Waves and Stability in Continuous Media

PREFACE.

CONFERENCE DATA.

TABLE OF CONTENTS.

In this paper we investigate the linear stability behavior of a 1D flow in a poroelastic material with respect to longitudinal as well as transversal disturbances with mass exchange. Fields are assumed to be a superposition of a stationary (nonuniform) 1D solution and of infinitesimal disturbances in the form of a one- or two-dimensional linear wave ansatz. We solve numerically the eigenvalue problem for the first step field equations using a finite-difference-scheme.

We perform a multiple time scale, single space scale analysis of a compressible fluid in a time-dependent domain, when the time variations of the boudary are small with respect to the acoustic velocity. We introduce an average operator with respect to the fast time. The averaged leading order variables satisfy modified incompressible equations, which are coupled to linear acoustic equations with respect to the fast time.

We consider the inverse problem of determining the potential in the one-dimensional Schrödinger equation from surface dynamical observations, namely from the range values of the Neumann to Dirichlet map. Dynamical boundary data have not been used in the inverse problem for the Schrödinger equation, since the traditional Gelfand-Levitan-Marchenko approach reconstructs the potential from spectral or scattering data. Here we show that one can completely recover the spectral data from the dynamical data. The construction of the spectral data uses a new result on spectral controllability for the Schrödinger equation, which we obtain by using the properties of exponential Riesz bases (nonharmonic Fourier series). From the spectral data, we solve the inverse problem by the Boundary Control method.

We review some recent results concerning possible applications of wavelets and, more specifically, of multi-resolution analysis (MRA) to the problem of constructing the ground state of a two-dimensional electron gas in a strong magnetic field. We show how to produce wavelet-like orthonormal bases in the lowest Landau level (LLL). Further, we prove that any MRA produces a Slater determinant belonging to the LLL and viceversa.

No abstract received.

We consider a problem of stationary heat conduction in a monatomic gas between two co-axial cylinders which are at rest in a non-inertial frame. Instead of the usual Navier-Stokes-Fourier theory, we employ extended thermodynamics with 13 moments. Three results follow: There is a tangential heat flux between the cylinders, no rigid rotation of the heat conducting gas is possible and there is a normal pressure field in the axial direction.

A Grad's closure strategy is applied to the moment equations relevant to a four species gas mixture undergoing a bimolecular chemical reaction. The resulting robust approximation is shown to satisfy an entropy inequality. Numerical results are presented, taking into account the chemical activation energies.

Asymptotic stability of the homogeneous stationary solution to the hydrodynamical model of charge transport in semiconductors based on the maximum entropy principle (MEP)^1,2, is proved in the linear case in the limit of small electric fields.

This paper derives spatial decay bounds in a dynamical problem of mixtures of thermoelastic solids defined on a semi-infinite cylindrical region. Previous results for isothermal elastodynamics and for the parabolic heat equation lead us to suspect that the solution of the problem should tend to zero faster than a decaying exponential of the distance from the finite end of the cylinder. We prove that an energy expression is actually bounded above by a decaying exponential of a quadratic polynomial of the distance.

For a general hyperbolic conservative system in three-dimensional space variables the Lax condition of genuine non-linearity depends on the normal to the wave front. When it fails for some directions of the normal (exceptional directions) we study the possibility of existence of a characteristic shock and give its explicit expression. Conditions for the boundedness of the shock are given in the case of the Euler's variational equations.

No abstract received.

Linear stability analysis in combination with the Boussinesq approximation is usually used in order to determine the critical conditions for the onset of convection for the Rayleigh-Bénard problem. In this way a non-dimensional number, the Rayleigh number, is obtained, which reflects whether the fluid is at rest (stability) or in motion (instability). From a thermodynamic point of view the Boussinesq approximation has a shortcoming that contradicts thermodynamic stability. Therefore a linear stability analysis for the compressible Navier-Stokes-Fourier equations has been carried out numerically for rigid-rigid, rigid-free and free-free boundaries. The results show that the critical value of the Rayleigh number is not constant anymore as in the Boussinesq case, but that it depends on the thickness of the fluid layer.

