Heteroclinic Cycles and Homoclinic Closures for Generic Diffeomorphisms

We prove some C ¹ generic results about orbit-connecting, in particular about heteroclinic cycles and homoclinic closures. As a consequence we obtain a three-ways C ¹ density theorem: Diffeomorphisms with either infinitely many weakly transitive components or a heterodimensional cycle are C ¹ dense in the complement of the C ¹ closure of Axiom A and no-cycle diffeomorphisms.     

Periodically Expansive Homoclinic Classes

In this paper we introduce the notion of periodically expansiveness and discuss the homoclinic classes exhibit the property in a persistent way. More precisely, we prove that if a homoclinic class H(p, f) of a diffeomorphisms f is C ¹-persistently periodically expansive then it admits a dominated splitting ${E \bigoplus F}$ with dim?(E)?=?index?(p). We also prove that C ¹-generically any locally maximal periodically expansive homoclinic class is hyperbolic.     

Regularity Issues in the Problem of Fluid Structure Interaction

We investigate the evolution of rigid bodies in a viscous incompressible fluid. The flow is governed by the 2D Navier–Stokes equations, set in a bounded domain with Dirichlet boundary conditions. The boundaries of the solids and the domain have Hölder regularity C ^1,α , 0 < α ≦ 1. First, we show the existence and uniqueness of strong solutions up to the collision. A key ingredient is a BMO bound on the velocity gradient, which substitutes to the standard H ² estimate for smoother domains. Then, we study the asymptotic behaviour of one C ^1,α body falling over a flat surface. We show that a collision is possible in finite time if and only if α < 1/2.     

Stationary problem of third-grade fluids in two and three dimensions: existence and uniqueness

This paper is devoted to the stationary problem of third-grade fluids in two and three dimensions. In two dimensions, we show existence of solutions and uniqueness, for a boundary of class C^2,1 and small data, by generalizing the method used by J.M. Bernard for the stationary problem of second-grade fluids (we deal with a polynomial of four degrees instead of two degrees). Contrary to the case of two dimensions, the resolution of the problem of third-grade fluids in three dimensions requires the physical condition |α₁+α₂|<(24νβ)^1/2. From this condition, we derive two “pseudo ellipticities” for the operator ν|A(u)|²+(α₁+α₂)tr(A(u)³)+β|A(u)|⁴, where A(u) is a 3-order symmetric matrix such that tr(A(u))=0. Thus, with, in addition, a sharp estimate of the scalar product (|A(u)|²A(u)-|A(v)|²A(v),A(u)-A(v)), we are able to prove existence of solutions and uniqueness, for a boundary of class C^2,1 and small data, in three dimensions.

Résumé

Cet article est consacré au problème stationnaire des fluides de grade trois en dimension deux et trois. En dimension deux, nous montrons l’existence de solutions et l’unicité, pour une frontière de classe C^2,1 et une donnée petite, en généralisant la méthode utilisée par J.M. Bernard pour le problème stationnaire des fluides de grade deux (nous avons affaire à un polynôme de degré quatre au lieu de deux). Contrairement au cas de la dimension deux, la résolution du problème des fluides de grade trois en dimension trois requière la condition physique |α₁+α₂|<(24νβ)^1/2. De cette condition, nous déduisons deux “pseudo matrice” pour l’opérateur ν|A(u)|²+(α₁+α₂)tr(A(u)³)+β|A(u)|⁴, où A(u) est une matice symétrique d’ordre 3 à trace nulle. De là, avec, en plus, une fine estimation du produit scalaire (|A(u)|²A(u)-|A(v)|²A(v),A(u)-A(v)), nous sommes capables de prouver l’existence de solutions et l’unicité, pour une frontière de classe C^2,1 et une donnée petite, en dimension trois.     

Strong L p -Solutions of the T α-Type Navier–Stokes Equation and the Regularizing-Decay Rate Estimation

In the present paper, we investigate t ^α-type Navier–Stokes equations introduced in a previous paper of Tu–Zhai. The existence and uniqueness results are given in L ^p space. Moreover, the regularizing decay rate estimates and high order approximation are derived for these solutions.     

Incompressible laminar 2D steady thermal boundary layers with temperature-dependent kinematic viscosity and thermal diffusivity

We prove that the incompressible 2D steady thermal boundary layer equations with temperature-dependent kinematic viscosity ν and thermal diffusivity α is maximally symmetric provided the Prantl number Pr=ν/α is constant and or ν=K₂(AT+B)^K₁ if we neglect energy dissipation and if we take into account dissipation. This result corroborates assumptions often made in applications. When we disregard dissipation, the symmetry Lie algebra assumes the forms L^r⊕L^∞, where L^∞ is an infinite-dimensional Lie algebra and L^r is an r-dimensional Lie algebra with r∈{3,4,5,6}. If we include dissipation, r∈{2,3}. We notice that dissipation has a symmetry breaking effect.We also show how the symmetries can be employed for the calculation of invariant solutions.     

