Thermodynamically compatible conservation laws in the model of heat conducting radiating gas

Thermodynamic compatibility of the mass, momentum, and energy conservation laws that describe the motion of heat conducting gas in the presence of radiation heat exchange is considered. The study is based on the one-velocity two-component mathematical model of continuous compressible medium with the gas and radiation components. The work uses experimental data for radiation and other experimental data of modern physics.     

Dynamics of non-equilibrium phase boundaries in a heat conducting non-linearly elastic medium

The phase transformation of the first kind in a non-linearly elastic heat conducting medium is simulated by the relationships on a strong discontinuity. A generalization of the Stefan formulation is given. An existence condition for stationary flow, analogous to the Gibbs phase equilibrium condition, is obtained for non-equilibrium phase boundaries. A pure dilatational phase transition in a compressible fluid and pure shear transformation of the twinning type in non-linearly elastic crystals are considered as model examples. The problem of the structure is solved for closure of the system of relationships on the shock.

A phase transformation ordinarily turns out to be localized in a narrow domain of space and it can be simulated in terms of the conditions on a strong discontinuity /1/. Formulation of the problem of the static equilibrium of liquid phases as well as of liquid and (non-linearly elastic) solid phases was given by Gibbs, who proposed a phase equilibrium criterion and formulated appropriate conditions on the shock; the extension of the Gibbs conditions to the case of the equilibrium of two solid phases is known in both the linear /2/ and non-linear /3/ theories of elasticity. The dynamic problem of the propagation of the equilibrium phase boundary is considered in the Stefan formulation as a rule, including the assumption about the continuity of the density (the strain tensor component) on the shock; the thermal problem is here separated from the mechanical one. Simulating the interphasal surface on the shock the temperature fields are merged by using the well-known Stefan conditions as well as the phase equilibrium condition that reduces to giving the temperature on the front.

The purpose of this paper is to extend the Stefan-Gibbs formulation to the case of the motion of a coherent isothermal phase boundary in a non-linearly elastic heat conducting medium and to derive the dynamic analogue of the phase equilibrium condition (and the Stefan conditions) with possible dissipation at the transformation front. Two dissipative mechanisms are examined, viscous and kinetic. The case of equilibrium phase boundaries was investigated in /4–6/.     

A linearized heat‐conducting fluid and its quasi‐static approximation

A linearized flow of a compressible inviscid heat‐conducting fluid is considered and a comparison is made with its coupled/quasi‐static approximation. Copyright © 2005 John Wiley & Sons, Ltd.     

A model of heat conducting collisionless plasma

A model of heat conducting diluted and collisionless plasma embedded in a strong magnetic field is proposed in the framework of extended irreversible thermodynamics. Finally in the special case of polytropic fluid the dispersion relation for weak, plane disturbances is derived.     

Stationary Solutions of Compressible Navier–Stokes Equations with Slip Boundary Conditions

We consider the Navier–Stokes equations for a compressible, viscous fluid with heat–conduction in a bounded domain of IR² or IR³. Under the assumption that the external force field and the external heat supply are small we prove the existence and local uniqueness of a stationary solution satisfying a slip boundary condition. For the temperature we assume a Dirichlet or an oblique boundary condition.     

Global strong solutions for nonhomogeneous heat conducting Navier‐Stokes equations

We study the Cauchy problem of the 3‐dimensional nonhomogeneous heat conducting Navier‐Stokes equations with nonnegative density. First of all, we show that for the initial density allowing vacuum, the strong solution to the problem exists globally if the velocity satisfies the Serrin's condition. Then, under some smallness condition, we prove that there is a unique global strong solution to the 3D viscous nonhomogeneous heat conducting Navier‐Stokes flows. Our method relies upon the delicate energy estimates.     

