Best constants in Sobolev inequalities for ultraspherical polynomials

A family of sharp Sobolev-type inequalities for functions on the classical measure spaces associated with the ultraspherical or Gegenbauer polynomials is obtained. These estimates generalize the Sobolev inequalities for the n-sphere S ⁿ given by Beckner, and are derived from a sharp Sobolev inequality for functions on the real line. Spectral considerations allow these estimates to be expressed as multiplier inequalities for functions which have expansions in terms of Gegenbauer polynomials.     

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.     

Weighted L∞ bounds and uniqueness for the Boltzmann BGK model

We prove rather general L bounds for hydrodynamical fields in terms of weighted L norms of the kinetic density. We use these estimates to prove L bounds and uniqueness for the solution of the BGK Equation, which is a simple relaxation model introduced by Bhatnagar, Gross & Krook to mimic Boltzmann flows.     

Propagating phase boundaries: Formulation of the problem and existence via the Glimm method

This paper treats the hyperbolic-elliptic system of two conservation laws which describes the dynamics of an elastic material having a non-monotone strain-stress function. FollowingAbeyaratne &Knowles, we propose a notion of admissible weak solution for this system in the class of functions of bounded variation. The formulation includes an entropy inequality, a kinetic relation (imposed along any subsonic phase boundary) and an initiation criterion (for the appearance of new phase boundaries). We prove theL ¹-continuous dependence of the solution to the Riemann problem. Our main result yields the existence and the stability of propagating phase boundaries. The proofs are based onGlimm's scheme and in particular on the techniques ofGlimm andLax. In order to deal with the kinetic relation, we prove a result of pointwise convergence of the phase boundary.     

Existence of Global Strong Solutions in Critical Spaces for Barotropic Viscous Fluids

This paper is dedicated to the study of viscous compressible barotropic fluids in dimension N ≧ 2. We address the question of the global existence of strong solutions for initial data close to a constant state having critical Besov regularity. First, this article shows the recent results of Charve and Danchin (Arch Ration Mech Anal 198(1):233–271, 2010) and Chen et al. (Commun Pure Appl Math 63:1173–1224, 2010) with a new proof. Our result relies on a new a priori estimate for the velocity that we derive via the intermediary of the effective velocity, which allows us to cancel out the coupling between the density and the velocity as in Haspot (Well-posedness in critical spaces for barotropic viscous fluids, 2009). Second, we improve the results of Charve and Danchin (2010) and Chen et al. (2010) by adding as in Charve and Danchin (2010) some regularity on the initial data in low frequencies. In this case we obtain global strong solutions for a class of large initial data which rely on the results of Hoff (Arch Rational Mech Anal 139:303–354, 1997), Hoff (Commun Pure Appl Math 55(11):1365–1407, 2002), and Hoff (J Math Fluid Mech 7(3):315–338, 2005) and those of Charve and Danchin (2010) and Chen et al. (2010). We conclude by generalizing these results for general viscosity coefficients.     

Existence and Stability of Viscoelastic Shock Profiles

We investigate existence and stability of viscoelastic shock profiles for a class of planar models including the incompressible shear case studied by Antman and Malek-Madani. We establish that the resulting equations fall into the class of symmetrizable hyperbolic–parabolic systems, hence spectral stability implies linearized and nonlinear stability with sharp rates of decay. The new contributions are treatment of the compressible case, formulation of a rigorous nonlinear stability theory, including verification of stability of small-amplitude Lax shocks, and the systematic incorporation in our investigations of numerical Evans function computations determining stability of large-amplitude and nonclassical type shock profiles.     