W2,p Estimates for the Parabolic¶Monge-Ampère Equation

When u is a solution to the equation ?u _t det D _x ² u=f with f positive, continuous, and f _t satisfying certain growth conditions, we establish estimates in L ^∞ for u _t and show that D _x ² u satisfies uniform interior estimates in L ^p for 0相似文献   

Differential Systems with Feedback: Time Discretizations and Lyapunov Functions

In this paper we examine a class of Eulerian time discretizations for a monotone cyclic feedback system with a time delay; see Mallet-Paret and Sell (1996a, 1996b) for background information. We construct an integer-valued function V for the discrete-time problem. The Main Theorem shows that V is a Lyapunov function, that is, V(x ⁿ⁺¹)≤V(x ⁿ ) along a solution {x ⁿ }^∞ _n=0, where the time steps can be relatively large.     

Universal Attractors¶for the Navier-Stokes Equations¶of Compressible and Heat-Conductive Fluid¶in Bounded Annular Domains in ℝn

This paper is concerned with the dynamics for the Navier-Stokes equations for a polytropic viscous heat-conductive ideal gas in bounded annular domains Ω _n in ? ⁿ (n= 2, 3). One of the important features of this problem is that the metric spaces H ⁽¹⁾ and H ⁽²⁾ we work with are two incomplete metric spaces, as can be seen from the constraints θ >0 and u> 0, withθ and u being absolute temperature and specific volume respectively. For any constants δ₁, δ₂, δ₃, δ₄, δ₅ satisfying certain conditions, two sequences of closed subspaces H ^{( i )} _δ?H ^{( i )} (i= 1,2) are found, and the existence of two (maximal) universal attractors in H ⁽¹⁾ _δ and H ⁽²⁾ _δ is proved.     

On the Transformation Property of the Deformation Gradient under a Change of Frame

If the deformation gradients are denoted by F and F ^* respectively before and after a change of frame, they are related by the transformation formula, F ^*=QF, where Q is the orthogonal transformation associated with the change of frame. Although it has been pointed out that this relation is valid “provided that the reference configuration be unaffected by the change of frame” (see p. 308 of [1]), this formula is found in most textbook of Continuum Mechanics, and is used, without further justification, in deriving the condition of material frame-indifference, ?(QF)=Q?(F)Q ^T for the constitutive function ? of the stress tensor of an elastic body. In this note, we shall analyze the effect of change of frame on the transformation property of the deformation gradient, and show that the above transformation formula is not valid in general. However, we shall confirm the validity of the above well-known condition of material frame-indifference without the assumption that the reference configuration be unaffected by the change of frame.     

The L-pseudo-solution using stochastic algorithm of Landweber

In this paper, we consider a linear equation Ax=u. A is an operator with an unbounded inverse in a Hilbert space. The right side u does not belong to the range of A. Obviously, a solution in classical sense does not exist and A ^?1 u does not have a sense. To solve this problem arising from many experimental fields of science, where the second member u stems from measurements, we propose a recurrent procedure which converges almost completely and in quadratic mean to L-pseudo-solution and for which we build up a confidence interval. To check the validity of our results, a numerical example which is standard in rheology is proposed.     

A note on periodic solutions of Riccati equations

In this note, we show that under certain assumptions the scalar Riccati differential equation x′=a(t)x+b(t)x ²+c(t) with periodic coefficients admits at least one periodic solution. Also, we give two illustrative examples in order to indicate the validity of the assumptions.     

On the correspondence between the three- and four-dimensional parameters of the three-dimensional rotation group

A fundamental kinematic theorem due to Euler permits synthesizing a series of three- and four-dimensional orientation parameters that correspond to each other in spaces of the same dimension. We use the theorem about the homeomorphism of two topological spaces (the three-dimensional sphere S ³ ? R ⁴ with a single punctured (removed) point and the three-dimensional space R ³) to establish a one-to-one mutually continuous correspondence between the four- and three-dimensional kinematic parameters prescribed in these spaces. The latter can be proved using the stereographic projection of points of the sphere S ³ onto the hyperplane R ³. For the normalized (Hamiltonian) Rodrigues-Hamilton parameters, we present a method of stereographic projection of a point belonging to the three-dimensional sphere S ³ onto the oriented space R ³. We present a family of local kinematic parameters obtained by the method of mapping four symmetric kinematic parameters of the space R ⁴ onto the oriented real space R ³. In contrast to the well-known four symmetric global parameters of the Rodrigues-Hamilton orientation, the synthesized three-dimensional orientation parameters are local (have two singular points ±360°). The differential equations of rotation in the three-dimensional orientation parameters are obtained by the projection method. We present the three-dimensional parameters corresponding to the classical Hamiltonian quaternions defined in the four-dimensional vector space R ⁴.     

Design of analog variable fractional order differentiator and integrator

This paper deals with the design of analog variable fractional order differentiator s ^m and integrator s ^?m , for 0<m<1, for a given frequency band, a subject that has not been yet investigated. The main feature of this analog variable fractional order integrator or differentiator is that its frequency characteristics can be changed without redesigning a new one. First, analog rational function approximation of the fractional order differentiator s ^m and integrator s ^?m are derived with the new idea to keep all its poles to be independent of the fractional orders?m. Next, we have used the polynomial interpolation method to design the variable fractional order analog integrator and differentiator that can be implemented by an analog structure like the digital Farrow structure. Finally, some examples are presented to illustrate the efficiency and the effectiveness of the proposed design method.     