On the existence of global weak solutions to the Navier–Stokes equations for viscous compressible and heat conducting fluids

The purpose of this work is to investigate the problem of global in time existence of sequences of weak solutions to the Navier–Stokes equations for viscous compressible and heat conducting fluids. A class of density and temperature dependent viscosity and conductivity coefficients is considered. This result extends P.-L. Lions' work in 1993 [P.-L. Lions, Compacité des solutions des équations de Navier–Stokes compressibles isentropiques, C. R. Acad. Sci. Paris, Sér. I 317 (1993) 115–120] restricted to barotropic flows, and provides weak solutions “à la Leray” to the full compressible model that includes internal energy evolution equation with thermal conduction effects. A partial answer is therefore given to this currently widely open problem, described for instance in P.-L. Lions' book [P.-L. Lions, Mathematical Topics in Fluid Dynamics, vol. 2, Compressible Models, Oxford Science Publication, Oxford, 1998]. The proof uses the generalization to the temperature dependent case, of a new mathematical entropy equality derived by the authors in [D. Bresch, B. Desjardins, Some diffusive capillary models of Korteweg type, C. R. Acad. Sci., Paris, Section Mécanique 332 (11) (2004) 881–886]. The construction scheme of approximate solutions, using on additional regularizing effects such as capillarity, is provided in [D. Bresch, B. Desjardins, On the construction of approximate solutions for 2D viscous shallow water model and for compressible Navier–Stokes models, J. Math. Pures Appl. 86 (4) (2006) 362–368], and allows to use the stability arguments of this paper.     

Evolution free boundary problem for equations of viscous compressible heat‐conducting capillary fluids

In the paper the global motion of a viscous compressible heat conducting capillary fluid in a domain bounded by a free surface is considered. Assuming that the initial data are sufficiently close to a constant state and the external force vanishes we prove the existence of a global‐in‐time solution which is close to the constant state for any moment of time. The solution is obtained in such Sobolev–Slobodetskii spaces that the velocity, the temperature and the density of the fluid have $W_2^{2+\alpha,1+\alpha/2}$\nopagenumbers\end , $W_2^{2+\alpha,1+\alpha/2}$\nopagenumbers\end and $W_2^{1+\alpha,1/2+\alpha/2}$\nopagenumbers\end —regularity with α∈(¾, 1), respectively. Copyright © 2001 John Wiley & Sons, Ltd.     

Global solutions for a one-dimensional problem in conducting fluids

A mathematical model of the motion of conducting fluids is studied in this paper. The dynamics of such fluids is described by the equations of compressible fluids coupled to the Maxwell’s equations. We prove global existence of strong solution for a one-dimensional initial-boundary value problem of this model (plane conducting flows) with general large data.     

Relativistic heat conducting magnetizable fluids: An extended thermodynamical approach

In the framework of the extended irreversible thermodynamics we examine the coupling between the magnetic field and the thermal field and their influence on the evolution of a heat conducting relativistic magnetizable fluid.     

Compactness of weak solutions to the three-dimensional compressible magnetohydrodynamic equations

The compactness of weak solutions to the magnetohydrodynamic equations for the viscous, compressible, heat conducting fluids is considered in both the three-dimensional space R³ and the three-dimensional periodic domains. The viscosities, the heat conductivity as well as the magnetic coefficient are allowed to depend on the density, and may vanish on the vacuum. This paper provides a different idea from [X. Hu, D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys. (2008), in press] to show the compactness of solutions of viscous, compressible, heat conducting magnetohydrodynamic flows, derives a new entropy identity, and shows that the limit of a sequence of weak solutions is still a weak solution to the compressible magnetohydrodynamic equations.     

Measure-valued Solutions of the Euler Equations for Ideal Compressible Polytropic Fluids

The existence of global measure-valued solutions to the Euler equations describing the motion of an ideal compressible and heat conducting fluid is proved. The motion is considered in a bounded domain Ω⊂ℝ³ with impermeable boundary. The solution is a limit of an approximate solution obtained by adding the sixth-order elliptic operator in the equation of momentum.     

