Waves and Stability in Continuous Media

PREFACE.

CONFERENCE DATA.

CONTENTS.

The solution of the Cauchy problem for the Camassa–Holm equation φ_t + 3φφ_x = ε²(φ_xxt + 2φ_xφ_xx + φφ_xxx) − 2νφ_x, in the limit ε → 0 is characterized by a zone of rapid modulated oscillations. This situation is very similar to solution of the Korteweg de Vries equation in the small dispersion limit. In this latter case the modulated oscillations are known to be approximately described by the Whitham equations. The analogous problem is, on the contrary, still open in the case of the Camassa–Holm equation. In this paper, we present some preliminary work in the direction of numerically compare the solution to the Cauchy problem of the Camassa–Holm equation in the limit ε → 0 and the approximate solution which should be described by the corresponding Whitham equations.

We consider an inverse problem which arises in the framework of identification of doping profiles for semiconductor devices, based on current measures for varying voltage. We set formally the inverse problem, and study and discuss the main properties of the resulting problem.

In this note we review some recent results on hyperbolic systems with boundary and their approximations by uniformly parabolic quasilinear systems. In particular we consider systems with oscillatory boundary and no relations among the characteristic hyperbolic speed and the speed of the boundary.

We review here some recent results concerning an operatorial approach to a stock market which is described and analyzed using the same framework adopted in the description of a gas of interacting bosons.

In this paper we propose some numerical experiments performed on a kinetic model proposed in the recent literature, describing the behavior of a mixture of polytropic gases undergoing chemical reversible reactions. Numerical experiments are made both at the mesoscopic scale, by using a particle method, and at the macroscopic scale, numerically solving the partial differential equations describing the hydrodynamical limit. We focus on the approximate solution of a Riemann problem, considering both a reacting and a non-reacting mixture.

Grad's 13-moment theory, appropriate for rarefied gases, implies that a gas cannot perform a rigid rotation, if it conducts heat. Indeed, stationary heat conduction in a gas between coaxial cylinders at rest in a non-inertial frame exhibits azimuthal components of velocity and of heat flux. These effects are due to Coriolis terms in all transfer equations that result from the Boltzmann equation.

In this paper we study the nonlinear stability of uniform steady states of a fourth-order reaction diffusion equation in two spatial dimensions. An important consequence of this stability is that the equation preserves the sign of a set of its solutions provided some appropriate restrictions on the parameters and the initial data are met. This equation and related ones are also used in the contexts of population dynamics, where obviously the solutions must be nonnegative functions. They have a very rich dynamics including travelling waves, periodic patterns and localised structures.

The non-relativistic limit of Relativistic Extended Thermodynamics with 14 moments can be found in a paper by Dreyer and Weiss, which has been widely appreciated. In particular it suggest a particular structure for the classical counterpart of the theory, in particular that developed by Kremer, instead of the previous one with 13 moments. The same thing needs to be obtained with an arbitrary but fixed number of moments, and this is the object of the present paper. Also our results predict a particular structure for the classical counterpart with many moments, and it is not the simpler one. Moreover, from the passages here involved, it is evident that the deviation from the dominant parts of the equations is of the second order with respect to 1/c, with c the speed of light. This is interesting also for future applications; it suffices now to remember that the Maxwell equations are linear with respect to 1/c.

In this paper a hydrodynamic set of equations is derived from a Schrödinger-like model for the dynamics of electrons in a two-band semiconductor, via the Madelung ansatz. A diffusive scaling allows to attain a drift-diffusion formulation.

In this paper we suppose that the initial datum for the 2D Navier–Stokes equations are of the vortex layer type, in the sense that there is a rapid variation in the tangential component across a curve. The variation occurs through a distance which is of the same order of the square root of the viscosity. Assuming the initial as well the matching (with the outer flow) data analytic, we show that our model equations are well posed. Another necessary assumption is that the radius of curvature of the curve is much larger than the thickness of the layer.

The authors study the small oscillations of a pendulum completely filled by an inviscid, incompressible, almost homogeneous liquid. The main object of the paper is to prove that the spectrum is formed by one isolated eigenvalue and an essential spectrum, which fills an interval.

Penetrative convection for density quadratically depending on temperature and linear internal heating is studied.

The penetrative convection in a horizontal binary fluid mixture layer with an internal heating due to a quadratic concentration source, according to the Darcy - Oberbeque - Boussinesq model, is considered. Conditions guaranteeing the onset of penetrative convection are obtained.

