The Anisotropic Lagrangian Averaged Euler and Navier-Stokes Equations

 The purpose of this paper is twofold. First, we give a derivation of the Lagrangian averaged Euler (LAE-α) and Navier-Stokes (LANS-α) equations. This theory involves a spatial scale α and the equations are designed to accurately capture the dynamics of the Euler and Navier-Stokes equations at length scales larger than α, while averaging the motion at scales smaller than α. The derivation involves an averaging procedure that combines ideas from both the material (Lagrangian) and spatial (Eulerian) viewpoints. This framework allows the use of a variant of G. I. Taylor's ``frozen turbulence' hypothesis as the foundation for the model equations; more precisely, the derivation is based on the strong physical assumption that fluctutations are frozen into the mean flow. In this article, we use this hypothesis to derive the averaged Lagrangian for the theory, and all the terms up to and including order α² are accounted for. The equations come in both an isotropic and anisotropic version. The anisotropic equations are a coupled system of PDEs (partial differential equations) for the mean velocity field and the Lagrangian covariance tensor. In earlier works by Foias, Holm & Titi [10], and ourselves [16], an analysis of the isotropic equations has been given. In the second part of this paper, we establish local in time well-posedness of the anisotropic LANS-α equations using quasilinear PDE type methods. (Accepted September 2, 2002) Published online November 26, 2002 Dedicated to Stuart Antman on the occasion of his 60th birthday Communicated by S. MüLLER     

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     

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].     

Energy minimization and the formation of microstructure in dynamic anti-plane shear

We investigate the behavior of a continuum model designed to provide insight into the dynamical development of microstructures observed during displacive phase transformations in certain materials. The model is presented within the framework of nonlinear viscoelasticity and is also of interest as an example of a strongly dissipative infinite-dimensional dynamical system whose forward orbits need not lie on a finite-dimensional attracting set, and which can display a subtle dependence on initial conditions quite different from that of classical finite-dimensional chaos.We study the problem of dynamical (two-dimensional) anti-plane shear with linear viscoelastic damping. Within the framework of nonlinear hyperelasticity, we consider both isotropic and anisotropic constitutive laws which can allow different phases and we characterize their ability to deliver minimizers and minimizing sequences of the stored elastic energy (Theorem 2.3). Using a transformation due to Rybka, we recast the problem as a semilinear degenerate parabolic system, thereby allowing the application of semigroup theory to establish existence, uniqueness and regularity of solutions in L ^p spaces (Theorem 3.1). We also discuss the issues of energy minimization and propagation of strain discontinuities. We comment on the difficulties encountered in trying to exploit the geometrical properties of specific constitutive laws. In particular, we are unable to obtain analogues of the absence of minimizers and of the non-propagation of strain discontinuities found by Ball, Holmes, James, Pego & Swart [1991] for a one-dimensional model problem.Several numerical experiments are presented, which prompt the following conclusions. It appears that the absence of an absolute minimizer may prevent energy minimization, thereby providing a dynamical mechanism to limit the fineness of observed microstructure, as has been proved in the one-dimensional case. Similarly, viscoelastic damping appears to prevent the propagation of strain discontinuities. During the extremely slow development of fine structure, solutions are observed to display local refinement in an effort to overcome incompatibility with boundary and initial conditions, with the distribution and shape of the resulting finer scales displaying a subtle dependence on initial conditions.     

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|.     

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.     

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.     

An upper bound to the free energy of n-variant polycrystalline shape memory alloys

In the last two decades, the problem of computing the elastic energy of phase transforming materials has been studied by a variety of research groups. Due to the non-quasiconvexity of the underlying multi-well landscape, different relaxation methods have been used in order to estimate the quasiconvex envelope of the energy density, for which no explicit expression is known at present.This paper combines a recently developed lamination bound for monocrystalline shape memory alloys which relies on martensitic twinned microstructures with the work of Smyshlyaev and Willis [1998a. A ‘non-local’ variational approach to the elastic energy minimization of martensitic polycrystals. Proc. R. Soc. London A 454, 1573–1613]. As a result, a lamination upper bound for n-variant polycrystalline martensitic materials is obtained.The lamination bound is then compared with Reuß- and Taylor-type estimates. While, for given volume fractions, good agreement of lamination upper and convexification lower bounds is obtained, a comparison using energy-minimizing volume fractions computed from the various bounds yields larger differences. Finally, we also investigate the influence of the polycrystal's texture. For a strong ellipsoidal texture, we observe even better agreement of upper and lower bounds than for the case of isotropic statistics.     

