Waves and Stability in Continuous Media

Preface.

Conference Data.

We consider a linear RLC network which contains a semiconductor device. In a multi-physics approach, the network is modeled by Modified Nodal Analysis, while a detailed drift-diffusion model is used for the semiconductor device. We discuss the coupling of the two models and address briefly the well-posedness of the resulting partial differential-algebraic equations, both for steady and transient drift-diffusion equations.

We present some considerations on time-oscillatory phenomena in hydrodynamical models for semiconductors and study the existence of periodic solutions. For the one-dimensional, viscous, isentropic model, written in Lagrangian mass coordinates, we state a first existence result and give a sketch of the proof.

In this work we present a fluid dynamical model for electron transport in silicon which takes direct account of highly energetic electrons by introducing macroscopic quantities averaged over the tail electron population. The model is based on the maximum entropy principle and is free of any fitting parameter.

We give necessary and sufficient conditions for a derivation on a topological quasi *-algebras to be spatial, that is to be implemented by a (self-adjoint) operator, and we discuss some applications to quantum systems with infinite degrees of freedom.

No abstract received.

The Proper Orthogonal Decomposition (POD) is used as a very efficient method to find low–dimensional models that accurately describe the periodic dynamics of a N–dimensional system. Starting from a reduced model obtained by POD, the possible existence of an unstable equilibrium, "related" to the orbit, can be investigated, the appearance of the periodic behaviour by Hopf bifurcation clarified and the equilibrium loosing stability found. The procedure can also be a substantial step toward center manifold and normal form calculations of the Hopf bifurcation.

We consider three inverse problems concerning the physical structure of interstellar clouds. In the problems the unknowns are, alternatively, the dimension of the cloud, its total cross section or the photon source. Here we solve numerically, with a similar procedure, the first two problems, since the third requires an additional analysis, reported in Part II (see next paper by Belleni-Morante and Riganti). To determine the unknowns one needs additional data, namely measured values of the photon density outside the cloud. Numerical results are provided by using a suitable iterative scheme.

A model for the spatial distribution of a UV–photon source inside a slab representing an interstellar cloud is proposed. Inverse problems for the related quasi static photon transport equation are studied, in order to determine the intensity and the location of the source by using measured data of the photon densities at points very far from the slab. Numerical simulations of the obtained results are also presented.

Mathematical models describing interband tunneling in semiconductor heterostructure devices (superlattices) are discussed and compared to a model here rigorously derived from the Schrödinger equation. The continuity equation for the probability density is also derived.

No abstract received.

A first approach to the hydrodynamic limit for a gas mixture undergoing moderately fast bimolecular reactions is presented by a Chapman-Enskog asymptotic procedure applied to Grad's 13–moment equations.

Relativistic Extended Thermodynamics is a very interesting theory which is widely appreciated. Here its methods are applied to ultrarelativistic gases, and an arbitrary, but fixed, number of moments is considered. The kinetic approach has already been developed in literature; then, the macroscopic approach is here considered and the constitutive functions appearing in the balance equations are determined up to whatever order with respect to thermodynamical equilibrium. The results of the kinetic approach are a particular case of the present ones.

The Riemann problem for a binary mixture of Euler fluids without chemical reaction is numerically studied. This system is a particular case of hyperbolic systems with relaxation and the interest of such analysis could be found in the fact that, since now, there is no systematic theory about Riemann problem for this kind of equations. For relaxation terms not too big, we found that the solutions of this Riemann problem are a composition of shocks, constant states and "deformed" rarefaction waves. It was already shown that the mixture model is simplified under the strong assumption that the constituent fluids have equal masses. The validity of this approximation is also studied for different binary mixtures.

The nonlinear stability analysis of the motionless state for a binary fluid mixture in a porous layer, under horizontal periodic temperature and concentration gradients, is performed.

In this note, we summarize some recent results concerning a study a non-local variational inequality. The non-locality appears in the coefficients of the operator through some integral representing some elastic energy. It appears also in the constraints which have been called soft in the literature⁷.

This paper deals with the Riemann problem for a gas mixture undergoing irreversible bimolecular reactions governed by a suitable closure at Euler level of the Boltzmann equation. The aim of the paper is to propose a 'simplified' solution procedure for the Riemann problem in absence of chemical reactions (conservation laws), providing its validation by means of numerical simulations, and to investigate the influence of chemical reactions, as well as of the initial data, on the space-time evolution of the initial jump.