Given a monoparametric family of trajectories f(x,y) = c in a two-dimensional perfect fluid in planar steady motion, we find two compatibility conditions among the function f(x,y), the pressure p(x,y), the density ρ(x,y) and the components X(x,y), Y(x,y), of the body force acting on the fluid. These conditions serve to determine, apart from some arbitrary functions, the pressure and density of the fluid if, f, X, Y are known.

Many models in Mathematical Physics and Continuum Mechanics describe the motion of a continuum of identical particles (or extended objects such as strings or membranes) that are subject to various kind of interactions (collisions, repulsive or attractive forces…). Some of them can be approximated by very simple dynamical systems involving permutations of the particles labels. Several examples will be discussed, including the simplest model of adhesion dynamics, linked to one-dimensional scalar conservation laws, the Euler equations of inviscid incompressible fluids, some models in Electrodynamics and Geophysics (such as the semi-geostrophic equations for atmospheric fronts) and, finally, isothermal gas dynamics equations through the concept of harmonic functions "up to rearrangement".

From the Boltzmann equations it is possible to derive truncated and closed finite moment systems through the procedures of moment theory and Extended Thermodynamics. In particular the closure of the system is provided by the entropy principle which is surely one of the main requirement of Extended Thermodynamics. For the sake of simplicity, the corresponding truncated distribution function is usually expanded with respect to the non equilibrium variables in the neighborhood of an equilibrium state. For the approximated system we have shown that in general the entropy principle requirement fails if all the non-equilibrium variables are of the same order of magnitude. This fact constitutes a selection criterion for the possible solutions and suggests a possible way to select the so called non-controllable boundary data. The results are also illustrated through some simple examples.

No abstract received.

We discuss the existence of global generating functions describing Lagrangian submanifolds connected with evolution problems for Hamilton-Jacobi equations. By using Viterbo's version of the Amann-Conley-Zehnder reduction, we compute, for generic (in a suitable sense) Hamiltonian functions and initial data, global space-time generating functions with finite parameters for geometric solutions of a H-J equations of evolution kind.

In a 1884 paper, Helmholtz showed that for a one–dimensional mechanical system with convex potential energy φ depending on a parameter V it is possible to define the temperature T, the pressure p and the entropy S verifying the Gibbs relation TdS = de + pdV where e = kin.en. + φ. In the paper we propose an extension of the Helmholtz construction to general natural mechanical systems endowed with a fibration over the (possibly multidimensional) space of macroscopic parameters v. Moreover, we show that, for certain discrete mechanical systems with non convex potential energy, used as models for phase transitions in solids, the above introduced thermodynamic pressure p = p(e, V) provides a single–valued macroscopic stress–strain relation.

A nonlinear equilibrium problem in elasticity is here studied. In particular, the model adopted consists of two elastic bodies which interact with each other: the first one represents the elastic body of interest and the latter the environment which is surrounding it. Accordingly, they are said to form a body-environment pair. The equilibrium problem is considered in the case when the interaction body-environment is live, that is the energy functional depends not only on the position and on the deformation which takes place at that position, but also on deformation gradients. In particular, when a material is of grade 1, the energy functional depends on the first deformation gradient and, when a material is of grade 2 it depends on the first deformation gradient and, in addition, on the second deformation gradient. Here, the environment is assumed to be a simple material, namely of grade 1, while the body immersed into it, of grade 1 or 2. In the two different cases, respectively, the equilibrium conditions are written under the assumption that the body boundary is not a regular one, but is obtained as the union of two regular surfaces which intersect each other on a regular line. The latter, represents a wedge discontinuity line of the body boundary; that is, a line on which no outer normal unit vector is defined. The two different models, termed also First-Order and Second-Order Surface Interaction Potentials, in turn, are analyzed under this assumption on the body boundary. Thus, it is shown that further conditions need to be imposed. A comparison between the conditions in the two different cases is provided.

