In this article, we study some fundamental properties of nonlinear waves and the Riemann problem of Euler’s relativistic system when the constitutive equation for energy is that of Synge for a ...monatomic rarefied gas or its generalization for diatomic gas. These constitutive equations are the only ones compatible with the relativistic kinetic theory for massive particles in the whole range from the classical to the ultra-relativistic regime. They involve modified Bessel functions of the second kind and this makes Euler’s relativistic system rather complex. Based on delicate estimates of the Bessel functions, we prove: (i) a limit on the speed of sound of
1
/
3
times the speed of light (which a fortiori implies subluminality, that is causality), (ii) the genuine non-linearity of the acoustic waves, (iii) the compatibility of Rankine–Hugoniot relations with the second law of thermodynamics (entropy growth through all Lax shocks), and (iv) the unique resolvability of the initial value problem of Riemann (if we include the possibility of vacuum as in the non-relativistic context).
This paper is concerned with the existence, uniqueness, and nonlinear stability of stationary solutions to the Cauchy problem of the full compressible Navier–Stokes–Korteweg system effected by the ...given mass source, the external force of general form, and the energy source in R3. Based on the weighted L2-method and some delicate L∞ estimates on solutions to the linearized problem, the existence and uniqueness of stationary solution are obtained by the contraction mapping principle. The proof of the stability result is given by an elementary energy method and relies on some intrinsic properties of the full compressible Navier–Stokes–Korteweg system.
Le sujet de cet article porte sur l'existence, l'unicité et la stabilité non linéaire des solutions stationnaires du problème de Cauchy d'un système compressible de Navier–Stokes–Korteweg, affecté par la source de masse, la force extérieure de forme générale, et la source d'énergie dans R3. Basées sur la méthode L2 à poids et certaines estimations délicates sur les solutions du système linéarisé, l'existence et l'unicité de solutions stationnaires sont obtenues par le principe d'une application contractant. La démonstration du résultat sur la stabilité est donnée par une méthode d'énergie élémentaire et dépend de certaines propriétés intrinsèques du système compressible de Navier–Stokes–Korteweg.
In this paper we consider the global existence and large-time behavior of solutions of one-dimensional viscous radiative and reactive gas with large initial data. In precedent studies the case for ...the initial-boundary value problem in bounded domain has been investigated and our main purpose focuses on the corresponding problem in unbounded domain. If the initial data is a large perturbation of a non-vacuum constant equilibrium state and is assumed to be without vacuum, mass concentrations, or vanishing temperatures, then we can show that its Cauchy problem admits a unique global smooth non-vacuum solution which tends time-asymptotically to such an equilibrium state. The key point in the analysis is to deduce the uniform positive lower and upper bounds on the specific volume and the absolute temperature.
Dynamics of the macroscopic motion of a compressible, viscous and heat-conducting, one-dimensional monatomic ionized gas can be described by a compressible Navier-Stokes type system. Compared with ...the corresponding compressible Navier-Stokes type systems modeling a viscous and heat conducting ideal polytropic gas 7,10,13–16,18,19,23,24,26,28,31,32, the real gases 17, nonlinear thermoviscoelasticity 5,6, and a viscous radiative and reactive gas 11,12,20,21,30, the dependence of the state of an ionized gas on the degree of ionization leads to the loss of concavity of the physical entropy in some small bounded domain. Such a property of the ionized gas makes it impossible to deduce the dissipative estimates on the first-order spatial derivative of the bulk velocity and the absolute temperature from the basic energy estimates obtained through the normalized entropy and, as a consequence, makes the construction of its global large solutions and the study of their large time behavior more difficult.
By making full use of the intrinsic structure of equations, the existence of global large smooth solutions to two types of initial boundary value problems of one-dimensional, compressible Navier-Stokes system for a viscous and heat-conducting ionized gas is established in 22 for a class of density and/or temperature dependent transport coefficients, but since the estimates obtained there depend on the time variable, the problem on their large time behavior remains unsolved and the main purpose of this paper is to give the precise description of the large time behavior of the global solutions constructed in 22. The key ingredient in our analysis is to deduce the uniform-in-time positive lower and upper bounds on the specific volume and the absolute temperature.
Although the decay in time estimates of the semi-group generated by the linearized Boltzmann operator without forcing have been well established, there is no corresponding result for the case with ...general external force. This paper is mainly concerned with the optimal decay estimates on the solution operator in some weighted Sobolev spaces for the linearized Boltzmann equation with a time dependent external force. No time decay assumption is made on the force. The proof is based on both the energy method through the macro-micro decomposition and the
L
p
-
L
q
estimates from the spectral analysis. The decay estimates thus obtained are applied to the study on the global existence of the Cauchy problem to the nonlinear Boltzmann equation with time dependent external force and source. Precisely, for space dimension
n
≥ 3, the global existence and decay rates of solutions to the Cauchy problem are obtained under the condition that the force and source decay in time with some rates. This time decay restriction can be removed for space dimension
n
≥ 5. Moreover, the existence and asymptotic stability of the time periodic solution are given for space dimension
n
≥ 5 when the force and source are time periodic with the same period.
We establish the global existence of small-amplitude solutions near a global Maxwellian to the Cauchy problem of the Vlasov-Maxwell-Boltzmann system for non-cutoff soft potentials with weak angular ...singularity. This extends the work of Duan et al.（2013）, in which the case of strong angular singularity is considered, to the case of weak angular singularity.
This paper is concerned with the construction of globally smooth solutions near a given global Maxwellian to the Cauchy problem of tile Vlasov-Poisson-Boltzmann system for non-cutoff hard potentials ...in three space dimensions without the neutral condition imposed on the initial perturbation. Our analysis is based on the time-weighted energy method and some delicate estimates.
We are concerned with the time-asymptotic behavior toward rarefaction waves for strong non-vacuum solutions to the Cauchy problem of the one-dimensional compressible Navier-Stokes equations with ...degenerate density-dependent viscosity. The case when the pressure p(ρ)=ργ and the viscosity coefficient μ(ρ)=ρα for some parameters α,γ∈R is considered. For α≥0, γ≥max{1,α}, if the initial data is assumed to be sufficiently regular, without vacuum and mass concentrations, we show that the Cauchy problem of the one-dimensional compressible Navier-Stokes equations admits a unique global strong non-vacuum solution, which tends to the rarefaction waves as time goes to infinity. Here both the initial perturbation and the strength of the rarefaction waves can be arbitrarily large. The proof is established via a delicate energy method and the key ingredient in our analysis is to derive the uniform-in-time positive lower and upper bounds on the specific volume.
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