For planar (x,z coordinates) discrete velocity models (DVMs), we present binary mixtures (masses m = 1, M > 1 for light and heavy species) for a gas in a half-space (x, z ≠ 0) , filling with the same geometrical structure, all integer coordinates of the half-plane. The interface (without velocities) is located at z = 0 and only a z spatial dependence of the gas (densities with (±x, z) are equal). We prove that the models are "physical", with only physical invariants: masses, energy and momentum along the z-axis. First we obtain the same geometrical structure in the plane (|x| + |z| odd or even) for an unbounded of heavy mass values: M even, odd divided by 2^q. Second, we present M = (2p + 1)/5 models with the heavy species in dodecagons, while the light fills all empty sites.

For a hyperbolic model describing tumour growth an approach is developed to construct travelling wave-like solutions which are discontinuous across a strong shock line in the phase plane. Connection of the resulting equilibria is also investigated.

The hydrodynamical equations governing the irrotational flow of an isothermal and inviscid fluid whose pressure tensor depends on the density and its first and second space derivatives are shown to describe a capillary fluid with a suitable form of the free energy. Moreover, such a system is equivalent to a nonlinear Schrödinger-like equation, with logarithmic nonlinearity, that admits soliton-like solutions.

We describe here some of the aspects of a general method enabling to yield quantitative estimates about the rate of convergence toward the equilibrium. We are interested in solutions of PDEs in which dissipative effects can be easily obtained only with respect to part of the variables. Then, one has to use in a subtle way the interplay between the different variables in order to conclude. The example of spatially inhomogeneous kinetic equations (in a confining potential, or in a box with boundary conditions) is detailed.

In this paper we propose a model of a sole larval population. We distinguish two time scales: at the fast time scale we have the migration dynamics and at the slow time scale the demographic process. At a first step, we study the so called "aggregated" system by means of the semigroups theory. Then, we study the model by using the Chapman-Enskog procedure. We show that the solution of the aggregated system is a good approximation to the solution of the exact model.

No abstract received.

We consider a discrete model of a pseudoelastic material formed by a chain of bistable elements (snaps springs). This model, considered in e.g.^1,2,3 exhibits non monotone stres—strain relation due to the non convexity of the potential energy and it can mimic the hysteretic behaviour observed under cyclic loading in a hard device. We compute all the possible stress states compatible with a given total strain and we apply the Maximum Entropy Principle. Given an arbitrary hysteresis cycle, we are able to infer the evolution of the phase fraction and the (information) entropy during the cycle.

An outcome of allelopathic competitions in nature often consists in the coexistence of species. Here we show that such a result can be obtained from mathematical models in two ways. When the rate of toxins production is constant, the survival of both species holds if their nutrient uptaking model is of the Andrews type. This result subsists if toxins are of inhibitory type as well as they are of lethal type. When the toxicant is of lethal type and nutrient uptaking is modelled by the Michaelis-Menten function coexistence can be obtained by including in the model the so called "quorum sensing" mechanism. Analytical and numerical investigations, performed on the stability properties of the systems considered in this paper, show very rich dynamical behaviors and confirm that the rate of toxicity can always be considered as a bifurcation parameter.

An iterative formula due to Cattaneo concerning discontinuities of any order on a given hypersurface is analysed in detail. Applications are considered in the framework of relativistic wave propagation (either for ordinary and non ordinary waves) where this formula is of special interest. Explicit examples are also given.

This paper considers the nonlinear diffusion equation where the diffusivity depends on the dependent variable. Unsteady and steady states, corresponding to Dirichlet boundary conditions independent of time, are addressed. The rate of convergence, as the time t → ∞, of the unsteady to the steady state is studied by obtaining an upper estimate for a Liapunov functional governing the perturbation. A similar estimate is obtained for the i.b.v.p. for the perturbation backwards in time, and it is proved that the solution fails to exist for sufficiently large time. In all of the foregoing a diffusivity appropriate to a porous medium is assumed. An analogous issue is considered for steady state diffusion in a right cylinder with Dirichlet boundary conditions on its lateral surface, independent of the axial coordinate. The rate of convergence of the solution to the corresponding two-dimensional solution - as one recedes from the plane ends - is studied, using a methodology similar to that used in the previous context.

Two different reduction approaches are outlined in order to obtain exact wave–like solutions of a quasilinear parabolic second order equation in one space variable, which is of wide use in diverse physical applications. Both methods provide a mathematical vehicle for defining appropriate model forms of the material response functions involved in the governing equation and which permit to solve classes of initial value problems.

A PDEs system, describing the expansive growth of a benign tumor and the phenomenon of encapsulation, is studied via a group analysis approach. A weak equivalence classification is obtained and the original PDEs system is reduced to an ODEs system. Numerical simulations are performed both for ODEs and PDEs, which turn out to be in perfect agreement between each other, showing a realistic enough description of the biological process.