A local existence and uniqueness theorem for a K-BKZ-fluid

We consider the three-dimensional motion of a viscoelastic liquid occupying all of space. The constitutive law is assumed to be of the form suggested by Kaye [13] and Bernstein, Kearsley & Zapas [2]. An existence and uniqueness result for solutions of the initial value problem on sufficiently short time intervals is proved using Kato's theory of quasilinear hyperbolic equations.Sponsored by the United States Army under Contract No. DAAG29-80-C-0041. This material is based upon work supported by the National Science Foundation under Grants Nos. MCS-8210950 and MCS-8215064.     

Mechanics on manifolds and the incorporation of spin into Nelson's Stochastic mechanics

Nelson's theory of universal Brownian motion is generalized to manifold-valued processes with Hamiltonian of the form H=(p–A) ²+. It is shown that a spin model of Bopp & Haag is such a process. We show that, as the radius of the sphere of this model approaches zero, we recover the Pauli equation. We analyze further the case of no external field, singling out a continuous random variable which we call the angular momentum and showing that this random variable has the quantum mechanical expectation values. We also prove an ergodic theorem to the effect that the average value of the angular momentum equals its time average along trajectories.     

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)     

Plane entry flows and energy estimates for the navier-stokes equations

One of the classic problems of laminar flow theory is the development of velocity profiles in the inlet regions of channels or pipes. Such entry flow problems have been investigated extensively, usually by approximate techniques. In a recent paper [4], Horgan & Wheeler have provided an alternative approach, based on an energy method for the stationary Navier-Stokes equations. In [4], concerned with laminar flow in a cylindrical pipe of arbitrary cross-section, an analogy is drawn between the end effect issue of concern here, called the end effect, and the celebrated Saint-Venant's Principle of the theory of elasticity.In this paper, I consider the two-dimensional analog of the problem treated in [4] with a view to providing a more explicit formulation of the energy approach to entry flow problems. The flow development in a semi-infinite channel with parallel-plates is analyzed within the framework of the stationary Navier-Stokes equations. Introduction of a stream function leads to a formulation in terms of a boundary-value problem for a single fourth order nonlinear elliptic equation. In the case of Stokes flow, this problem is formally equivalent to a boundary-value problem for the biharmonic equation considered by Knowles [5] in the analysis of Saint-Venant's Principle in plane elasticity. The main result is an explicit estimate which establishes the exponential spatial flow development and leads to an upper bound for an appropriately defined entrance length. These results are obtained using differential inequality techniques analogous to those developed in investigation of Saint-Venant's Principle.     

Convergence of biorthogonal series of biharmonic eigenfunctions by the method of titchmarsh

Canonical edge problems for the biharmonic equation can be solved by separating variables. The eigenvalues and eigenvectors arising in this separation are derived from a reduced system of ordinary differential equations along lines suggested in the excellent work of R. C. Smith (1952). We study the reduced system which is governed by a vector ordinary differential equation. A solution of the biharmonic problem, governed by a partial differential equation, can be found only if the prescribed data is restricted to a subspace of the space spanned by the eigenfunctions of the reduced problem. The theory leads to problems in generalized harmonic analysis which seek conditions under which arbitrary vector fields f(y) with values in ² can be represented in terms of eigenvectors of the reduced problem. This paper adds new theorems and conjectures to the theory. We extend Smith's generalization to fourth-order problems of the methods introduced by Titchmarsh (1946) to study eigenfunction expansions associated with second-order problems. We use this method to prove that, if f(y)=[(f ₁(y), f ₂y)], -1y1, f(y) C¹[-1, 1], f L₂[-1, 1], then the series expressing f(y) converges uniformly to f(y) in the open interval (-1, 1), uniformly in [-1, 1] if f ₁(±1)=0 and, in any case, to [0, f ₂(±1)-f ₁(±1)] at y=±1. This is unlike Fourier series, which converge to the mean value of the periodic extension of a function. The series exhibits a Gibbs phenomenon near the end points of discontinuity when f ₁(±1) 0.The Gibbs undershoot and overshoot for the step function vector [1, 0] and ramp function vector [y, 0] are computed numerically. The undershoot and overshoot are much larger than in the case of Fourier series and, unlike Fourier series, the Gibbs oscillations do not appear to be entirely suppressed by Féjer's method of summing Cesaro sums. We show that, when f(y) has interior points of discontinuity, the series for f(y) diverges and we present numerical results which indicate that, in this divergent case, the Cesaro sums converge to f(y) apparently with Gibbs oscillations near the point of discontinuity.     