No abstract received.

In part I of this article, Borghero, Demontis and Pennisi have obtained the limits for light speed c going to infty, of the balance equations in Relativistic Extended Thermodynamics with many moments. In order to obtain independent equations, they have taken a suitable linear combination of the equations, before taking the limit. What happens with this procedure to the relativistic conservation laws of mass, momentum and energy? Obviously, they transform in their classical counterparts; but proof of this property is not easy and is treated in this part II of the article.

The paper presents a recent result by the author concerning Maxwell molecules, without any cutoff in the collision kernel, in the one-dimensional case. Conservation of energy also holds.

No abstract received.

The role of gradient dependent constitutive spaces is investigated on the example of Extended Thermodynamics of rigid heat conductors. First order nonlocality is developed and the consequences of some additional constitutive assumptions are analyzed.

We prove the existence of a strongly continuous semigroup of solutions associated with the Cauchy problem for the Degasperis-Procesi equation with initial conditions in L² ∩ L⁴.

A 3-D time-dependent numerical model for the analysis of a dc transferred argon arc is presented. The model allows to investigate the influence on the behavior of the arc and on the evolution of the temperature inside the anode of an external magnetic field induced by a current flowing in an external conductor. If the currents of the arc and of the external conductor have opposite directions, the magnetic fields mutually repel and the arc is deflected. Such a practice to obtain a deflected arc finds applications in surface treatment of metallic substrates.

Two sets of hydrodynamic equations describing fast and slow chemical reactions in a mixture of four gases are discussed in the framework of dissipative quasi-linear hyperbolic systems compatible with an entropy principle.

Most evolution equations are partially integrable and, in order to explicitly integrate all possible cases, there exist several methods of complex analysis, but none is optimal. The theory of Nevanlinna and Wiman-Valiron on the growth of the meromorphic solutions gives predictions and bounds, but it is not constructive and restricted to meromorphic solutions. The Painlevé approach via the a priori singularities of the solutions gives no bounds but it is often (not always) constructive. It seems that an adequate combination of the two methods could yield much more output in terms of explicit (i.e. closed form) analytic solutions. We review this question, mainly taking as an example the chaotic equation of Kuramoto and Sivashinsky νu′′′ + bu′′ + μu′ + u²/2 + A = 0, ν ≠ 0, with (ν, b, μ, A) constants.

We are concerned with the problem of the energy stability of steady motions of an incompressible viscous fluid. It is assumed that the motion takes place in an unbounded smooth region of the three-dimensional Euclidean space with a priori non-compact boundary and it is governed by the Navier-Stokes equations. We study the asymptotic behavior in time of the kinetic energy of perturbations; the class of unperturbed motions is, in some sense, physically reasonable and certainly non empty. Precisely, we prove that the kinetic energy of the perturbation goes to zero and, moreover, we give a decay rate.

The propagation of acceleration waves in two isotropic solids in contact with one another is studied in the case of plane symmetry. The transmitted and reflected waves across an interface are evidentiated and their amplitudes are determined.

A solid-fluid mixture is generally modelled assuming that the state of stress in the reference configuration is identically equal to zero. However, such an assumption is not always appropriate to take into account some instability phenomena occurring in Nature. In this contribution, the continuum mechanics point of view is used and the reference configuration of the solid-fluid mixture has a state of stress, i.e. the pre-stress is different from zero. The instability of the mixture with respect to the perturbation fields given by a general plane wave is then studied.

Aim of this paper is to furnish further arguments on the naturalness of the work “A new method to exploit the Entropy Principle and Galilean invariance in the macroscopic approach of Extended Thermodynamics” by Pennisi and Ruggeri; in particular, it will be shown how it was potentially included in a previous work on Galileanity, by Ruggeri. Therefore, the salient points of this work will be here revised, taking care to show the above mentioned application. The notation will be useful also for the paper “The Galilean Relativity Principle as non-relativistic limit of Einstein's one in Extended Thermodynamics” by Carrisi, Pennisi and Scanu, where the same results will be obtained starting from the Einstein' s relativity principle.

We study the viscoelastic second grade solid, for which the constitutive equation consists in the sum of a purely elastic part and a viscoelastic part; this latter part is specified by two microstructural coefficients α₁ and α₂, in addition to the usual Newtonian viscosity ν. We show via some exact solutions that such solids may describe some interesting dispersive effects. The solutions under investigation belong to special classes of standing waves and of circularly-polarized finite-amplitude waves.

