The maximal regularity and its application to a multi-dimensional non-conservative viscous compressible two-fluid model with capillarity effects in $L^{ p}$-type framework
The present paper is the continuation of work \cite{XC}, devoted to extending it to a critical functional framework which is not related to the energy space. Employing the special dissipative structure of the non-conservative viscous compressible two-fluid model with capillarity effects, we first exploit the maximal regularity estimates for the corresponding linearized system in all frequencies which behaves like the heat equation. Then we construct the global well-posedness for the multi-dimensional model when the initial data are close to a stable equilibrium state in the sense of suitable $L^{ p}$-type Besov norms. As a consequence, this allows us to work in the framework of Besov space with negative regularity indices and this fact is particularly important when the initial data are large highly oscillating in physical dimensions $N= 2, 3$. Furthermore, based on a refined time weighted inequalities in the Fourier spaces, we also establish optimal time decay rates for the constructed global solutions under a mild additional decay assumption involving only the low frequencies of the initial data.
The present paper deals with the Cauchy problem of a compressible generic two-fluid model with capillarity effects in any dimension $N\geq2$. We first study the unique global solvability of the model in spaces with critical regularity indices with respect to the scaling of the associated equations. Due to the presence of the capillary terms, we exploit the parabolic properties of the linearized system for all frequencies which enables us to apply contraction mapping principle to show the unique global solvability of strong solutions close to a stable equilibrium state. Furthermore, under a mil…
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 …
In this paper we study existence and regularity of solutions to the capillarity problem for compressible liquids in a tube. We introduce an appropriate space of functions of bounded variation, in which the energy functional recently introduced by Robert Finn can be defined. We prove existence of a locally Lipschitz minimiser in this class.
We are concerned with an isothermal model of viscous and capillary compressible fluids derived by J. E. Dunn and J. Serrin (1985), which can be used as a phase transition model. Compared with the classical compressible Navier-Stokes equations, there is a smoothing effect on the density that comes from the capillary terms. First, we prove that the global solutions with critical regularity that have been constructed in [11] by the second author and B. Desjardins (2001), are Gevrey analytic. Second, we extend that result to a more general critical L p framework. As a consequence, we obtain algebr…
We consider a nonlocal version of the quasi-static Navier-Stokes-Korteweg equations with a non-monotone pressure law. This system governs the low-Reynolds number dynamics of a compressible viscous fluid that may take either a liquid or a vapour state. For a porous domain that is perforated by cavities with diameter proportional to their mutual distance the homogenization limit is analyzed. We extend the results for compressible one-phase flow with polytropic pressure laws and prove that the effective motion is governed by a nonlocal version of the Cahn-Hilliard equation. Crucial for the analys…
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…
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 ex…
In this article, we consider the compressible Navier-Stokes equation with density dependent viscosity coefficients and a term of capillarity introduced by Coquel et al in \cite{5CR}. This model includes at the same time the barotropic Navier-Stokes equations with variable viscosity coefficients, shallow-water system and the model of Rohde. We first study the well-posedness of the model in critical regularity spaces with respect to the scaling of the associated equations. In a functional setting as close as possibleto the physical energy spaces, we prove global existence of solutions close to a…
In the first part of this paper, we prove the existence of global strong solution for Korteweg system in one dimension. In the second part, motivated by the processes of vanishing capillarity-viscosity limit in order to select the physically relevant solutions for a hyperbolic system, we show that the global strong solution of the Korteweg system converges in the case of a $γ$ law for the pressure ($P(ρ)=aρ^γ$, $γ>1$) to entropic solution of the compressible Euler equations. In particular it justifies that the Korteweg system is suitable for selecting the physical solutions in the case where t…
This work is devoted to prove existence of global weak solutions for a general isothermal model of capillary fluids derived by J.- E Dunn and J. Serrin (1985) [6], which can be used as a phase transition model. We improve the results of [5] by showing the existence of global weak solution in dimension two for initial data in the energy space, close to a stable equilibrium and with specific choices on the capillary coefficients. In particular we are interested in capillary coefficients approximating a constant capillarity coefficient. To finish we show the existence of global weak solution in d…
We are concerned with the time decay rates of strong solutions to a non-conservative compressible viscous two-phase fluid model in the whole space R3. Compared to the previous related works, the main novelty of this paper lies in the fact that it provides a general framework that can be used to extract the optimal decay rates of the solution as well as its all-order spatial derivatives from one-order to the highest-order, which are the same as those of the heat equation. Furthermore, for well-chosen initial data, we also show the lower bounds on the decay rates. Our methods mainly consist of H…
The present paper deals with the Cauchy problem of a multi-dimensional non-conservative viscous compressible two-fluid system. We first study the well-posedness of the model in spaces with critical regularity indices with respect to the scaling of the associated equations. In the functional setting as close as possible to the physical energy spaces, we prove the unique global solvability of strong solutions close to a stable equilibrium state. Furthermore, under a mild additional decay assumption involving only the low frequencies of the data, we establish the time decay rates for the construc…
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…
In this paper we are interested in the barotropic compressible Navier-Stokes system endowed with a non-local capillarity tensor depending on a small parameter $ε$ such that it heuristically tends to the local Korteweg system. After giving some physical motivations related to the theory of non-classical shocks (see [28]) we prove global well-posedness (in the whole space $R^d$ with $d\geq 2$) for the non-local model and we also prove the convergence, as $ε$ goes to zero, to the solution of the local Korteweg system.
