In this note, we propose the first mathematical derivation of a macroscopic Baer-Nunziato type system for compressible two-phase flows allowing two pressure state laws depending on the different phases. By doing so, we extend the results obtained by the first author and M. Hillairet (Annales ENS 2019) to cover this important physical situation. A relaxation term in the mass fraction equation is obtained without closure assumptions contrarily to theoretical-physics literature dedicated to mixture theory, see for instance "Thermo-Fluid dynamics of Two Fluid Flows" by M. Ishii. The relaxation parameter is linked to the viscosities of the different fluids (which may be small for applications) and the relaxed quantity is linked to the laws chosen at interfaces of the two-fluid system at a mesoscale.
The Euler-Maxwell system as a hydrodynamic model for plasma physics to describe the dynamics of the compressible electrons in a constant charged non-moving ion background is studied. The global smooth flow with small amplitude is constructed in three space dimensions when the electron velocity relaxation is present. The speed of the electrons flow trending to uniform equilibrium is obtained. The pointwise behavior of solutions to the linearized homogeneous system in the frequency space is also investigated in detail.
The work is devoted to the relaxation limit in larger Besov spaces for compressible Euler equations, which contains previous results in Sobolev spaces and Besov spaces with critical regularity. Such an extension depends on a revision of commutator estimates and an elementary fact which indicates the connection between homogeneous and inhomogeneous Chemin-Lerner spaces.
The relaxation limit in critical Besov spaces for the multidimensional compressible Euler equations is considered. As the first step of this justification, the uniform (global) classical solutions to the Cauchy problem with initial data close to an equilibrium state are constructed in the Chemin-Lerner's spaces with critical regularity. Furthermore, it is shown that the density converges towards the solution to the porous medium equation, as the relaxation time tends to zero. Several important estimates are achieved, including a crucial estimate of commutator.
In this paper, we rigorously derive a new compressible multifluid system from compressible Navier-Stokes equations with density-dependent viscosity in the one-dimensional in space setting. More precisely, we propose and mathematically derive a generalization of the usual one velocity Baer-Nunziato model with a new relaxation term in the PDE governing the volume fractions. This new relaxation term encodes the change of viscosity and pressure between the different fluids. For the reader's convenience, we first establish a formal derivation in the bifluid setting using a WKB decomposition and the…
The asymptotic stability of rarefaction wave for 1-d relaxed compressible isentropic Navier-Stokes equations is established. For initial data with different far-field values, we show that there exists a unique global in time solution. Moreover, as time goes to infinity, the obtained solutions are shown to converge uniformly to rarefaction wave solution of $p$-system with corresponding Riemann initial data. The proof is based on $L^2$ energy methods.
In this paper, we study the relaxation limit of the relaxing Cauchy problem for non-isentropic compressible Euler equations with damping in multi-dimensions. We prove that the velocity of the relaxing equations converges weakly to that of the relaxed equations, while other variables of the relaxing equations converges strongly to the corresponding variables of the relaxed equations. We show that as relaxation time approaches 0, there exists an initial layer for the ill-prepared data, the convergence of the velocity is strong outside the layer; while there is no initial layer for the well-prepa…
In this paper, the compressible immiscible two-phase flow with relaxation is investigated, this model can be regarded as a natural modification of Jin-Xin relaxation scheme proposed and developed by S.Jin and Z.P.Xin([Comm.Pure Appl.Math., 48,1995]) in view of the numerical approximation of conservation laws. Given any entropy solution consists of two different families of shocks interacting at some positive time for the standard two-phase compressible Euler equations, it is proved that such entropy solution is the sharp interface limit for a family global strong solutions of the modified Jin-…
On the relaxation towards mechanical equilibrium for two-pressure compressible flows
Université Paris Cité（巴黎西岱大学）；Sorbonne Université（索邦大学）；CNRS（法国国家科学研究中心）；IMJ-PRG（数学研究所巴黎分部）；Université Paul Sabatier（图卢兹第三大学）；Institut de Mathématiques de Toulouse（图卢兹数学研究所）；Université Claude Bernard Lyon 1（里昂第一大学）；Institut Camille Jordan（卡米尔·若尔当研究所）
This paper is concerned with the 1-D compressible Euler-Poisson equations with moving physical vacuum boundary condition. It is usually used to describe the motion of a self-gravitating inviscid gaseous star. The local well-posedness of classical solutions is established in the case of the adiabatic index $1<γ<3$.