We are interested in obtaining explicit rates of convergence toward the equilibrium for equations having a Lyapounov functional (entropy). We assume that the dissipation of entropy dominates the entropy itself. However, instead of being independant of time, we only suppose that the rate of this domination deteriorates slightly when the time goes to infinity. Such a situation was described first by Toscani and Villani in the context of the Boltzmann equation (and its variants when grazing collisions are predominant). We show here how the estimate of convergence toward equilibrium obtained there can be used to establish new (global in time) a priori estimates for the equation under study, which, in turn, sometimes enable to precise the rate of convergence toward equilibrium. Examples of applications of thise ideas are taken from works in collaboration with Mouhot (¹⁰) and Fellner (⁸), respectively for homogeneous kinetic equations and reaction-diffusion systems.

The governing equations for a 1D case of the Mindlin model for microstructured materials are derived and analysed. These equations exhibit hierarchical properties assigning the wave operators to internal scales. The dispersion of waves is characterized by higher-order derivatives including also the mixed derivatives with respect to coordinate and time.

An allelopathic competition between two populations of microorganisms, taking place in a chemostat-like environment, is analyzed. The allelochemicals production by one of the two species is supposed delayed. The same allelochemical compound is also introduced as an external input concentration. Meaningful steady-state solutions and their stability properties are analyzed. The survival of the producing species is, in particular, studied in the special case of an exponential delay kernel.

A nonlinear double diffusive system is the subject of this article: we consider two simultaneous p.d.e.'s in two dependent variables, first order in time and second order in the spatial variables. Dirichlet boundary conditions, independent of time, are assumed.

The principal concern of the article is the stability of the steady states together with the convergence of the unsteady states thereto. Novel Liapunov functionals are used to this end. Both linear and nonlinear stability are discussed together with some aspects of (linear) instability.

Prior to these considerations, some relevant remarks are made concerning uniqueness and nonuniqueness of the steady states.

Riemann problems for a class of 2 × 2 hyperbolic systems involving source–like terms are considered. By means of reduction techniques it is shown that the searched rarefaction wave is given by an exact solution which, in fact, represents a deformation of the rarefaction wave admitted by the corresponding homogeneous governing model. Hence the solution of the Riemann problem of interest is provided.

This paper addresses the control of the chaotic Chen system via a feedback technique. We first present a nonlinear feedback controller which drives the trajectories of the Chen system to a given point for any initial conditions. Then, we design a linear feedback controller which still assures the global stability of the Chen system. We moreover achieve the tracking of a reference signal. Numerical simulations are provided to show the effectiveness of the developed controllers.

In this paper we consider a (2 + 1)-dimensional integrable Calogero-Degasperis-Fokas equation. We apply the nonclassical method in order to obtain new symmetry reductions. From these partial differential equations in (1 + 1) dimensions by further reductions, we get second order ordinary differential equations. These ODE's provide several new solutions; all of them are expressible in terms of known functions, some of them are expressible in terms of the third Painleve trascendents. The corresponding solutions of the (2 + 1)-dimensional equation, involve up to three arbitrary smooth functions. Consequently the solutions exhibit a rich variety of qualitative behaviour.

A linear stability analysis of the Bénard problem for deep convection is performed. An estimate of the critical Rayleigh number that reduces to the classical one for vanishing depth parameter is obtained.

We discuss multicomponent reactive flow models derived from the kinetic theory of gases. We review the evaluation of multicomponent transport coefficients as well as the mathematical hyperbolic-parabolic structure of the resulting systems of partial differential equations. We address the anchored wave problem, various numerical algorithms specifically devoted to complex chemistry flows, and recent extensions to chemical equilibrium flows and ionized/magnetized mixtures.

A non linear model associated with a Landau-Ginzburg-like behavior in mean field approximation forecasts phase transition waves and solitary kinks near the critical point. The behavior of isothermal waves is different of the one of isentropic waves as well in conservative cases as in dissipative cases.

The problem of an hydrodynamic closure at the Euler level of the complicated set of integro–differential non–linear Boltzmann–like equations describing the evolution of a chemically reactive gas mixture at the kinetic level is addressed. In comparison to other physical situations, the case in which the process is driven by mechanical collisions between molecules of the same species is worked out and discussed.

