On the equations of age-dependent population dynamics

The aim of this work is to give a direct and constructive proof of existence and uniqueness of a global solution to the equations of age-dependent population dynamics introduced and considered by M. E. Gurtin & R. C. MacCamy in [3]. The linear theory was developed by F. R. Sharpe & A. J. Lotka [10] and A. G. McKendrick [8] (see also [1], [9]) and extended to the nonlinear case by M. E. Gurtin & R. C. MacCamy in [3] (see also [4] [5] [6]). In [3], the key of the proof of existence and uniqueness was to reduce the problem to a pair of integral equations. In fact, as we shall see, the problem can also be solved by a simple fixed point argument. To outline more clearly the ideas of the proof, we will first discuss the setting and the resolution of the linear case, and then we will generalize the results of [3].     

A group theoretical approach to a certain class of perturbations of the Kepler problem

We study a particular class of perturbations of the classical Kepler Hamiltonian, first in two, then in three and finally in n dimensions. At every stage of our investigation the group theoretical nature of our constructions is fully exposed.In particular we present a new regularization of the n-dimensional Kepler problem which is based on previous constructions of Guillemin & Sternberg (see [8]). This regularization is similarily related to Moser's (see [9]) as is Kustaanheimo-Stiefel's (see [4]) in three dimensions.     

On the admissibility and existence of unsteady retarded plane Poiseuille flow of Simple Fluids with Fading Memory. I. Admissibility

The retarded histories of unsteady plane parallel (Poiseuille) flows of Simple Fluids with Fading Memory between two parallel plates of infinite extent at a finite distance apart are shown to be admissible, in the sense that they satisfy the equations of motion at arbitrary time t = 0 to any order of approximation in the retardation parameter according to the scheme of approximation of Coleman & Noll [2]. The result obtained by Coleman & Mizel [6] for second-order fluids is reinterpreted in the above context.     

The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations

We prove that viscosity solutions in W ^1, of the second order, fully nonlinear, equation F(D ² u, Du, u) = 0 are unique when (i) F is degenerate elliptic and decreasing in u or (ii) F is uniformly elliptic and nonincreasing in u. We do not assume that F is convex. The method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions, respectively. This method is completely different from that used in Lions [8, 9] for second order problems with F convex in D ² u and from that used by Crandall & Lions [3] and Crandall, Evans & Lions [2] for fully nonlinear first order problems.The research reported here was supported in part by grants from the Alfred P. Sloan Foundation and the National Science Foundation.     

A microstructure of martensite which is not a minimiser of energy: the X-interface

In this paper we investigate the X-interface, a microstructure observed in Indium-Thallium byBasinski &Christian [5].Ball &James [3] have shown howsimple martensitic microstructures can be represented by sequences of elastic deformations which minimise a free-energy functional. In contrast we show that the X-interfacecannot be represented by such a sequence. In an attempt to understand this result we develop a less restrictive theory based onEricksen's ideas about low-energy modes of deformation for martensitic materials. This theory has some interesting conclusions for the X-interface and Indium-Thallium, for the wedge-like microstructures analysed recently byBhattacharya [6], and for the general problem of microstructures which cannot be represented by minimising sequences. The calculations in this paper apply only to cubic-to-tetragonal transformations.     

Capillary gravity waves on the free surface of an inviscid fluid of infinite depth. Existence of solitary waves

Permanent capillary gravity waves on the free surface of a two dimensional inviscid fluid of infinite depth are investigated. An application of the hodograph transform converts the free boundary-value problem into a boundary-value problem for the Cauchy-Riemann equations in the lower halfplane with nonlinear differential boundary conditions. This can be converted to an integro-differential equation with symbol –k ²+4|k|–4(1+), where is a bifurcation parameter. A normal-form analysis is presented which shows that the boundary-value problem can be reduced to an integrable system of ordinary differential equations plus a remainder term containing nonlocal terms of higher order for || small. This normal form system has been studied thoroughly by several authors (Iooss &Kirchgässner [8],Iooss &Pérouème [10],Dias &Iooss [5]). It admits a pair of solitary-wave solutions which are reversible in the sense ofKirchgässner [11]. By applying a method introduced in [11], it is shown that this pair of reversible solitary waves persists for the boundary-value problem, and that the decay at infinity of these solitary waves is at least like 1/|x|.     

Convergence of Solutions to the Boltzmann Equation in the Incompressible Euler Limit

 We consider here the problem of deriving rigorously, for well-prepared initial data and without any additional assumption, dissipative or smooth solutions of the incompressible Euler equations from renormalized solutions of the Boltzmann equation. This completes the partial results obtained by Golse [B. Perthame and L. Desvillettes eds., Series in Applied Mathematics 4 (2000), Gauthier-Villars, Paris] and Lions & Masmoudi [Arch. Rational Mech. Anal. 158 (2001), 195–211]. (Accepted June 6, 2002) Published online December 3, 2002 Communicated by Y. BRENIER     

Eventual partition of conserved quantities for Maxwell's equations

In [4], Dassios uses a method introduced by Duffin in [5] to show that if a classical solution to Maxwell's equations has compact spatial support at some time, its energy is partitioned after a finite time into exactly equal electric and magnetic parts. Here we apply the same method to conserved quantities other than the energy. These quantities are shown to be eventually partitioned into a number of linear or quadratic functions of the time with constant coefficients.     