A combined incompressible and vanishing capillarity limit in the barotropic compressible Navier-Stokes equations for smooth solutions is proved. The equations are considered on the two-dimensional torus with well prepared initial data. The momentum equation contains a rotational term originating from a Coriolis force, a general Korteweg-type tensor modeling capillary effects, and a density-dependent viscosity. The limiting model is the viscous quasi-geostrophic equation for the "rotated" velocity potential. The proof of the singular limit is based on the modulated energy method with a careful …
In this work we study the solution of the Riemann problem for the barotropic version of the conservative symmetric hyperbolic and thermodynamically compatible (SHTC) two-phase flow model introduced in \cite{Romenski2007,Romenski2009}. All characteristic fields are carefully studied and explicit expressions are derived for the Riemann invariants and the Rankine-Hugoniot conditions. Due to the presence of multiple characteristics in the system under consideration, non-standard wave phenomena can occur. Therefore we briefly review admissibility conditions for discontinuities and then discuss poss…
The Cauchy problem of the bipolar Navier-Stokes-Poisson system (1.1) in dimension three is considered. We obtain the pointwise estimates of the time-asymptotic shape of the solution, which exhibit generalized Huygens' principle as the Navier-Stokes system. This phenomenon is the the most important difference from the unipolar Navier-Stokes-Poisson system. Due to non-conservative structure of the system (1.1) and interplay of two carriers which counteracts the influence of electric field (a nonlocal term), some new observations are essential for the proof. We make full use of the conservative s…
In the present article we are interested in further investigations for the barotropic compressible Navier-Stokes system endowed with a non-local capillarity we studied in [7]. Thanks to an accurate study of the associated linear system using a Lagrangian change of coordinates, we provide more precise energy estimates in terms of hybrid Besov spaces naturally depending on a threshold frequency (which is determined in function of the physical parameter) distinguishing the low and the high regimes. It allows us in particular to prove the convergence of the solutions from the non-local to the loca…
We present a novel technique for the imposition of non-linear entropy conservative and entropy stable solid wall boundary conditions for the compressible Navier-Stokes equations in the presence of an adiabatic wall, or a wall with a prescribed heat entropy flow. The procedure relies on the formalism and mimetic properties of diagonal-norm, summation-by-parts, and simultaneous-approximation-term operators, and is a generalization of previous works on discontinuous interface coupling [1] and solid wall boundary conditions [2]. Using the method of lines, a semi-discrete entropy estimate for the e…
This paper addresses the issue of global existence of so-called $κ$-entropy solutions to the Navier-Stokes equations for viscous compressible and barotropic fluids with degenerate viscosities. We consider the three dimensional space domain with periodic boundary conditions. Our solutions satisfy the weak formulation of the mass and momentum conservation equations and also a generalization of the BD-entropy identity called: $κ$-entropy. This new entropy involves a mixture parameter $κ\in (0,1)$ between the two velocities $\bf{u}$ and $\bf{u} + 2\nablaφ(\varrho)$ (the latter was introduced by th…
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
Homogenization of the compressible Navier-Stokes equations via two-scale convergence in perforated domains
Augsburg University（奥格斯堡大学）；Institute of Mathematics of the Czech Academy of Sciences（捷克科学院数学研究所）；Friedrich-Alexander-Universität Erlangen-Nürnberg（埃尔朗根-纽伦堡弗里德里希·亚历山大大学）
We investigate global existence and optimal decay rates of a generic non-conservative compressible two--fluid model with general constant viscosities and capillary coefficients.The main novelty of this work is three--fold: First, for any integer $\ell\geq3$, we show that the densities and velocities converge to their corresponding equilibrium states at the $L^2$ rate $(1+t)^{-\frac{3}{4}}$, and the $k$($\in [1, \ell]$)--order spatial derivatives of them converge to zero at the $L^2$ rate $(1+t)^{-\frac{3}{4}-\frac{k}{2}}$, which are the same as ones of the compressible Navier--Stokes system, N…
In this article we present a novel staggered semi-implicit hybrid finite-volume/finite-element (FV/FE) method for the resolution of weakly compressible flows in two and three space dimensions. The pressure-based methodology introduced in Bermúdez et al. 2014 and Busto et al. 2018 for viscous incompressible flows is extended here to solve the compressible Navier-Stokes equations. Instead of considering the classical system including the energy conservation equation, we replace it by the pressure evolution equation written in non-conservative form. To ease the discretization of complex spatial d…