Consider the following nonlinear boundary value problem in the exterior space of the unit sphere S: Given a vector field D : S → ℝ³ we ask for all harmonic vector fields which decay at least as fast as a dipole field at infinity and are parallel to D on S, i.e. there is f : S → ℝ such that B = f D. This problem is related to the problem of reconstructing the geomagnetic field outside the earth from directional data measured on the earth's surface. It is shown in this note that in any neighbourhood of a single multipole field (restricted on S) , there are direction fields , close to , in any reasonable norm, for which the above problem has no solution.

We show that under certain assumptions a general model of nonlocal nonlinear response in 1 + 1−dimension is equivalent to the model considered by Królikowski and Bang for a Kerr-type medium. We derive the limit of weak nonlocality in high frequency regime and discuss the integrable cases.

The relaxation process towards mechanical and chemical equilibrium of a reacting gas mixture is modeled starting from the generalized Boltzmann equation which is linearized following a BGK-type approach. A first-order perturbation scheme of Chapman-Enskog type is performed in a flow regime close to chemical equilibrium and the reactive Navier-Stokes equations of the model are presented in the hydrodynamic form. Moreover, the transport coefficients of diffusion, shear viscosity and thermal conductivity are characterized in an explicit form which shows the dependence on elastic and chemical contributions.

We study the properties of a n²-dimensional Lotka-Volterra system describing competition among species with behaviorally adaptive abilities, in which one species is made ecologically differentiated with respect to the others by carrying capacity and intrinsic growth rate. The case in which one species has a carrying capacity higher than the others is considered here. Stability of equilibria and time-dependent regimes have been investigated in the case of four species: an interesting example of chaotic window and period-adding sequences is presented and discussed.

No abstract received.

In bipolar electronic devices both charge carriers contribute to the total current. Here we present a complete hydrodynamical model of hole and electron coupled transport in silicon semiconductors based on the maximum entropy principle, following an approach already used for unipolar devices¹. Generation-recombination effects are taken into account. We employ this model for studying a p-n junction.

In this paper we study a mathematical model for the interaction of algae with light consisting in two integro-differential equations. In particular, we prove existence, uniqueness and positivity of the mild solution, in a suitable Banach space. Estimates for the number of photons and the algal biomass concentration are also given.

We use a mixed spectral/finite-difference numerical method to investigate the possibility of a finite time blow-up of the solutions of Prandtl's equations for the case of the impulsively started cylinder. Our tool is the complex singularity tracking method. We show that a cubic root singularity seems to develop, in a time that can be made arbitrarily short, from a class of data uniformly bounded in H¹.

In this paper nonlocal boundary conditions for the Navier–Stokes equations are derived, starting from the Boltzmann equation in the hydrodynamic limit. Basing on phenomenological arguments, two scattering kernels which model non–local interactions between the gas molecules and the wall boundary are proposed. They satisfy the global mass conservation and a generalized reciprocity relation. The asymptotic expansion of the boundary value problem for the Boltzmann equation, provides, in the continuum limit, the Navier–Stokes equations associated with a new class of nonlocal boundary conditions.

We consider dissipative hyperbolic systems of balance laws in which a block of equations are conservation laws. In this case, a coupling condition firstly introduced by Shizuta and Kawashima (K-condition) plays a fundamental role for the global existence of smooth solutions for small initial data and for the stability of constant states. Nevertheless the example by Zeng proves that the K-condition is only a sufficient condition. Using acceleration waves, we prove the necessity of the weak K-condition in which the condition is required only for the genuine non linear characteristic velocities and not for the linear degenerate ones.

We prove some results of pointwise continuous dependence and of pointwise stability of steady solutions of the Navier–Stokes system. The perturbation at the initial instant is just assumed a continuous function, small in a such a way that for any t > 0 the perturbation is a smooth solution. The new results stated in this note are part of a forthcoming paper.

For reduced resistive magnetohydrodynamics we consider the class of symmetric equilibria with no motion and analyse the sequence of bifurcations varying the aspect ratio ∈ of the domain as a control parameter of the magnetic shear. The stability threshold in the inviscid case ∈₀ scales linearly with α. Destabilization happens because of a symmetry-breaking bifurcation, occurring at ∈_c < ∈₀ because of viscosity, which originates a stable equilibrium with a small magnetic island and vortices. For α = 1 the equilibrium disappears by tangent bifurcation at a critical value ∈_P. Above ∈_P, no stationary solutions with small island exist and the system very rapidly develops an island of macroscopic size. Differently, for α = 2, i.e. for higher magnetic shear at x = 0, the equilibrium with small island does not disappear, but becomes unstable at ∈ _P.= 2.437. Below this value a stable equilibrium of weaker symmetry has been detected, with magnetic island of essentially the same size but velocity field with different topology and strongly enhanced. The phenomenology is observed for the first time in RRMHD and is of interest because connected with experimental evidences in fusion plasmas.