This paper deals with the oscillations of a mechanical system in presence of relay forces (string with monotone unilateral friction). The fundamental equation is studied in the MRA (Multi Resolution Analysis) framework, representing functions and differential operators with Haar wavelet bases and with the spline derivative of Haar series². The nonlinear equation of the mechanical system is transformed into a linear ordinary differential system, in the wavelet coefficients and a numerical approximate solution is eventually given.

We sketch the derivation of continuum equations from the relaxation model of kinetic theory. The equations found involve a pressure tensor and a heat flux whose approximate expressions contain the time derivatives of the macroscopic fields. We may eliminate those derivatives by using the fluid equations themselves. If we use the Euler equations for this, as in the Chapman-Enskog procedure, we obtain the Navier-Stokes equations. However, we advocate using the full fluid equations and so are led to a generalization of the Navier-Stokes equations that provides improved agreement with experiment in the case of long mean free paths of particles.

Starting from the Boltzmann equation for a reacting gas mixture, we formulate the problem of the propagation of a steady detonation wave. Numerical solutions of the derived macroscopic equations are provided in order to show detonation profiles and to recover the relevant physical quantities of the chemical process.

For two ClassesI-II of p-th squares Physical DVMs, planar D = 2 Discrete Velocity Models (generalizations of previous models^1–3), we try to see whether they can satisfy two continuous relations deduced from Maxwellian equilibrium states.

A system of two equations governing the irrotational flow of an isothermal fluid whose pressure tensor depends, in a suitable form, on the density and its first and second space derivatives, is shwon to reduce to a nonlinear Schrödinger equation.

In this paper we present two new models, consisting of Delay Differential Equations with parameters independent or dependent on delay. The first system, in the Population Dynamics context, is actually a new type of polluted chemostat model, describing the allelopathic competition of two algal phytoplanktonic species. The second one, belongs to the context of Innovation Diffusion Theory and models the problem of the reaction of two different social systems to the offer of a new product. Stability properties of steady state solutions have been investigated and, by means of geometric criteria recently proposed in the literature, the problem of their stability switches has been solved. The results, based both on analytical and numerical computation, confirm that large size delays are ultimately stabilizing for systems with delay dependent parameters.

The paper commences with a comparison between the Liapunov method for stability in connection with parabolic systems and the cross-sectional method for elliptic systems. The two approaches are illustrated using essentially the same Liapunov functional in two different, yet related, contexts: unsteady and steady nonlinear diffusion. The paper proceeds to use the cross-sectional method in two different contexts in elasticity: A traction problem for a semi-infinite, smoothly varying inhomogeneous, isotropic strip is considered, and a cross-sectional estimate is obtained reflecting Saint-Venant's Principle; cross-sectional estimates are also obtained for a displacement type problem for an isotropic right cylinder, both homogeneous, and smoothly varying inhomogeneous incompressible, material.

A reduction method is worked out for determining a class of exact solutions to nonlinear second order PDEs involving two independent variables. The approach in point is based upon appending suitable constraint equations to the governing equations to be satisfied by the dependent field variables. That permits to introduce a subsystem of equations which rules the evolution of a "reduced field". The integration of this auxiliary subsystem provides the searched solution to the full governing model. In the process appropriate classes of material response functions can be determined in order to solve in a closed form initial and/or boundary value problems associated to nonlinear equations of interest in wave propagation. The present method can be also used for determining exact wave-like solutions to mathematical models exhibiting a "wave hierarchy" structure, typically accounting for relaxation effects in the field evolution, as those arising from the theory of irreversible processes.

The penetrative convection in a fluid-saturated porous medium with a cubic equation of state for the density is considered, according to the Darcy model. Here, the stability analysis of the conduction solution by using normal modes analysis and the Lyapunov direct method is performed.

