经典局部Korteweg系统（涉及三阶导数）或非局部系统（也由Rohde在Van der Waals的作品之后介绍，并且涉及卷积算子）的情况下遇到的数值困难。我们证明了这个系统有一个唯一的全球解决方案的初始数据接近平衡，我们精确地研究了这个解决方案的收敛到当地Korteweg模型。
Global existence and uniqueness of the density-dependent incompressible Navier-Stokes-Korteweg system with variable capillarity and viscosity coefficients
Analyticity and asymptotic behavior of solutions to the compressible Navier-Stokes-Korteweg equations with the zero sound speed in scaling critical spaces
Graduate School of Engineering Science, The University of Osaka（大阪大学工学研究科）；Faculty of Engineering, Shinshu University（信州大学工学部）
Vanishing capillarity limit of the Navier-Stokes-Korteweg system in one dimension with degenerate viscosity coefficient and discontinuous initial density
在本文的第一个主要结果中，我们证明了当毛细参数趋于0时，可以用一维Navier-Stokes-Korteweg方程组的解来逼近一维Navier-Stokes方程组的间断解。此外，我们还允许粘性系数$μ$ = $μ$（$ρ$）在真空附近退化。为了获得这一结果，我们提出了两个主要的技术创新。首先，我们提供了一个上界的密度验证NSK不退化时，毛细系数趋于0。其次，我们能够证明，有效速度的正部分是有界的一致w.r.t.毛细管系数。这在为密度提供一个下界时是至关重要的。第二个主要结果是在$ρ$ 0，1/ $ρ$ 0 $\in$ L $\infty$的条件下，给出了一维Navier-Stokes方程组唯一的有限能量整体强解的存在性.这最后一个结果发现自己在多相流的数学建模的上下文中的一个自然的应用。
On dynamic stability of energetically stable equilibria of the Navier-Stokes-Korteweg flows
Graduate School of Mathematical Sciences The University of Tokyo（东京大学大学院数理科学研究科）；Applied Physics Course, Department of Electrical, Electronic and Computer Engineering Gifu University（岐阜大学电气电子计算机工学科应用物理课程）；Center for Fundamental Education Kyushu Sangyo University（九州产业大学基础教育中心）