Nonlinear stability of Jeffery-Hamel flows in a wedge is studied. By using weighted energy methods, conditions guaranteeing nonlinear stability in the L²-norm are found.

Isothermal interfacial zones are investigated starting from a local energy which can be considered as the sum of two terms: one corresponding to a medium with a uniform composition equal to the local one and a second one associated with the non-uniformity of the fluid. The additional term can be approximated by a gradient expansion, typically truncated to the second order. A representation of the energy near the critical point therefore allows the study of interfaces of non-molecular size. Capillary layer and bulk phases are not considered independently. Obviously, this model is simpler than models associated with the renormalization-group theory. Nevertheless, it has the advantage of extending easily well-known results for equilibrium cases to the dynamics of interfaces. The equation of state of a one-component system may be expressed as a relation among the energy, entropy and matter density, α, s and ρ in the form α = α(ρ, s). Now, let α(ρ, s) be the analytic α as it might be given by a mean-field theory. In the simplest case, in an extended van der Waals theory¹⁸, the volumic internal energy ε is proposed with a gradient expansion depending not only on grad ρ but also on grad s (the associated fluid is called thermocapillary fluid³):

With an energy in the form (l), we obtain the equations of motion of conservative movements for nonhomogeneous fluid near its critical point. For such media, it is not possible to obtain shock waves. The idea of studying interface motions as localized travelling waves in a multi-gradient theory is not new and can be traced throughout many problems of condensed matter and phase-transition physics⁸. Here, adiabatic waves are considered and a new kind of waves appears. The waves are associated with the spatial second derivatives of entropy and matter density. In Cahn and Hilliard's model¹, the direction of solitary waves was along the gradient of density. For this new kind of adiabatic waves, the direction of propagation is normal to the gradient of densities. In the case of a thick interface, the waves are tangential to the interface and the wave celerity is expressed depending on thermodynamic conditions at the critical point.

No abstract received.

An exact steady solution to a class of nonlinear evolution problems, arising in the study of the diffusion of some test particles interacting between themselves and against other field particles, is presented. The test particles are emitted by an external spatially uniform source at constant rate in time. The test particles so injected diffuse in the given background of field particles by binary collisions either against the field particles and between themselves, each collision resulting in either scattering or removal events.

On using a rational function approximation scheme for the response functions of hyperelastic isotropic materials we propose a new model to describe the mechanical behaviour of atactic polymers. The new model takes into account the finite chain extensibility effects characteristic of this class of materials.

The common approach to the design of a non linear control strategy for the attitude of a platform equipped with a complex system of rotating fly-wheels, is somehow faulty as it uses, as control variables, the gimbal angles and the wheel angles rather than the mechanical torques exerted along the gimbal axes and the wheel spin axis. The basis for the design of a "torque based" control law are laid down in this article. An infinite number of possible strategies is found and a criteria is introduced to select between all this strategies an optimal one in the sense of minimal electrical power used. It is discussed how such a criteria leads to highly undesirable control laws and has therefore to be disregarded.

We consider a delayed distributed control for the Bènard convection problem. The control function is chosen to be an external heat source. We construct a finite dimensional approximation of the PDE system of the Lorenz–type. For a certain parameters range and in absence of any control, it is well known that the system shows transition to a chaotic regime, characterized by a large number of unstable low periodic orbits. We investigate the possibility of controlling the flow using a delayed nonlinear feedback control. We show that, if the time lag is small, the system is stabilized close to a periodic solution.

In this work we investigate the low field and the high field Γ, L and total electron mobilities in GaAs. Comparisons are made with the Caughey-Thomas formula and the Einstein relation between diffusivity and mobility is analized.

The definition of velocity and the transformation laws of coordinates are discussed for models of deformed Special Relativity with two fundamental scales, inspired by quantum gravity. A consistent implementation of the Lorentz transformations requires the introduction of noncanonical Poisson bracket in the phase space of a free particle.

We first study a mathematical model describing diffusion of the oxygen that is supplied to a sample of living tissue by blood flowing in a capillary. Hence, through a homogenization procedure, we model blood perfusion of a muscle by micro-circulation as a diffusion-consumption problem in a continuum possessing suitably averaged properties.

In this paper we report some numerical simulations for a mixture of gases undergoing chemical reactions of the oxygen/hydrogen chain. The simulations are performed integrating a set of time-space evolution equations at the hydrodynamic scale, proposed in a previous paper. At this end we use an appropriate second order non oscillatory central scheme for conservation equations extended to balance laws.

