A free boundary problem in an absorbing porous material with saturation dependent permeability

We prove a well posedness result for a free boundary problem describing the filtration of an incompressible viscous fluid in a porous medium containing water absorbing granules.?The location of the wetting front (the free boundary) is described by a Stefan like problem for a parabolic equation which is coupled with an hyperbolic equation describing the absorption kinetic of the granules. Received December 1999     

Scaling limits for the Ruijgrok–Wu model of the Boltzmann equation

The Ruijgrok–Wu model of the kinetic theory of rarefied gases is investigated both in the fluid‐dynamic and hydro‐dynamic scalings. It is shown that the first limit equation is a first order quasilinear conservation law, whereas the limit equation in the diffusive scaling is the viscous Burgers equation. The main difficulties came from initial layers that we handle here. Copyright © 2001 John Wiley & Sons, Ltd.     

The Boltzmann equation and its fluid dynamic limits: Numerical studies

We introduce a Boltzmann equation on discrete lattices and demonstrate its applicability for the numerical simulation of flows in the transition regime to fluid dynamics. The application concerns an evaporation condensation where the fluid dynamic flow is ruled by a thin kinetic boundary layer. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homogenization of dynamic laminates

This paper addresses the study of the homogenization problem associated with propagation of long wave disturbances in materials whose properties exhibit not only spacial but also temporal inhomogeneities (called dynamic materials). The study was initiated by Lurie in his pioneering work of 1997. Homogenization theory is employed to replace an equation with oscillating coefficients by a homogenized equation. Two typical examples of periodic homogenization are considered: the wave equation and Maxwell's system coefficients oscillating rapidly not only in space but also in time. Conditions that generate applicability of the homogenization procedure to dynamic materials composites are developed. In particular, we examine a cell problem for periodic composites as well as the laminate formulae. The effective tensors of rank-one laminates for one-dimensional wave equation and the full Maxwell's system are computed explicitly. We also note some dramatic differences between the hyperbolic and the elliptic cases.     

Iterative Method of Solving the Inverse Problem for a Quasilinear Hyperbolic Equation

We consider the inverse problem for the quasilinear hyperbolic equation connected with the mathematical model of adsorption dynamics with a concentration-dependent kinetic coefficient. An iterative method is proposed that reduces the inverse problem to a nonlinear operator equation.__________Translated from Prikladnaya Matematika i Informatika, No. 17, pp. 5–12, 2004.     

Extended thermodynamics—a theory of symmetric hyperbolic field equations

Extended thermodynamics is based on a set of equations of balance which are supplemented by local and instantaneous constitutive equations so that the field equations are quasi-linear differential equations of first order. If the constitutive functions are subject to the requirements of the entropy principle, one may write them in symmetric hyperbolic form by a suitable choice of fields. The kinetic theory of gases, or the moment theories based on the Boltzmann equation, provide an explicit example for extended thermodynamics. The theory proves its usefulness and practicality in the successful treatment of light scattering in rarefied gases. It would seem that extended thermodynamics is worthy of the attention of mathematicians. It may offer them a non-trivial field of study concerning hyperbolic equations, if ever they get tired of the Burgers equation. Dedicated to Jürgen Sprekels on the occasion of his 60th birthday     

Fourier Integral Operators Defined by Classical Symbols with Exit Behaviour

We present the detailed construction of the classical version of the calculus for Fourier Integral Operators (FIOs) in the class of symbols with exit behaviour ( SG symbols). In particular, we analyse what happens when one restricts the choice of amplitude and phase functions to the subset of the classical SG symbols. It turns out that the main composition theorem, obtained in the environment of general SG classes, has a “classical” counterpart. As an application, we study the Cauchy problem for classical hyperbolic operators of order (1, 1), refining the known results about the analogous problem for general SG hyperbolic operators. The theory developed here will be used in forthcoming papers to study the propagation of singularities and the Weyl formula for suitableclasses of operators defined on manifolds with cylindrical ends.     

Dynamic PDE parametric curves

Dynamic partial differential equation (PDE) parametric curves which can be expressed as a coupled system of two hyperbolic equations are developed. In curve design, dynamic PDE parametric curves can be modified intuitively and are more flexible than ordinary differential equation (ODE) curves. The calculation of dynamic PDE parametric curves must recur to numerical methods and a three-level finite difference scheme is proposed. Approximation and stability properties for the scheme are proved and convergence property is derived. An example of interpolating PDE curves is presented as an application of dynamic PDE parametric curves.     

Two-gap elliptic solutions to integrable nonlinear equations

We study spectral surfaces associated with elliptic two-gap solutions to the nonlinear Schrödinger equation (NLS), the Korteweg-de Vries equation (KdV), and the sine-Gordon equation (SG). It is shown that elliptic solutions to the NLS and SG equations, as well as solutions to the KdV equation elliptic with respect tot, can be assigned to any hyperelliptic surface of genus 2 that forms a covering over an elliptic surface.     

Some existence-uniqueness results for a class of one-dimensional nonlinear Biot models

The wave propagation in a poro-elastic medium is generally described by a Biot model. This model couples the displacement in the solid structure with the fluid pressure and the most complete system involves coupled equations which are mixed hyperbolic–parabolic. In this paper, we are interested in including nonlinearities in the displacement equation. We restrict our study to the one-dimensional case and we establish existence and uniqueness results in Sobolev spaces using Galerkin approximants. The quasi-static case is also investigated. The hyperbolic character is then suppressed and we get the well-posedness of the system with data less regular than the complete model. But, we also prove that the complete model may be considered as an approximation of the quasi-static model.     