The spreading of fluids on solid surfaces constitutes a significant field of research into the processes met in nature, biology and modern industry. It is noteworthy that since the publication of Young on capillarity, the understanding of this phenomenom has remained incomplete. The equation of the contact-line motion is the key of our study. An example of non-Newtonian fluid motions exhibits a flat liquid interface moving steadily over a flat solid in creeping flow approximation and theory of capillarity. The kinematics of liquids at the contact-line and equations of motion are revisited. Adherence conditions are required except at the contact-line. The velocity field appears to be realistic but at some instant a liquid material point may leave the surface. The velocity field discontinuous at the contact-line generates a new concept of line viscosity but stresses and viscous dissipation remain bounded. A Young-Dupré equation for the dynamic contact angle between the interface and the solid is proposed.

A new sequence of nonlinear evolution systems satisfying the zero curvature property is constructed, by using the invariant singularity analysis. All these systems are completely integrable and a pseudo-potential (linearization) is explicitly determined for each of them. The second system of the sequence is the Broer-Kaup system, which, as is well known, corresponds to the higher order Boussinesq approximation in describing shallow water waves.

It is pointed out the special meaning of some parameters which are present in the constitutive relations in Continuum Mechanics. Questions related to the stress in the rigid bodies, the incompressibility, the heat propagation and to the microstructures theory are considered (see also Grioli¹).

A perturbative method, in order to generate explicit solutions to a nonlinear equation of the particle transport theory in the stochastic models, is developed.

The method of generalized energy functionals is discussed with respect to its applicability to channel flows. We present functionals which provide better stability boundaries than so far known. However, the rigid boundary conditions turn out to be incompatible with the standard procedure to arrive at a nonlinear stability result. Thus, the method does not seem to be appropriate for channel flows.

We study a mathematical one-dimensional model of photon transport in a homogeneous interstellar cloud (nebula) with a source of photons inside (e.g. a star). Existence, uniqueness and positivity of the solution are proved by using the theory of semigroups of operators. Moreover, we define the quasi-static solution and we show that it is a good approximation to the exact solution. A numerical estimate of the relative error is also given.

The nonlinear stability of a horizontal layer of a binary fluid mixture in an isotropic and homogeneous porous medium heated and salted from below is studied, for the Oberbeck-Boussinesq - Darcy and Oberbeck-Boussinesq - Brinkman-Forchheimer models, through the Lyapunov direct method. This is an interesting geophysical case because the solute concentration gradient is stabilizing while heating from below provides a destabilizing effect. Unconditional nonlinear stability is found for any value of the porosity and the Lewis number. In particular, if the normalised porosity ∊ is equal to 1, a necessary and sufficient condition for nonlinear stability is proved: in this case the critical linear and nonlinear Rayleigh numbers coincide. For other values of ∊ a conditional stability theorem is shown and the coincidence of the critical parameters holds whenever the Principle of Exchange of Stabilities is valid.

The existence and uniqueness of the mild solution of the boundary layer (BL) equation is proved assuming analyticity of the data with respect to the tangential variable. Moreover we use the well-posedness of the BL equation to perform an asymptotic expansion of the Navier-Stokes equations on a bounded domain.

The fundamental solutions (Green functions) for the Cauchy problems of the space-time fractional diffusion equation are investigated with respect to their scaling and similarity properties, starting from their composite Fourier-Laplace representation. By using the Mellin transform, a general representation of the Green functions in terms of Mellin-Barnes integrals in the complex plane is presented, that allows us to obtain their computational form in the space-time domain and to analyse their probability interpretation.