On the instability of equilibrium when the potential has a non-strict local minimum

Instability is studied for a mechanical system with two degrees of freedom whose potential has a non-strict local minimum at the origin. Under suitable conditions of smoothness it is shown that the corresponding equilibrium is unstable when the potential has a local minimum along a curve passing through the origin. As a particular case, we get instability if the potential is analytic and has a non-isolated minimum at the origin. These results improve those of G. Hamel [1] and Silla [5] on the same subject.

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Résumé On étudie l'instabilité d'un système mécanique à deux degrés de liberté quand le potentiel a un minimum non isolé a l'origine. On montre que l'équilibre correspondant est instable si le potentiel est assez régulier et a un minimum local en chaque point d'une courbe passant par l'origine. Comme cas particulier, on prouve l'instabilité de l'origine si le potentiel est analytique et y présente un minimum non isolé. Ces résultats améliorent ceux de G. Hamel [1] et L. Silla [5] sur le même sujet.

     

Exact Relations for Effective Tensors of Polycrystals. I. Necessary Conditions

The set of all effective moduli of a polycrystal usually has a nonempty interior. When it does not, we say that there is an exact relation for effective moduli. This can indeed happen as evidenced by recent results [4, 10, 12] on polycrystals. In this paper we describe a general method for finding such relations for effective moduli of laminates. The method is applicable to any physical setting that can be put into the Hilbert space framework developed by Milton[13]. The idea is to use the W-function of Milton[13] that transforms a lamination formula into a convex combination. The method reduces the problem of finding exact relations to a problem from representation theory of SO(d)(d= 2 or 3) corresponding to a particular physical setting. When this last problem is solved, there is a finite amount of calculation required to be done in order to answer the question completely. At present, each candidate relation has to be examined separately in order to confirm the stability under homogenization. We apply our general theory to the settings of conductivity and two‐dimensional elasticity. (Accepted April 4, 1997)     

On the Passage from Atomic to Continuum Theory for Thin Films

We give a rigorous derivation of a continuum theory from atomic models for thin films. This scheme has been proposed by Friesecke and James in [J. Mech. Phys. Solids 48, 1519–1540 (2000)]. The resulting continuum energy expression is obtained by integrating a stored energy density which not only depends on the deformation gradient, but also on ν-1 director fields when ν is the (fixed) number of atomic film layers.     

Conservation laws in classical mechanics for quasi-coordinates

Noether's theorem and Noether's inverse theorem for mechanical systems with gauge-variant Lagrangians under symmetric infinitesimal transformations and whose motion is described by quasi-coordinates are established. The existence of first integrals depends on the existence of solutions of the system of partial differential equations — the so-called Killing equations. Non-holonomic mechanical systems are analysed separately and their special properties are pointed out. By use of this theory, the transformation which corresponds to Ko Valevskaya first integral in rigid-body dynamics is found. Also, the nature of the energy integral in non-holonomic mechanics is shown and a few new first integrals for non-conservative problems are obtained. Finally, these integrals are used in constructing Lyapunov's function and in the stability analyses of nonautonomous systems. The theory is based on the concept of a mechanical system, but the results obtained can be applied to all problems in mathematical physics admitting a Lagrangian function.     

Poincaré and the shortcomings of the Hamilton-Jacobi method for classical or quantized mechanics

A great theorem was proven by H. Poincaré in celestial mechanics. It states that, in the most general problems of mechanics, the total energy of the system is the only well behaved first integral of the system, while other so-called integrals cannot be represented by uniform and convergent series. This very important result can be explained and visualized by comparison with standard methods of discussion, as, for example, the Hamilton-Jacobi procedure. The discussion shows that there are serious limitations to the use of this procedure, which collapses in the most general problems (Poincaré theorem) and can be used only for “almost separated” variables. The Poincaré theorem appears to provide the distinction between determinism in mechanics and statistical mechanics according to Boltzmann. The research presented here done under Contract Nonr 266(56) and was first described in a Quarterly Report dated July 31, 1959.     

An asymptotic method for a singular hyperbolic equation

Ho & Meyer [1] have sketched an argument for arriving at asymptotic properties of solutions of a pair of non-linear conservation laws with floating, shocktype boundary condition of importance in gas dynamics and oceanography. To clarify the mathematical basis of their method of inequalities, a proof of their main asymptotic results is given here. It depends on substantially weakened hypotheses.     