We show how to obtain general nonlinear aggregation-diffusion models, including Keller-Segel type models with nonlinear diffusions, as relaxations from nonlocal compressible Euler-type hydrodynamic systems via the relative entropy method. We discuss the assumptions on the confinement and interaction potentials depending on the relative energy of the free energy functional allowing for this relaxation limit to hold. We deal with weak solutions for the nonlocal compressible Euler-type systems and strong solutions for the limiting aggregation-diffusion equations. Finally, we show the existence of…
The paper addresses a two-temperature model for simulating compressible two-phase flow taking into account diffusion processes related to the heat conduction and viscosity of the phases. This model is reduced from the two-phase Baer-Nunziato model in the limit of complete velocity relaxation and consists of the phase mass and energy balance equations, the mixture momentum equation, and a transport equation for the volume fraction.Terms describing effects of mechanical relaxation, temperature relaxation, and thermal conduction on volume fraction evolution are derived and demonstrated to be sign…
This work considers a compressible, viscous, heat-conducting fluid exhibiting thermal relaxation according to Christov's constitutive heat transfer law (C. I. Christov, Mech. Research Comm. 36 (2009)), which is of Cattaneo type. The resulting evolution equations are known as Cattaneo-Christov systems. In this contribution, it is shown that Cattaneo-Christov systems for one-dimensional compressible fluid flow are strictly dissipative. The proof is based on the verification of a genuine coupling condition for hyperbolic-parabolic systems with viscous and relaxation effects combined as well as on…
We study the geodesic problem on the group of diffeomorphism of a domain M$\subset$Rd, equipped with the H(div) metric. The geodesic equations coincide with the Camassa-Holm equation when d=1, and represent one of its possible multi-dimensional generalizations when d>1. We propose a relaxation {à} la Brenier of this problem, in which solutions are represented as probability measures on the space of continuous paths on the cone over M. We use this relaxation to prove that smooth H(div) geodesics are globally length minimizing for short times. We also prove that there exists a unique pressure fi…
The present article proposes a diffuse interface model for compressible multicomponent flows with transport phenomena of mass, momentum and energy (i.e., mass diffusion, viscous dissipation and heat conduction). The model is reduced from the seven-equation Baer-Nuziato type model with asymptotic analysis in the limit of instantaneous mechanical relaxations. The main difference between the present model and the Kapila's five-equation model consists in that different time scales for pressure and velocity relaxations are assumed, the former being much smaller than the latter. Thanks to this assum…
This short paper is an introduction of the memoir recently written by the two authors (see [D.Bresch., P.--E. Jabin, arXiv:1507.04629, (2015)]) which concerns the resolution of two longstanding problems: Global existence of weak solutions for compressible Navier-Stokes equations with thermodynamically unstable pressure and with anisotropic stress tensor. We focus here on a Stokes-like system which can for instance model flows in a compressible tissue in biology or in a compressible porous media in petroleum engineering. This allows us to explain, on a simpler but still relevant and important s…
A new concept of semi-compressible fluids is introduced for slightly compressible visco-elastic fluids (typically rather liquids than gasses) where mass density variations are negligible in some sense, while being directly controlled by pressure which is very small in comparison with the elastic bulk modulus. The physically consistent fully Eulerian models with specific dispersion of pressure-wave speed are devised. This contrasts to the so-called quasi-incompressible fluids which are described not physically consistently and, in fact, only approximate ideally incompressible ones in the limit.…
An important problem in the theory of compressible gas flows is to understand the singular behavior of vacuum states. The main difficulty lies in the fact that the system becomes degenerate at the vacuum boundary, where the characteristics coincide and have unbounded derivative. In this paper, we overcome this difficulty by presenting a new formulation and new energy spaces. We establish the local in time well-posedness of one-dimensional compressible Euler equations for isentropic flows with the physical vacuum singularity in some spaces adapted to the singularity.
We proposed rigorous definitions of Radon measure solutions for boundary value problems of steady compressible Euler equations which modeling hypersonic-limit inviscid flows passing two-dimensional ramps, and their interactions with still gas and pressureless jets. We proved the Newton-Busemann pressure law of drags of body in hypersonic flow, and constructed various physically interesting measure solutions with density containing Dirac measures supported on curves, also exhibited examples of blow up of certain measure solutions. This established a mathematical foundation for applications in e…
We prove a priori estimates for the three-dimensional compressible Euler equations with moving {\it physical} vacuum boundary, with an equation of state given by $p(ρ) = C_γρ^γ$ for $γ>1$. The vacuum condition necessitates the vanishing of the pressure, and hence density, on the dynamic boundary, which creates a degenerate and characteristic hyperbolic {\it free-boundary} system to which standard methods of symmetrizable hyperbolic equations cannot be applied.
This paper is concerned with the three dimensional compressible Euler--Poisson equations with moving physical vacuum boundary condition. This fluid system is usually used to describe the motion of a self-gravitating inviscid gaseous star. The local existence of classical solutions for initial data in certain weighted Sobolev spaces is established in the case that the adiabatic index satisfies $1 < γ< 3$.