Intrinsic Geometry and Analysis of Diffusion Processes and L ∞-Variational Problems

The aim of this paper is twofold. First, we obtain a better understanding of the intrinsic distance of diffusion processes. Precisely, (a) for all n ≧ 1, the diffusion matrix A is weak upper semicontinuous on Ω if and only if the intrinsic differential and the local intrinsic distance structures coincide; (b) if n = 1, or if n ≧ 2 and A is weak upper semicontinuous on Ω, the intrinsic distance and differential structures always coincide; (c) if n ≧ 2 and A fails to be weak upper semicontinuous on Ω, the (non-)coincidence of the intrinsic distance and differential structures depend on the geometry of the non-weak-upper-semicontinuity set of A. Second, for an arbitrary diffusion matrix A, we show that the intrinsic distance completely determines the absolute minimizer of the corresponding L ^∞-variational problem, and then obtain the existence and uniqueness for given boundary data. We also give an example of a diffusion matrix A for which there is an absolute minimizer that is not of class C ¹. When A is continuous, we also obtain the linear approximation property of the absolute minimizer.     

Statistical description of a cloud of compressible bubbles

We use the methods of statistical mechanics to describe the interaction of N compressible gas bubbles in an incompressible, inviscid and irrotational liquid. The governing equations for bubble positions, radii and corresponding momenta form a Hamiltonian system depending on the virtual mass matrix. An explicit expression of the virtual mass matrix is presented, which is calculated with accuracy (b/d)³, where b and d are respectively the mean bubble radius and the mean inter-bubble distance. We study two limit cases: the limit of moving rigid spheres and the limit of immobile oscillating bubbles. In each case, we construct a canonical ensemble partition function. In the limit of rigid spheres, we improve results by Yurkovetsky and Brady (phys Fluids 8(4): 881–895, 1996). In particular, we derive an analytic expression for the “attractive” potential which may be responsible for the clustering effect, and show why the accuracy (b/d)³ is not sufficient to characterize the “repulsive potential” . In the limit of immobile oscillating bubbles, we prove the existence of a long range repulsive potential.     

On the symmetry of deformable crystals

We prove existence, uniqueness, and regularity properties for a solution u of the Bellman-Dirichlet equation of dynamic programming: (1) $$\left\{ \begin{gathered} \max {\text{ }}\{ L^i u + f^i = 0{\text{ in }}\Omega \hfill \\ i{\text{ = 1,2 }} \hfill \\ u{\text{ = 0 on }}\partial \Omega , \hfill \\ \end{gathered} \right.$$ where L ¹ and L ² are two second order, uniformly elliptic operators. The method of proof is to rephrase (1) as a variational inequality for the operator K=L ²(L ¹)^?1 in L ²(Ω) and to invoke known existence theorems. For sufficiently nice f ¹ and f ² we prove in addition that u is in H ³(Ω)?C ^2,α(Ω) (for some 0<α<1) and hence is a classical solution of (1).     

Double Obstacle Problems with Obstacles Given by Non-C 2 Hamilton–Jacobi Equations

We prove optimal regularity for double obstacle problems when obstacles are given by solutions to Hamilton–Jacobi equations that are not C ². When the Hamilton–Jacobi equation is not C ² then the standard Bernstein technique fails and we lose the usual semi-concavity estimates. Using a non-homogeneous scaling (different speeds in different directions) we develop a new pointwise regularity theory for Hamilton–Jacobi equations at points where the solution touches the obstacle. A consequence of our result is that C ¹-solutions to the Hamilton–Jacobi equation $$\pm |\nabla h-a(x)|^2=\pm 1\,{\rm in}\,B_1,\quad h=f \,{\rm on}\, \partial B_1$$ , are, in fact, C ^1,α/2, provided that ${a \in C^\alpha}$ . This result is optimal and, to the authors’ best knowledge, new.     

Asymptotics for the eigenelements of vibrating membranes with very heavy thin inclusionsComportement asymptotique des éléments propres pour des membranes vibrantes avec des couches très minces et très lourdes

We consider the vibrations of a membrane that contains a very thin and heavy inclusion around a curve γ. We assume that the membrane occupies a domain $\text{Ω}$ of $\text{R}{}^2$. The inclusion occupies a layer-like domain $\text{ω}{}_{\mathrm{\epsilon }}\text{?Ω}$ of width 2ε and it has a density of order O(ε^?3). The density is of order O(1) outside this inclusion, the concentrated mass around the curve γ. ε is a positive parameter, ε∈(0,1). By means of asymptotic expansions, we describe the behaviour, as ε→0, of the eigenelements (λ^ε,u^ε) of the associated spectral problem. We provide complete asymptotic series for the low frequencies λ^ε=O(ε²), the medium frequencies λ^ε=O(ε) and the corresponding eigenfunctions u^ε. To cite this article: Y. Golovaty et al., C. R. Mecanique 330 (2002) 777–782.