In this work we have analyzed the sequence of bifurcated equilibria in two-dimensional reduced resistive magnetohydrodynamics. We have considered classes of symmetric equilibria and shown that, as in a previous work¹⁵, after a first symmetry-breaking bifurcation, leading to an equilibrium with a small size magnetic island, the system undergoes a tangent bifurcation for a critical value of the aspect ratio. Above this value no equilibrium with a small island exists and the system rapidly develops an island of macroscopic size. This general property can be proposed as a basic ingredient of the intermittent events observed in magnetically confined plasmas in more general situations. The case of equilibria with a symmetric magnetic field plus a small island perturbation is also considered. The bifurcations in the system are investigated as a function of the aspect ratio ∊ and of the "error" parameter δ measuring the perturbation. When ∊ is decreased from one, also in this case equilibria with a small island disappear because of a tangent bifurcation. The range in ∊ for which the small island equilibria exists becomes smaller and smaller when δ is increased. The possibility to extend investigation to the case of different boundary conditions is also discussed.

The aim of this paper is to study the solutions of the thermoelasticity equations in the case of unbounded domains and unbounded solutions. We obtain existence, uniqueness and qualitative behaviour results of the solutions in the themoelasticity of type III. Thermoelasticity without energy dissipation is also considered.

In this paper the Compton cooling of a radiating fluid is studied using the method of moments and the maximum entropy principle. Numerical comparisons are presented which show agreement between the maximum entropy results and Monte Carlo calculations.

In previous works a macroscopic monofluid model of liquid helium II, which is based on Extended Thermodynamics, was formulated,where the time evolution of the non equilibrium stress tensor was neglected, putting zero the relaxation times τ₀ and τ₂ of the non equilibrium pressure and of the stress deviator. In this work, the complete model with 14 fields is studied in the linear approximation: a dispersion relation is obtained and the solutions of this equation are determined perturbing the solutions obtained in the cases τ₀ = τ₂ = 0. The corresponding modes are also discussed.

The existence of a material internal constraint in a continuous body generally implies the existence of an indetermination in the reaction stress. One may think that it can be eliminated by the prescriptions of the external contact force and body force: 'The precise value of N ('the reaction stress') depends on the external body forces and the boundary tractions.' ([1], p. 133). In [2] the aforementioned indetermination is studied for a linearly elastic, prestressed body B subject to any internal material constraint, with regard to the initial/boundary value-problem; it is pointed out that different external actions can sustain body processes with the same motion and different reaction stresses; hence the notion of physically equivalent processes is introduced. A suitable vector space, l_τ, of the reaction stresses that cannot be detected by means of processes which are solution of some ibvp for B related with a given portion τ of ∂B, is defined. This space coincides with the class of indetermination of r in the solution (u,r) of any initial/boundary value-problem related with τ, where u is the displacement field and r is the list of the reaction stress multipliers. If the internal constraints are explicitly defined, this indetermination can be computed. Here we show that in some cases it fails to vanish; thus the aforementioned assertion of [1] is not true.

The tools of game theory for the determination of the expected gain from a competition of hawks and doves for a resource are employed to determine the conditions under which the population is integrated or segregated. The birds are supposed to have two contest strategies to choose from and the price of the resource determines which strategy they prefer. The resulting strategy diagram bears a strong resemblance to the phase diagrams of a binary solution or an alloy in different phases and with a miscibility gap in the liquid phase.

In this paper we study the second sound propagation in superfluid helium using a model performed in the case of a rigid conductor. The theoretical basis of our arguments is given by a recent paper ¹ in which is proved that the differential system of a binary Euler's fluid can be written as a system for a single heat conducting fluid. Then the phenomenon of second sound propagation in crystals and in Helium II and, more in general the phenomenology arising in these materials at low temperatures, can be described within a unique framework.

A BGK approximation of the Boltzmann Transport Equation is used for simulating carrier transport in silicon semiconductors. An analytic solution has been obtained in the stationary and homogeneous case. Some properties of this solution have been discussed and its validity have been assessed by Monte Carlo simulations. The BGK model has been also used for simulating a n⁺ - n - n⁺ silicon diode.

We consider the equations governing a continuum with scalar microstructure, and look for solutions having the features of progressive waves by means of a suitable asymptotic expansion of the field variables. The result of the procedure shows that the wave process can be studied by means of nonlinear evolution equations that capture the complexity of the considered models. Some examples are provided.