On the stiffness of materials containing a disordered array of microscopic holes or hard inclusions

We study the macroscopic mechanical behavior of materials with microscopic holes or hard inclusions. Specifically, we deal with the effective elastic moduli of composites whose microgeometry consists of either soft or hard isolated inclusions surrounded by an elastic matrix. We approach this problem by taking the stiffness of the inclusion phase to be a complex variable, which we eventually evaluate at the soft or hard limits. Our main result states that there is a certain class of non-physical, negative-definite values of the elastic moduli of the inclusion phase for which the effective tensor does not have infinities or become otherwise singular.We present applications of this result to the estimation of effective moduli and to homogenization theorems. The first application involves using complexanalytic methods to obtain rigorous and accurate bounds on the effective moduli of the high-contrast composites under consideration. We also discuss the variational estimates of Rubenfeld & Keller, which yield a complementary set of bounds on these moduli. The best bounds are given by a combination of the analytical and variational results. As a second application, we show that certain known theorems of homogenization for materials with holes are simple consequences of our main result, and in this connection we establish corresponding new theorems for materials with hard inclusions. While our rederivation of the homogenization theorems for materials with holes can be closely related to other known constructions, it appears that certain elements provided by our main result are essential in the proof of homogenization for the hard-inclusion case.     

Functional and measure-valued solutions of the euler equations for flows of incompressible fluids

We consider the notion of a functional solution of the Euler equations for incompressible fluid flows. We show that a functional solution can be constructed under very weak a priori estimates on approximate solution sequences of the equation; an estimate uniform in L _loc ¹ together with weak consistency with the equation is sufficient to construct a solution. We prove that if we have an estimate uniform in L _loc ² available for the approximate solution sequence, then the structured functional solution just described becomes a measure-valued solution in the sense of DiPerna & Majda. We also show that a functional solution can be obtained from a measure-valued solution. We give an example showing that a much higher concentration of energy than in the case of measure-valued solutions is allowed by the approximation procedure of a functional solution.     

Existence of entropy as a consequence of asymptotic stability

For a class of physical systems whose temporal evolution is governed by ordinary differential equations, the consequences of an assumption of asymptotic stability for equilibrium states in isolation remarkably resemble various forms of the second law of thermodynamics. Here we apply a known converse to Lyapunov's stability theorem to motivate both Gibbs' theory of thermostatics and the use of the Clausius-Duhem inequality for systems which are out of equilibrium and exchanging heat with their surroundings. We also discuss conditions under which the entropy of a system can be expressed as a sum of the entropies of its material points.     

Existence of homoenergetic affine flows for the Boltzmann equation

An existence theorem is proved for homoenergetic affine flows described by the Boltzmann equation. The result complements the analysis of Truesdell and of Galkin on the moment equations for a gas of Maxwellian molecules. Existence of the distribution function is established here for a large class of molecular models (hard sphere and angular cut-off interactions). Some of the data lead to an implosion and infinite density in a finite time, in agreement with the physical picture of the associated flows; for the remaining set of data, global existence is shown to hold.     

Stationary Stokes, Oseen and Navier–Stokes Equations with Singular Data

The concept of very weak solution introduced by Giga (Math Z 178:287–329, 1981) for the Stokes equations has hardly been studied in recent years for either the Navier–Stokes equations or the Navier–Stokes type equations. We treat the stationary Stokes, Oseen and Navier–Stokes systems in the case of a bounded open set, connected of class C^1,1{\mathcal{C}^{1,1}} of \mathbbR³{\mathbb{R}^3}. Taking up once again the duality method introduced by Lions and Magenes (Problèmes aus limites non-homogènes et applications, vols. 1 & 2, Dunod, Paris, 1968) and Giga (Math Z 178:287–329, 1981) for open sets of class C^￥{\mathcal{C}^{\infty}} [see also chapter 4 of Necas (Les méthodes directes en théorie des équations elliptiques. (French) Masson et Cie, éd., Paris; Academia, éditeurs, Prague, 1967), which considers the Hilbertian case p = 2 for general elliptic operators], we give a simpler proof of the existence of a very weak solution for stationary Oseen and Navier–Stokes equations when data are not regular enough, based on density arguments and a functional framework adequate for defining more rigourously the traces of non-regular vector fields. In the stationary Navier–Stokes case, the results will be valid for external forces not necessarily small, which lets us extend the uniqueness class of solutions for these equations. Considering more regular data, regularity results in fractional Sobolev spaces will also be discussed for the three systems. All these results can be extended to other dimensions.     