We recall the Gibbs paradox concerning the entropy of mixing of gases which to this day seems to be unresolved. There is also a pseudo-Gibbs-paradox that occurs in statistical mechanics of a gas and which predicts a non-additive entropy. The latter is due to an overzealous extrapolation of Boltzmann's formulae of the kinetic theory. The pseudo Gibbs paradox should never have occured in the first place and by now it is fully understood.

We study nonlinear stability of the zero solution of a dynamical system Eq. (1) with a linear operator which splits in a symmetric and skewsymmetric part. We give (for particular systems) an operative method (which rests upon the eigenvalues - eigenvectors method in ODE) to build a good Lyapunov function in order to obtain necessary and sufficient stability conditions. We apply this method to a reaction-diffusion system and to thermodiffusive convection for a mixture heated and salted from below.

The development of hydrodynamic models for the study of solid state devices is of great interest for industrial applications, especially in regimes far from thermal equilibrium. In this paper an Extended Hydrodynamic model for a 2D Bipolar Junction Transistor is introduced and validated by using Monte Carlo simulations.

In this paper we prove that the third order Monge-Ampère equation in one space dimension is a remarkable equation not only because it is a completely exceptional equation but also because it is uniquely characterized by its Lie point symmetries.

A multicomponent reacting gas with reversible reactions is studied at a kinetic level with the main objective of deriving the reactive Navier-Stokes equations in dependence on the macroscopic variables, and characterizing the dissipative terms related to shear viscosity, heat conduction and thermal diffusion. A step-by-step procedure, which can be applied to a quite large variety of reactive flows, is proposed in order to identify the transport coefficients, basically resorting to a first-order density approximation of Chapman-Enskog type.

No abstract received.

No abstract received.

In [1], we considered a stochastic model of the revised Enskog equation. Using the smearing function suggested by the work of Leegwater we applied the model to the repulsive Lennard-Jones potential and the inverse-power soft-sphere potential. The virial coefficients obtained from the equilibrium properties of the models are in excellent agreement with the known exact coefficients for these models. The transport coefficients for the repulsive Lennard-Jones (RLJ) model are also computed and appear to be of comparable accuracy to the Enskog-theory coefficients applied directly to a hard-sphere system. The pressure and the transport coefficients obtained from the model (shear viscosity, thermal conductivity, and self-diffusion) are compared with the pressure and the corresponding transport coefficients predicted by the Enskog and square-well kinetic theories. In this work we summarize the results for stochastic model of the revised Enskog equation found in [1] and also show how to extend the method to the full Lennard-Jones potential.

We present and discuss here some one–dimensional kinetic models with dissipative interactions. The first model is obtained from a suitable modification of Kac caricature of a Maxwellian gas, while the second, a fractional Fokker–Planck type equation, is obtained from the modified Kac equation with a singular kernel, in the so–called grazing collision limit. It is shown that both models admit global equilibria different from concentration, provided that we leave the usual assumption of finite energy. These equilibria are distributed like stable laws and attract initial densities which belong to the normal domain of attraction. We remark in this way the connections between the large–time behavior of dissipative models and the classical central limit theorem for stable laws. These connections enable us to make use of arguments typical of probability theory to recover the rate of decay towards equilibrium.

The nonlinear diffusion equation u_t = ΔF(u) is studied in exterior domains under Dirichlet boundary conditions. Asymptotic stability criteria for the steady states are proved. A criterion of pointwise asymptotic stability for the porous medium and horizontal filtration equations is obtained.

A group classification via equivalence transformations of a class of energy-transport models of semiconductors in the two dimensional stationary case is presented.

The entropy principle plays an important role on hyperbolic systems of balance laws: symmetrization, principal subsystems and nesting theories, equilibrium manifold. After a brief survey on these questions we present some recent results concerning the local and global well-posedness of the Cauchy problem for smooth solutions with particular attention to the genuine coupling Kawashima condition. These results are applied to the case of a binary mixture of ideal euler gas and we prove that the K-condition is satisfied only in presence of chemical reaction with consequence that there may exist global smooth solutions for small initial data and the solution converge to a constat equilibrium state. Viceversa, if the mixture is not chemically reacting, the problem of global existence remains an open question.

In this work we analyze propagation of non linear waves in mixtures of ideal Euler fluids. If the difference between molecular masses is negligible, we can separate the properties resembling the single fluid case from the ones peculiar to mixtures. We also showed that diffusive k-shock is locally exceptional.

Using Extended Thermodynamics in the 13-moments approach, we write the hydrodynamic balance system for Fermi and Bose gases. We evaluated numerically the characteristic polynomial and the hyperbolicity region. Particular attention is devoted to the completely degenerate case when Fermi gas reaches the 0 °K and when the Bose gas is close to the transition temperature T_c.