The purpose of this work is to deduce the system of Navier-Stokes equations specialized for a quite general discrete kinetic model of a gas mixture with a general reversible chemical reaction. A perturbation scheme of Chapman-Enskog type is applied to the kinetic equations leading to the constitutive laws of the model. In this frame the reactive Euler and Navier-Stokes equations of the model have been formally deduced in dependence on the macroscopic variables.

We present a survey of recent work on the lowest end of the eigenmode spectrum of the shallow water equations in a rotating reference frame. The results are complemented with numerical simulations of the fully nonlinear equations. Having care of using physically correct, mass-conserving boundary conditions, a description in terms of slow eigenmodes seems to be a key component in the explanation of climate's decadal variability.

A further extension of Relativistic Extended Thermodynamics is here investigated by considering 30 independent variables, instead of 14. The two models are compared by seeing their implications on a well known iterative procedure.

The paper deals with the numerical study of the appearance of several blow-ups in time which may appear in the evolution of ferromagnets and the heat flow of harmonic maps. By the implementation of a semi-implicit numerical scheme the role of the dissipative term and the effects of the boundary condition on the development of singular solutions are shown.

We give an example of how a kinematics of multilattices, constructed to keep track in a detailed way of the symmetry changes in deformable complex crystals and of describing correctly the invariance of their constitutive equations, are useful in the analysis of certain weak phase changes, either configurational or structural.

Existence of an H-theorem for a simple model of chemically reacting dense gases is obtained. The model amounts to a coloring process of the four component mixture of hard-spheres with probability 0 < α_R ≤ 1.

We consider a parabolic-elliptic system of partial differential equations modelling the chemotaxis. We assume that the concentration of the organisms cannot exceed a limit value . Consequently, a free boundary can exist separating a region where from the region where . In this paper we generalize the results of our privious study of the one-dimensional free boundary problem to the two- and three-dimensional radial symmetric cases.

No abstract received.

In this note we study a transport linear integro-differential equation in a time-dependent domain with slab geometry. A singular perturbations technique is applied in order to test the error between the exact solution and its quasi-static approximation, which satisfies a simpler equation.

The nonlinear reaction-diffusion equation u_t = ΔF(x, u) -g(x, u) in an unbounded domain is considered. In the case of Dirichlet boundary data and for solutions which – a priori – can grow at large spatial distances matters relating to the asymptotic behaviour as t → ∞ are studied. Conditions ensuring the existence of absorbing sets in a suitably weighted L² space are obtained.

It is established that in spatial hydrodynamic motions with velocity q = qt subject to the geometric restriction div t = 0, the t-field is constrained by the integrable Heisenberg spin equation on individual constant pressure surfaces. The same geometric formulation is then applied to the kinematics of ideal fibre-reinforced fluids to obtain necessary geometric constraints on the motions and again a connection to the Heisenberg spin equation.

The interest of researchers towards Rayleigh waves is justified by the importance that they have in engineering applications such as geotechnical soil characterization and dynamic response of structures^1,2. In this work the frequencies of resonance of a layered half-space will be presented, i.e the frequencies for which the vertical displacements due to travelling Rayleigh waves are maximised. Successively a sensitivity analysis for a simple system has allowed for an approximate relationship to be found among the frequencies of resonance and the properties of the system.

The symmetry classification of a class of energy-transport models, arising in hydrodynamical modeling of charge transport in semiconductors, is presented. Optimal systems of one dimensional Lie subalgebras are obtained.

Governing differential equations in conservative form are formulated for the process of inelastic deformation. Three different definitions of the rate of inelastic deformation are analysed. One of them defines the rate of inelastic deformation by the nonlinear kinetics of Maxwell relaxation model. The other one is based on the introducing the stress diffusion. The third one assumes that the rate of inelastic deformation is proportional to the vector of density of defects of structure. This vector is managed by a conservation law in which the flux of defects of structure is combined with another balance law. In this connection the field of defects of structure can exchange by an energy with the stress field and generate stress waves, but the total energy is not changed in such a process. Thermodynamical properties and symmetric form of governing equations are discussed.