Position dependent mass Scarf Hamiltonians generated via the Riccati equation

The construction of position dependent mass Scarf Hamiltonians of the trigonometric as well as the hyperbolic types is addressed by means of the factorization method and the Riccati equation. These Hamiltonians are shown to be independent of the ordering parameter of the kinetic term. Additionally, new families of Hamiltonians with the Scarf spectrum are also determined by supersymmetry. Some examples for masses with and without singularities are considered to illustrate our results.     

关于非守恒形式差分格式的能量守恒问题

在建立流体力学方程组的差分格式时,对能量方程有两种不同的选择:一种是采用关于总能量(即内能与动能之和)的守恒形式的方程;另一种是采用关于内能的非守恒形式的方程.对于守恒形式的方程,容易建立能量守恒的差分格式(下面称之为守恒形式的差分格式),而对非守恒形式的方程建立的格式(下面称之为非守恒形式的差分格式),则在     

Quantum electronic transport in graphene: A kinetic and fluid‐dynamic approach

We derive a fluid‐dynamic model for electron transport near a Dirac point in graphene. Starting from a kinetic model, based on spinorial Wigner functions, the derivation of the fluid model is based on the minimum entropy principle, which is exploited to close the moment system deduced from the Wigner equation. To this aim we make two main approximations: the usual semiclassical approximation (??1) and a new one, namely, the ‘strongly mixed state’ approximation, which allow to compute the closure explicitly. Particular solutions of the fluid‐dynamic equations are discussed which are of physical interest because of their connection with the Klein paradox phenomenon. Copyright © 2010 John Wiley & Sons, Ltd.     

On the limit behavior of the trajectory attractor of a nonlinear hyperbolic equation containing a small parameter at the highest derivative

We study the trajectory attractor of a nonlinear nonautonomous hyperbolic equation with dissipation depending on a small parameter. The nonlinear function appearing in this equation does not satisfy the Lipschitz condition. It is shown that, as the small parameter tends to zero, the trajectory attractor of the hyperbolic equation converges to the trajectory attractor of the limit parabolic equation in the corresponding topology.     

A kinetic scheme for unsteady pressurised flows in closed water pipes

The aim of this paper is to present a kinetic numerical scheme for the computations of transient pressurised flows in closed water pipes. Firstly, we detail the mathematical model written as a conservative hyperbolic partial differentiel system of equations, and then we recall how to obtain the corresponding kinetic formulation. Then we build the kinetic scheme ensuring an upwinding of the source term due to the topography performed in a close manner described by Perthame and Simeoni (2001) [1] and Botchorishvili et al. (2003) [2] using an energetic balance at microscopic level. The validation is lastly performed in the case of a water hammer in an uniform pipe: we compare the numerical results provided by an industrial code used at EDF-CIH (France), which solves the Allievi equation (the commonly used equation for pressurised flows in pipes) by the method of characteristics, with those of the kinetic scheme. It appears that they are in a very good agreement.     

Scattering of Inertial Waves by Smooth Convex Cylinders

A general method of solution of the hyperbolic partial differential equation governing the scattering of inertial waves in a rotating fluid is presented for the case of cylindrical scatterers with smooth convex cross-sections. The specific case of a parabolic cylinder is discussed in some detail.     

On a posteriori estimates for a linearized Oldroyd's problem

We study and give some a posteriori estimates for finite element approximation of a linearized Oldroyd's problem for the approximation of viscoelastic fluid flows. The major difficulty lies in the coupling between two different kinds of equations: a hyperbolic equation associated to the constitutive equation and Stokes problem. These estimates are then obtained combining two different techniques used for each kind of equation: the first technique is given in (First ICIAM, Paris, 1987) for Stokes's equations and the second in (C.R. Acad. Sci. Paris. Ser. I. 322 (1996) 493) for Friedrichs's system.     

Some recent methods for partial differential equations of divergence form

Some recent methods for solving second-order nonlinear partial differential equations of divergence form and related nonlinear problems are surveyed. These methods include entropy methods via the theory of divergence-measure fields for hyperbolic conservation laws, kinetic methods via kinetic formulations for degenerate parabolichyperbolic equations, and free-boundary methods via free-boundary iterations for multidimensional transonic shocks for nonlinear equation of mixed elliptic-hyperbolic type. Some recent trends in this direction are also discussed.Dedicated to IMPA on the occasion of its 50^th anniversary     

The nonlinear Klein-Gordon equation on an interval as a perturbed Sine-Gordon equation

We treat the nonlinear Klein-Gordon (NKG) equation as the Sine-Gordon (SG) equation, perturbed by a higher order term. It is proved that most small-amplitude finite-gap solutions of the SG equation, which satisfy either Dirichlet or Neumann boundary conditions, persist in the NKG equation and jointly form partial central manifolds, which are “Lipschitz manifolds with holes”. Our proof is based on an analysis of the finite-gap solutions of the boundary problems for SG equation by means of the Schottky uniformization approach, and an application of an infinite-dimensional KAM-theory. The first author was supported by the Alexander von Humbold Foundation and the Sonder-forschungsbereich 288.     

Existence of Strong Solutions for a Fully Hyperbolic Phase-Field Model Based on Type III Heat Conduction with a Logarithmic Nonlinear Term

In this paper, we prove the existence (and also the uniqueness) of strong solutions for a fully hyperbolic phase-field model based on type III heat conduction. The model consists of the hyperbolic relaxation of the usual equation for the temperature, coupled with the equation for the thermal displacement variable. We also study the limit problem, as the relaxation parameter goes to 0.