Two BGK-type models for reactive gas mixtures, derived in a previous paper, are here considered. The former is related to slow reactions, whereas the latter takes into account fast chemical processes. The paper, through numerical simulations carried out at the kinetic level for both fast and slow bimolecular reactions, is mainly devoted to show the characteristic behavior and features of the two corresponding chemical regimes.

We give an operative method to define optimal Lyapunov functionals to studying stability of solutions of a class of PDEs systems which includes reaction-diffusion systems and convection problems. By using the results of linearized instability analysis and the classical eigenvalue-eigenvectors method, we obtain new field variables and transform the system in an equivalent one in terms of the new fields. Then, we use the classical energy to define an optimal Lyapunov functional and we obtain the coincidence of the critical parameters reached by the linearized instability analysis R_C and with the Lyapunov method R_E.

We prove analytically the existence of high-energy tails for the Boltzmann transport equation for silicon semiconductors, in the stationary and homogeneous regime, in the quasi-parabolic band approximation, and acoustic and optical phonons scattering.

We prove the global well-posedness of the one-dimensional Zakharov-Rubenchik equation in the space H²(ℝ) × H¹(ℝ) × H¹(ℝ). We also prove the existence and the orbital stability of solitary wave solutions to this model.

In this paper we consider the discrete Boltzmann equation modelling a diatomic gas whose particles undergo elastic multiple collisions and chemical reactions of auto-catalytic type. We state that the one-dimensional initial-boundary value problem in the half-space possesses a global solution for small initial data in .

Within the framework of inverse Lie problems we give some non–trivial examples of Lie–remarkable equations, i.e., classes of partial differential equations that are in one–to–one correspondence with their Lie point symmetries. In particular, we prove that the second order Monge-Ampère equation in two independent variables is Lie–remarkable. The same property is shared by some classes of second order Monge-Ampère equations involving more than two independent variables, as well as by some classes of higher order Monge-Ampère equations in two independent variables. In closing, also the minimal surface equation in ℝ³ is considered.

We study, by the Lyapunov direct method, the Lyapunov stability of the conduction-diffusion solution of an anisotropic magnetic Bénard problem, for a partially ionized fluid. We determine the critical hypersurfaces ensuring linear Lyapunov stability recovering those obtained by solving the eigenvalue problem governing the linear instability. We show that, if the conduction diffusion solution is linearly stable, it is asymptotically non linearly stable.

This paper extends, to the case of variable depth, the recent derivation of the logical link between the Camassa-Holm (C-H) equation and shallow water theory. While solitons of the C-H equation have peak curvature adjustable independently of amplitude, the derivation emphasizes that ‘peakons’ are inappropriate to shallow water. Explicit solutions for C-H periodic and solitary travelling waves define equivalent shallow water profiles (and the associated fluid velocity field). A preliminary outline of modulation theory for variable depth is given.

It is well known that, in the relativistic context the relativity principle isn't imposed by separating variables into convective and non convective parts, but by imposing that the costitutive functions satisfy particular conditions; likely to this, the present considerations show that the same results are obtained also in the classical context. The result is achieved by taking the non-relativistic limit of Einstein's Relativity Principle. This fact furnishes further arguments on the naturalness of the work “A new method to exploit the Entropy Principle and Galilean invariance in the macroscopic approach of Extended Thermodynamics” by Pennisi and Ruggeri.

The recent paper “A new method to exploit the Entropy Principle and Galilean invariance in the macroscopic approach of Extended Thermodynamics”, by Pennisi and Ruggeri, shows a very interesting way to overcome the difficult calculations involved in imposing these principles. As tests for the validity of their procedure, they indicate 2 models: the 5 moments model and the 13 moments one. The first of these is exactly that previously known in literature, while for the second one the comparison was not possible, because in literature this model hasn't until now been exploited, up to whatever order with respect to equilibrium. This gap is here filled because we prove that also the traditional approach leads to the same result found by Pennisi and Ruggeri. Obviously, this doesn't overcome their paper; in fact, a matter is proving that given functions are the unique and exact solution of the concerned conditions (which proof is the object of the present paper), and another matter is to find the expressions of these functions (which is the result of Pennisi and Ruggeri's paper). So we aim here only to furnish a further proof which confirms their procedure.