The full compressible magnetohydrodynamic equations can be derived formally from the complete electromagnetic fluid system in some sense as the dielectric constant tends to zero. This process is usually referred as magnetohydrodynamic approximation in physical books. In this paper we justify this singular limit rigorously in the framework of smooth solutions for well-prepared initial data.
In this paper, we consider the initial-boundary value problem to the compressible Navier-Stokes equations for ideal gases without heat conduction in the half space or outside a fixed ball in $\mathbb R^N$, with $N\geq1$. We prove that any classical solutions $(ρ, u, θ)$, in the class $C^1([0,T]; H^m(Ω))$, $m>[\frac N2]+2$, with bounded from below initial entropy and compactly supported initial density, which allows to touch the physical boundary, must blow-up in finite time, as long as the initial mass is positive. This paper extends the classical reault by Xin [CPAM, 1998], in which the Cauch…
In this paper, we prove the leading term of time-asymptotics of the moving vacuum boundary for compressible inviscid flows with damping to be that for Barenblatt self-similar solutions to the corresponding porous media equations obtained by simplifying momentum equations via Darcy's law plus the possible shift due to the movement of the center of mass, in the one-dimensional and three-dimensional spherically symmetric motions, respectively. This gives a complete description of the large time asymptotic behavior of solutions to the corresponding vacuum free boundary problems. The results obtain…
An important problem in gas and fluid dynamics is to understand the behavior of vacuum states, namely the behavior of the system in the presence of vacuum. In particular, physical vacuum, in which the boundary moves with a nontrivial finite normal acceleration, naturally arises in the study of the motion of gaseous stars or shallow water. Despite its importance, there are only few mathematical results available near vacuum. The main difficulty lies in the fact that the physical systems become degenerate along the vacuum boundary. In this paper, we establish the local-in-time well-posedness of …
We study the energy balance for weak solutions of the three-dimensional compressible Navier--Stokes equations in a bounded domain. We establish an $L^p$-$L^q$ regularity conditions on the velocity field for the energy equality to hold, provided that the density is bounded and satisfies $\sqrtρ \in L^\infty_t H^1_x$. The main idea is to construct a global mollification combined with an independent boundary cut-off, and then take a double limit to prove the convergence of the resolved energy.
It is well-known that the rarefaction wave, one of the basic wave patterns to the hyperbolic conservation laws, is nonlinearly stable to the one-dimensional compressible Navier-Stokes equations (cf. [14,15,12,17]). In the present paper we proved the time-asymptotically nonlinear stability of the planar rarefaction wave to the two-dimensional compressible and isentropic Navier-Stokes equations, which gives the first stability result of the planar rarefaction wave to the multi-dimensional system with physical viscosities.
We study the motion of the steady compressible heat conducting viscous fluid in a bounded three dimensional domain governed by the compressible Navier-Stokes-Fourier system. Our main result is the existence of a weak solution to these equations for arbitrarily large data. A key element of the proof is a special approximation of the original system guaranteeing pointwise uniform boundedness of the density. Therefore the passage to the limit omits tedious technical tricks required by the standard theory. Basic estimates on the solutions are possible to obtain by a suitable choice of physically r…
This article is concerned with the local well-posedness problem for the compressible Euler equations in gas dynamics. For this system we consider the free boundary problem which corresponds to a physical vacuum. Despite the clear physical interest in this system, the prior work on this problemis limited to Lagrangian coordinates, in high regularity spaces. Instead, the objective of the present work is to provide a new, fully Eulerian approach to this problem, which provides a complete, Hadamard style well-posedness theory for this problem in low regularity Sobolev spaces. In particular we give…
We consider the non-isentropic compressible Navier-Stokes equations in two or three space dimensions for which the heat conduction of Fourier's law is replaced by Cattaneo's law and the classical Newtonian flow is replaced by a revised Maxwell flow. We show that a physical entropy exists for this new model. For two special cases, we show the global well-posedness of solutions with small initial data and the blow-up of solutions in finite time for a class of large initial data. Moreover, for vanishing relaxation parameters, the solutions (if it exists) are shown to converge to solutions of the …
In this paper, we establish a priori estimates for the three-dimensional compressible Euler equations with moving physical vacuum boundary, the $γ$-gas law equation of state for $γ=2$ and the general initial density $\ri \in H^5$. Because of the degeneracy of the initial density, we investigate the estimates of the horizontal spatial and time derivatives and then obtain the estimates of the normal or full derivatives through the elliptic-type estimates. We derive a mixed space-time interpolation inequality which play a vital role in our energy estimates and obtain some extra estimates for the …