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.     

Materials with Internal Variables and Relaxation to Conservation Laws

.The theory of materials with internal state variables of Coleman & Gurtin [CG] provides a natural framework to investigate the structure of relaxation approximations of conservation laws from the viewpoint of continuum thermomechanics. After reviewing the requirements imposed on constitutive theories by the principle of consistency with the Clausius‐Duhem inequality, we pursue two specific theories pertaining to stress relaxation and relaxation of internal energy. They each lead to a relaxation framework for the theory of thermoelastic non‐conductors of heat, equipped with globally defined “entropy” functions for the associated relaxation process. Next, we consider a semilinear model problem of stress relaxation. We discuss uniform stability and compactness for solutions of the relaxation system in the zero‐relaxation limit, and establish convergence to the system of isothermal elastodynamics by using compensated compactness. Finally, we prove a strong dissipation estimate for the relaxation approximations proposed in Jin & Xin [JX] when the limit system is equipped with a strictly convex entropy. (Accepted June 17, 1998)     

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.     

Isentropic Approximation¶of the Compressible Euler System¶in One Space Dimension

Using the stability results of Bressan & Colombo [BC] for strictly hyperbolic 2 × 2 systems in one space dimension, we prove that the solutions of isentropic and non-isentropic Euler equations in one space dimension with the respective initial data (ρ₀, u ₀) and (ρ₀, u ₀, &\theta;₀=ρ₀ ^{γ− 1}) remain close as soon as the total variation of (ρ₀, u ₀) is sufficiently small. Accepted April 25, 2000?Published online November 24, 2000     

Phenomenological Modeling of Electrorheological Fluids with an Extended Casson-Model

In this paper we propose a phenomenological theory for electrorheological fluids. In general these are suspensions which undergo dramatic changes in their material properties if they are exposed to an electric field. In the context of continuum mechanics these fluids can be modeled as non-Newtonian fluids. Recalling the governing equations of rational thermodynamics and electrodynamics of moving media (Maxwell-Minkowski-equations), we derive suitable governing equations of electrorheology using essentially two assumptions concerning magnetic quantities. Furthermore we introduce a 3-dimensional nonlinear constitutive equation for the Cauchy stress tensor which is an extension of the model proposed by Ružička (see [14]). Assuming a viscometric flow, we compare the shear stress of our model with other well known models and fit the parameters by using measurements that were obtained in a rotational viscometer. Excellent agreement between model and measurements is achieved. On the basis of these results we propose a 3-dimensional model, the so-called extended Casson -model. This model is investigated further for a channel flow configuration with a homogeneous electric field. We determine analytical solutions for the electric field, the velocity and the volumetric flow rate and illustrate the velocity profiles and the predicted pressure drop. The velocity profiles are flattened compared to parabolic profiles and become more flat if the electric field increases. Received March 21, 2000     

A linear theory of straight elastic rods

This paper is based on the work of Green & Laws who have given a general thermodynamical theory of rods which is valid for any material. Here, starting with the general non-linear theory of elastic rods, we derive a linear theory allowing for thermal effects. The resulting free energy as a quadratic function of kinematic variables is restricted by certain symmetry conditions. The basic equations then separate into four groups, two for flexure, one for torsion and one for extension of the rod with temperature effects occurring only in the latter group. Wave propagation along an infinite rod is considered. There are two wave speeds for each type of flexure, two for torsion and three for isothermal extension and all wave speeds depend on the wave length.     

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).