In the framework of Liapunov Direct Method, coincidence between linear and nonlinear stability is studied. The advantage of determining and using functionals depending (with their time derivative along the perturbations) directly on the eigenvalues of the involved linear operator is shown. Applications to spatial ecology end double diffusive convection in porous media are furnished.

The symmetry analysis of the one dimensional quantum drift-diffusion model for semiconductors, based on the Bohm potential, is performed and example of exact solutions are given.

Symmetries and reduction techniques are applied to a mathematical model describing one-dimensional motion in nonlinear dissipative media. Reduced equations through the optimal systems of subalgebras are performed and an application is shown.

We investigate group invariance properties of a (2 + 1) dimensional Burgers equation. We show that it is one of the higher dimensional nonlinear partial differential equations which does not admit Kac-Moody-Virasoro type sub-algebras. Through Lie symmetry analysis we derive a wide class of interesting solutions. Further, we deduce a transformation which maps the (2 + 1) dimensional Burgers equation to a (1 + 1) dimensional linear partial differential equation.

We study the transfer properties of a material body made of thin and thick layers of two different elastic constituents. The body is modelled under the assumption that layers in the first family have almost vanishing thickness when compared with those in the second family, as proposed in Ref.². Elastic waves are considered following the linearized dynamic model presented in Refs. ¹ and in two dimensions, and the study of the transfer properties of the system is made through the transfer matrix technique given in Refs.^3,4.

The analyses of linear and nonlinear wave propagation phenomena in solids at finite temperatures made so far on the basis of a model, which was proposed recently by the present author, are reviewed. The model is a finite-deformation continuum model for solids at finite temperatures, which is valid in a wide temperature range including the melting point as a limiting case. Some peculiar temperature-dependences of the phenomena near the melting point found in the analyses are summarized. Future problems to be solved are also pointed out and discussed.

In this paper the diffusion and shift models of the linear theory of elasticity of binary mixtures are considered. The basic properties of wave numbers of the plane harmonic waves are established. The existence theorems of eigenoscillation frequencies (eigenfrequencies) of interior boundary value problem (BVP) of steady vibrations are proved. The connection between plane waves and existence of eigenfrequencies is established. Lorentz's postulate on the asymptotic distribution of eigenfrequencies for binary mixtures is proved.

Within the maximum entropy principle (MEP) we present a general theory to obtain, under spatially homogeneous conditions, a closed set of balance HD equations for the ac small signal (dynamic) response using an arbitrary number of moments of the distribution function. The theoretical approach is applied to n-Si at 300 K and is validated by comparing numerical calculations with ensemble Monte Carlo simulations and with experimental data.

We study nonlinear dynamics of a periodic inhomogeneous DNA double helical chain under dynamic plane-base rotator model by considering angular rotation of bases in a plane normal to the helical axis. The dynamics is governed by a perturbed sine-Gordon equation. The perturbed soliton solution is obtained using a multiple scale soliton perturbation theory. The perturbed kink-antikink solitons represent formation of open state configuration with fluctuation in DNA.

A stage structure is incorporated into a prey-predator model in which predators are split into immature predators and mature predators. The local stability of equilibria are analyzed and the Hopf bifurcation is found. Sufficient conditions for the global stability of the positive equilibrium is studied by compound matrix.

Some models of chemotaxis are reviewed, particularly those involving three coupled nonlinear partial differential equations. It is shown how decay bounds may be formulated in these cases. Applications are considered, in particular to a model for glia aggregation, and the possible connection with Alzheimer's disease.

The choice of variables in extended thermodynamics is arbitrary and this leads to some poblems. In the version with consistend order, every variable has a certain order of magnitude and the set of variables taken into account contains only variables up to a choosen order. In this paper the determination of the order of magnitude is calculated with a method similar to the Chapman-Enskog method.

One-dimensional stationary heat conduction in a rarefied gas at rest is studied consistently by the 3rd-order theory of consistent-order extended thermodynamics. The temperature field is obtained in a series expansion form, which agrees quantitatively with numerical calculations. The consistency of such a series solution is specially checked. As illustrations, temperature jump and entropy production at the boundary are explicitly analyzed.