The free-boundary compressible 1-D Euler equations with moving physical vacuum boundary are a system of hyperbolic conservation laws which are both characteristic and degenerate. The physical vacuum singularity (or rate-of-degeneracy) requires the sound speed $c= γρ^{γ-1}$ to scale as the square-root of the distance to the vacuum boundary, and has attracted a great deal of attention in recent years. We establish the existence of unique solutions to this system on a short time-interval, which are smooth (in Sobolev spaces) all the way to the moving boundary. The proof is founded on a new higher…
For the physical vacuum free boundary problem with the sound speed being $C^{{1}/{2}}$-H$\ddot{\rm o}$lder continuous near vacuum boundaries of the three-dimensional compressible Euler equations with damping, the global existence of spherically symmetric smooth solutions is proved, which are shown to converge to Barenblatt self-similar solutions of the porous media equation with the same total masses when initial data are small perturbations of Barenblatt solutions. The pointwise convergence with a rate of density, the convergence rate of velocity in supreme norm and the precise expanding rate…
This paper concerns the well-posedness theory of the motion of physical vacuum for the compressible Euler equations with or without self-gravitation. First, a general uniqueness theorem of classical solutions is proved for the three dimensional general motion. Second, for the spherically symmetric motions, without imposing the compatibility condition of the first derivative being zero at the center of symmetry, a new local-in-time existence theory is established in a functional space involving less derivatives than those constructed for three-dimensional motions in \cite{10',7,16'} by construc…
We extend the weak-strong uniqueness principle to general models of compressible viscous fluids near/on the vacuum. In particular, the physically relevant case of positive density with polynomial decay at infinity is considered.
For the physical vacuum free boundary problem with the sound speed being $C^{{1}/{2}}$-H$\ddot{\rm o}$lder continuous near vacuum boundaries of the one-dimensional compressible Euler equations with damping, the global existence of the smooth solution is proved, which is shown to converge to the Barenblatt self-similar solution for the the porous media equation with the same total mass when the initial data is a small perturbation of the Barenblatt solution. The pointwise convergence with a rate of density, the convergence rate of velocity in supereme norm and the precise expanding rate of the …
We consider the Navier-Stokes system describing the time evolution of a compressible barotropic fluid confined to a bounded spatial domain in the 3-D physical space, supplemented with the Navier's slip boundary conditions. It is shown that the class of global in time strong solutions is robust with respect to small perturbations of the initial data. Explicit qualitative estimates are given also in terms of the shape of the underlying physical domain, with applications to problems posed on thin cylinders.
The global existence of smooth solutions to the vacuum free boundary problem with physical singularity of compressible Euler equations with damping and gravity is proved in space dimensions $n=1, 2, 3$, for the initial data being small perturbations of the stationary solution. Moreover, the exponential decay of the velocity is obtained for $n=1, 2, 3$. The exponentially fast convergence of the density and vacuum boundary to those of the stationary solution is shown for $n=1$, and it is proved for $n=2, 3$ that they stay close to those of the stationary solution if they do so initially. The pro…
We consider the coupling between the equations of motion of an inviscid compressible fluid in space with an objective Cattaneo-type extension for the heat flux. These equations are written in quasilinear form and we determine which of the given formulations for the heat flux allows for the hyperbolicity of the system. This feature is necessary for a physically acceptable sense of well-posedness for the Cauchy problem of such system of equations.
In this paper, the smooth solution of the physical vacuum problem for the one dimensional compressible Euler equations with time-dependent damping is considered. Near the vacuum boundary, the sound speed is $C^{1/2}$-Hölder continuous. The coefficient of the damping depends on time, given by this form $\fracμ{(1+t)^λ}$, $λ$, $μ>0$, which decays by order $-λ$ in time. Under the assumption that $0<λ<1$, $0<μ$ or $λ=1$, $2<μ$, we will prove the global existence of smooth solutions and convergence to the modified Barenblatt solution of the related porous media equation with time-dependent dissipat…
We consider the Navier-Stokes-Fourier system describing the motion of a compressible viscous fluid in a container with impermeable boundary subject to time periodic heating and under the action of a time periodic potential force. We show the existence of a time periodic weak solution for arbitrarily large physically admissible data.
A linear system of differential equations describing a joint motion of a thermoelastic porous body with a sufficiently large Lame's constants (absolutelty rigid body) and a thermofluid, occupying porous space, is considered. The rigorous justification, under various conditions imposed on physical parameters, is fulfilled for homogenization procedures as the dimensionless size of the pores tends to zero, while the porous body is geometrically periodic. As the results we derive Darcy's system of filtration for thermofluid, depending on ratios between physical parameters. The proofs are based on …