Physically nonlinear and related approximate theories of elasticity,and their invariance properties

The existing developments of physically nonlinear elasticity have several shortcomings. With the aim of remedying these deficiencies, a number of approximate theories of elasticity are discussed in the present paper and, in particular, a theory of physically nonlinear elasticity is systematically developed. More specifically, stress-strain relations for anisotropic physically nonlinear materials are derived. The method of Casey & Naghdi [1] is then applied to obtain properly invariant results. This method involves the use of auxiliary motions obtained by removing from any given motion the translation and rotation at any one particle, called a pivot. The auxiliary motions represent the original motions in the approximate theory. The connection between the transformation of fields under a change of pivot and invariance requirements associated with the auxiliary motions is investigated.     

A stability investigation for an incompressible simple fluid with fading memory

The nonlinear equations of motion for an incompressible simple fluid, occupying a fixed bounded container, are formulated on the basis of the finitelinear viscoelasticity theory for materials with fading memory; this formal boundary-initial value problem is then viewed as a nonlinear abstract evolution equation on a certain Hilbert space. It is shown that a linearized version of this evolution equation is associated with a linear dynamical system on this Hilbert space, and several results for stability and asymptotic behavior for this linearized problem are proved through the use of Liapunov stability methods. On the assumption that the original nonlinear evolution equation also is associated with some dynamical system on the same space, it is shown that the rest condition of the fluid is stable and all motions are bounded. The Liapunov function employed for this purpose can be interpreted as a mechanical energy function for the fluid.E. F. Infante's work was supported in part by the U.S. Office of Naval Research (grant N0014-76-C-0278P002), the U.S. National Science Foundation (grant MCS-76-07247 A03), and the U.S. Army Research Office (grant AROD 31-124-73-G-130); that of J. A. WALKER was supported in part by the U.S. National Science Foundation (grant ENG76-81570) and the U.S. Air Force (grant AFOSR-76-3063A).     

A Multi-Fluid Compressible System as the Limit of Weak Solutions of the Isentropic Compressible Navier–Stokes Equations

This paper mainly concerns the mathematical justification of a viscous compressible multi-fluid model linked to the Baer-Nunziato model used by engineers, see for instance Ishii (Thermo-fluid dynamic theory of two-phase flow, Eyrolles, Paris, 1975), under a “stratification” assumption. More precisely, we show that some approximate finite-energy weak solutions of the isentropic compressible Navier–Stokes equations converge, on a short time interval, to the strong solution of this viscous compressible multi-fluid model, provided the initial density sequence is uniformly bounded with corresponding Young measures which are linear convex combinations of m Dirac measures. To the authors’ knowledge, this provides, in the multidimensional in space case, a first positive answer to an open question, see Hillairet (J Math Fluid Mech 9:343–376, 2007), with a stratification assumption. The proof is based on the weak solutions constructed by Desjardins (Commun Partial Differ Equ 22(5–6):977–1008, 1997) and on the existence and uniqueness of a local strong solution for the multi-fluid model established by Hillairet assuming initial density to be far from vacuum. In a first step, adapting the ideas from Hoff and Santos (Arch Ration Mech Anal 188:509–543, 2008), we prove that the sequence of weak solutions built by Desjardins has extra regularity linked to the divergence of the velocity without any relation assumption between λ and μ. Coupled with the uniform bound of the density property, this allows us to use appropriate defect measures and their nice properties introduced and proved by Hillairet (Aspects interactifs de la m’ecanique des fluides, PhD Thesis, ENS Lyon, 2005) in order to prove that the Young measure associated to the weak limit is the convex combination of m Dirac measures. Finally, under a non-degeneracy assumption of this combination (“stratification” assumption), this provides a multi-fluid system. Using a weak–strong uniqueness argument, we prove that this convex combination is the one corresponding to the strong solution of the multi-fluid model built by Hillairet, if initial data are equal. We will briefly discuss this assumption. To complete the paper, we also present a blow-up criterion for this multi-fluid system following (Huang et al. in Serrin type criterion for the three-dimensional viscous compressible flows, arXiv, 2010).     

Zur mathematischen theorie akustischer Wellenfelder

Ohne Zusammenfassung Vorgelegt von C. Müller Zur praktischen Berechnung von U können nach Satz 2 die numerischen Methoden zur Auflösung Fredholmscher Integralgleichungen, etwa das Verfahren von Cl. Müller [10], verwendet werden.     

Graphentheorie des impulstransportes

Ohne Zusammenfassung Vorgelegt von M. M. Schiffer Diese Arbeit ist aus meiner Dissertation[3] hervorgegangen und vertieft die Ergebnisse derselben.     

Quasiconformal mappings as a tool to study certain two-dimensional G-closure problems

A recent theorem due to Astala establishes the best exponent for the area distortion of planar K-quasiconformal mappings. We use a refinement of Astala's theorem due to Eremenko and Hamilton to prove new bounds on the effective conductivity of two-dimensional composites. The bounds are valid for composites made of an arbitrary finite number n of possibly anisotropic phases in prescribed volume fractions. For n= 2 we prove the optimality of the bounds under certain additional assumptions on the G-closure parameters.