Flow of a non‐linear (density‐gradient‐dependent) viscous fluid with heat generation,viscous dissipation and radiation

In this paper, we study the flow of a compressible (density‐gradient‐dependent) non‐linear fluid down an inclined plane, subject to radiation boundary condition. The convective heat transfer is also considered where a source term, similar to the Arrhenius type reaction, is included. The non‐dimensional forms of the equations are solved numerically and the competing effects of conduction, dissipation, heat generation and radiation are discussed. Copyright © 2008 John Wiley & Sons, Ltd.     

On dynamics of fluids in meteorology

We consider the full Navier-Stokes-Fourier system of equations on an unbounded domain with prescribed nonvanishing boundary conditions for the density and temperature at infinity. The topic of this article continues author’s previous works on existence of the Navier-Stokes-Fourier system on nonsmooth domains. The procedure deeply relies on the techniques developed by Feireisl and others in the series of works on compressible, viscous and heat conducting fluids.     

Global existence and the low Mach number limit for the compressible magnetohydrodynamic equations in a bounded domain with perfectly conducting boundary

We study the compressible magnetohydrodynamic equations in a bounded smooth domain in ${{\mathbb{R}}^2}$ with perfectly conducting boundary, and prove the global existence and uniqueness of smooth solutions around a rest state. Moreover, the low Mach limit of the solutions is verified for all time, provided that the initial data are well prepared.     

Global weak solutions to a class of compressible non-Newtonian fluids with vacuum

The global weak solution of an initial-boundary value problem for a compressible non-Newtonian fluid is studied in a three-dimensional bounded domain. By the techniques of artificial pressure, a solution to the initial-boundary value problem is constructed through an approximation scheme and a weak convergence method. The existence of a global weak solution to the three-dimensional compressible non-Newtonian fluid with vacuum and large data is established.     

Well-posedness of thermal boundary layer equation in two-dimensional incompressible heat conducting flow with analytic datum

In this paper, we study the well-posedness of the thermal boundary layer equation in two-dimensional incompressible heat conducting flow. The thermal boundary layer equation describes the behavior of thermal layer and viscous layer for the two-dimensional incompressible viscous flow with heat conduction in the small viscosity and heat conductivity limit. When the initial datum are analytic, with respect to the tangential variable of the boundary, and without the monotonicity condition of the tangential velocity, by using the Littlewood-Paley theory, we obtain the local-in-time existence and uniqueness of solution to this thermal boundary layer problem.     

Self-similar convergence of a shock wave in a heat conducting gas

The problem of the convergence of a spherical shock wave (SW) to the centre, taking into account the thermal conductivity of the gas in front of the SW, is considered within the limits of a proposed approximate model of a heat conducting gas with an infinitely high thermal conductivity and a small temperature gradient, such that the heat flux is finite in a small region in front of the converging SW. In this model, there is a phase transition in the surface of the SW from a perfect gas to another gas with different constant specific heat and the heat outflow. The gas is polytropic and perfect behind the SW. Constraints are derived which are imposed on the self-similarity indices as a function of the adiabatic exponents on the two sides of the SW. In front of the SW, the temperature and density increase without limit. In the general case, a set of self-similar solutions with two self-similarity indices exists but, in the case of strong SW close to the limiting compression, there are two solutions, each of which is completely determined by the motion of the spherical piston causing the self-similar convergence of the SW.     

Global strong solutions for 3D viscous incompressible heat conducting Navier‐Stokes flows with the general external force

In this paper, we consider the initial boundary value problem for the nonhomogeneous heat–conducting fluids with non‐negative density and the general external force. We prove that there exists a unique global strong solution to the 3D viscous nonhomogeneous heat–conducting Navier‐Stokes flows if is suitably small.     

A bound from below for the temperature in compressible Navier–Stokes equations

We consider the full system of compressible Navier–Stokes equations for heat conducting fluid. We show that the temperature is uniformly positive for t ≥  t ₀ (for any t ₀ > 0) for any solutions with finite initial entropy. The assumptions on the viscosity and conductivity coefficients are minimal (for instance, the solutions constructed by Feireisl in (Oxford Lecture Series in Mathematics and its Applications, vol 26, 2004) verify all the requirements).