On a boundary layer problem for the nonlinear Boltzmann equation

This article deals with a boundary-layer problem arising in the kinetic theory of gases when the mean free path of molecules tends to zero. The model considered here is the stationary, nonlinear Boltzmann equation in one dimension with a slightly perturbed reflection boundary condition. We restrict our attention to the case of hard spheres collisions, with Grad's cutoff assumption. Existence, uniqueness and asymptotic behavior are derived by means of energy estimates.     

Long-Time Behaviour of Heat Flow: Global Estimates and Exact Asymptotics

We obtain global upper and lower bounds on the heat kernel of an elliptic second-order differential operator, which become sharp in certain long-time and large-space asymptotics. We prove a generalization of Aronson's Gaussian bounds which identifies correctly an effective drift for heat flow. In the case of periodic coefficients we give variational characterizations of the effective conductivity, which is then made to appear in heat kernel bounds. These results are for heat kernels with measurable coefficients. For differentiable coefficients we prove tighter estimates, in which the rate of homogenization is known to be optimal. (Accepted May 20, 1996)     

High-Frequency Asymptotics and One-Dimensional Stability of Zel’dovich–von Neumann–D?ring Detonations in the Small-Heat Release and High-Overdrive Limits

We establish one-dimensional spectral, or “normal modes”, stability of Zel’dovich–von Neumann–D?ring detonations in the small heat release limit and the related high overdrive limit with heat release and activation energy held fixed, verifying numerical observations made by Erpenbeck in the 1960s. The key technical points are a strategic rescaling of parameters converting the infinite overdrive limit to a finite, regular perturbation problem, and a careful high-frequency analysis depending uniformly on model parameters. The latter recovers and extends to arbitrary amplitudes the important result of high-frequency stability established by Erpenbeck by somewhat different techniques. Notably, the techniques used here yield quantitative estimates well suited for numerical stability investigation.     

Eigenfunction expansions for singular Schrodinger operators

We extend Ikebe's theory of eigenfunctions to a class of Schrodinger operators H = - + V on L ² (IR³), where the potential V is replaced by a measure which need not be absolutely continuous with respect to Lebesgue measure. Applications include the proof of the existence and completeness of the wave operators for a free particle which is partly reflected and partly transmitted by a compact 2-dimensional surface.     

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].     

Écoulements de fluides parfaits en deux variables indépendantes de type espace. Réflexion d'un choc plan par un diédre compressif

Résumé We consider the Euler equations of a perfect fluid having only two independent space-like variables, which account for the stationary 2-dimensional or axisymmetrical 3-dimensional cases as well as the 2-dimensional Riemann problem. We show that the pressure and the angle between an axis and the velocity field satisfy a first-order system which turns out to be elliptic in the subsonic zone. In particular, the pressure satisfies a maximum principle which has not been stated before, to the best of my knowledge. Using this and the Bernouilli law, we give various a priori estimates of the pressure, the density, the enthalpy, and the velocity in the problem of the reflection of a shock wave by a wedge. We also bound the size of the subsonic region and the force that the fluid applies to the boundary. Presenté par R. Kohn     

Bounds on Coarsening Rates for the Lifschitz–Slyozov–Wagner Equation

This paper is concerned with the large time behavior of solutions to the Lifschitz–Slyozov–Wagner (LSW) system of equations. Point-wise in time upper and lower bounds on the rate of coarsening are obtained for solutions with fairly general initial data. These bounds complement the time averaged upper bounds obtained by Dai and Pego, and the point-wise in time upper and lower bounds obtained by Niethammer and Velasquez for solutions with initial data close to a self-similar solution.