No abstract received.

In this talk a review is given of some modern shock capturing schemes for the numerical approximation of hyperbolic systems of balance laws. The focus is on finite volume central schemes, which can be easily applied to a variety of problems. After a brief introduction on conservative schemes and on second order central schemes, a way to construct high order schemes is presented. The treatment of systems with a stiff source term is also considered, and an application to Extended Thermodynamics is presented.

We study the long time behavior of a shallow water model introduced by Levermore and Sammartino to describe the motion of a viscous incompressible fluid confined in a basin with topography. Here we prove the existence of a global attractor and give an estimate on its Hausdorff and fractal dimension.

In this work we propose a new approach to model the drug effectiveness in the chemotherapy of the HIV infection. We introduce the drug resistance as a dynamical variable, and model its dynamics through a mechanism that has a simple biological interpretation in terms of the virus fitness. This new dynamical system is able to reproduce the lack of virus eradication. Stability of the equilibria of the systems is investigated. Numerical simulations have been performed to investigate the global dynamics of the model. Numerical and analytic results show a good agreement with the available clinical data.

The degenerate four-wave-mixing technique and laser-induced thermal acoustics represent interesting quantum optic phenomena. Two recently developed kinetic methods based on discrete velocity models and on a semi-continuous formulation of the Boltzmann equation provide deep insights into the dynamics of these phenomena. The results obtained in the fluid dynamic limit of the kinetic equations agree well with those gained from a purely fluid dynamic approach.

After recalling some recent results on the potential symmetries of a simplified nonlinear model for a reacting mixture we linearize the system and derive new solutions.

We study a material body made of thin and thick layers of two different elastic constituents, under the assumption that layers in the first family have almost vanishing thickness when compared with those in the second family.

We consider a wave propagating through the layers and derive the consequences of our assumptions. A two dimensional case is studied and a particular explict solution is reported.

In this article we review the state of the art concerning recent developments where unconditional nonlinear stability is proved in some problems in fluid dynamics. Open questions to fundamental problems are included and we detail several recent advances yielding unconditional stability results via new Lyapunov functionals. The problem of heating a fluid layer from below is highlighted when the viscosity varies with the temperature field.

The long time behaviour of the solutions of the equation u_t = ΔF(u) in exterior domains is studied.

We review some recent results on the large-time asymptotics of fourth-order nonlinear parabolic equations in two cases: 1) scalar problems in bounded domains; 2) scalar problems with confinement by a uniformly convex potential. The main analytical tool relies on the analysis of the entropy dissipation. A generalized Csizár-Kullback inequality allows for an estimation of the L¹-decay to equilibrium in terms of the relative entropy.

In this paper we get the equivalence transformations for a special model for heat conduction. We show how it is possible to get symmetries from equivalence transformations by applying a projection theorem.

By considering the macroscopic variables of interest we develope the Maximum Entropy Principle (MEP) including the full-band effects with a total energy scheme. Furthemore, under spatialy homogeneous conditions, a closed set of balance equations for the fluctuations of these variables is costruct and a systematic study of small-signal analysis is provided. By generalizing the results obtained in previous papers we analyze quantitatively the different coupling processes, as functions of the electric field and we prove that, for n-type Si material, the coupling between the different moments can lead to a strongly no exponential decay of the correspondig response functions.

In this paper, we consider the case of a spherical bridge built with an inviscid, incompressible liquid with density ρ. The study of the motion, which is assumed irrotational, is reduced to a variational equation on the function which gives the displacement of the free surface. We recognize that the stability of the equilibrium position depends essentially of the coerciveness of the bilinear form, which appears in this equation. The coerciveness can be studied by introducing an eigenvalue problem. We have shown in (⁷) that the smallest eigenvalue is always strictly greater than one, so that the bilinear form is coercive. Then, the existence of the eigenfrequencies of the problem is assured by a well-known method of functional analysis.