In recent years the artificial neural networks have improved the method for solving complex problem in many different areas such as pattern recognitions, image processing, function approximation etc.. In this paper a particular kind of neural network called Cellular Neural Network will be used as a new method for solving partial differential equations. The Cellular Neural Networks (CNNs) are a powerful tools based on some aspects of neurobiology and adapted to integrated circuits or computer simulation. The CNN offer a new computational model which has important potential applications in such areas as image processing, signal processing, and finally in partial differential equations (PDEs) solving. As example of previous assertion, a classic PDE will be considered in order to show how the CNN can be applied successfully. The PDE considered is The Heat Conduction Equation.

We study the limiting behavior of the Cauchy problem for a class of Carleman-like models in the diffusive scaling with data in L¹. We show that, in the limit, the solution of such models converges towards the solution of a nonlinear diffusion equation with initial values determined by the data of the hyperbolic system.

We consider the problem of the pricing a European option with transaction costs and non-constant volatility. Using a utility maximization procedure we obtain a fully non-linear Hamilton-Jacobi-Bellman equation. We solve this equation in the asymptotic regime of fast-mean reverting volatility and low transaction costs.

In this paper, we consider the two dimensional equations of thermohydraulics, i.e. the coupled system of equations of fluid and temperature in the Boussinesq approximation. We construct a family of approximate Inertial Manifolds whose order decreases exponentially fast with respect to the dimension of the manifold. We give the explicit expression of the order of the constructed manifolds.

Some of the exact solutions obtained in Ref.¹ for the equations of Ideal Magneto-Gas-Dynamics are here considered in connection to initial and boundary value problems of physical interest, and their linear stability is investigated.

A model is given of a general density law which permits oscillatory convection when an internal heat source is also present in a porous medium. Linear and nonlinear analyses are discussed.

A new continuum model of solids incorporating microscopic thermal vibration explicitly, which is valid over a wide temperature range including the melting temperature, is explained. Mechanical and thermal properties of solids can be studied in a unified and consistent way by the model. As an application of the model, linear wave propagation phenomena in solids at finite temperatures are analyzed and discussed. Propagation speeds and amplitude ratios of both longitudinal and transverse waves are derived from the model as the functions of the temperature. Furthermore, the local equilibrium assumption, that has usually been adopted in thermodynamics of irreversible processes, is reexamined in the course of the analysis.

There is interest in obtaining exact solutions of the Navier-Stokes equations. One of methods for constructing exact solutions is group analysis¹. By this method two classes of solutions can be found. One class is a class of invariant solutions. Another class is a class of partially invariant solutions. Study of partially invariant solutions is more difficult, since analysis of compatibility for them is more complicated. The main problem in obtaining partially invariant solutions consists of studying compatibility of a reduced system, obtained after substituting a representation of a partially invariant solution into the initial system of equations. Since the property of the group to be admitted is not involved in constructing a representation of a solution, this fact gave an idea for constructing partially invariant solutions for groups, which are not admitted by the initial system of differential equations. In the manuscript examples of such solutions for the Navier-Stokes equations are presented.

We investigate Electromagnetic-Wave propagation in continuous media in which the presence of nonlinearities gives rise to parametric mixing. The case of a laser pulse injected in an optical fiber has been analyzed. This propagation is described by means of the Nonlinear Schrödinger Equation (NLS). On one side, we integrate numerically the NLS with different injection conditions describing phase and amplitude modulation as well as wave mixing phenomena. On the other side we illustrate a simplified model of coupled-mode equations giving an Hamiltonian description of the problem which enable us to obtain analytical solutions. We compare and discuss the results obtained with the two different approaches relatively to the same input conditions.

By using the differential invariants, with respect to the equivalence transformation algebra of the class of wave equations u_tt - u_xx = f(u, u_t, u_x), we characterize a subclass of linearizable equations.

Acceleration waves propagating in isotropic solids at finite temperatures are studied on the basis of a new continuum model. The propagation speeds and the differential equations governing the time-variation of the wave amplitudes are derived. The analytical results, valid in a wide temperature range including the melting point, are evaluated numerically for several materials and their physical implications are discussed.

In this article the existence and uniqueness theory of stationary kinetic equations in L¹-spaces is developed for collision terms dominated in the norm by the collision frequency.

The aim of this paper is to study the relative asymptotic stability and to mix the method of Liapunov's functions families (Matrosov, Rouche, Salvadori) with the method of limiting equations (Sell, Artstein, Andreev).