A characteristics-mixed covolume method for a convection-dominated transport problem

In this paper, we propose a characteristics-mixed covolume method for approximating the solution to a convection dominated transport problem. The method is a combination of characteristic approximation to handle the convection term in time and mixed covolume method spatial approximation to deal with the diffusion term. The velocity and press are approximated by the lowest order Raviart-Thomas mixed finite element space on rectangles. The projection of a mixed covolume element is introduced. We prove its first order optimal rate of convergence for the approximate velocities in the L² norm as well as for the approximate pressures in the L² norm.     

Optimal decay rates for the compressible fluid models of Korteweg type

We consider the compressible Navier-Stokes-Korteweg system that models the motions of the compressible isothermal viscous capillary fluids. We prove the optimal L² and Lp, p?2 decay rates for the classical solutions and their spatial derivatives. In particular, the optimal L² decay rate of the second-order spatial derivatives is obtained. The proof is based on the detailed study of the linear decay estimates and nonlinear energy estimates.     

A Second-Order Model for Image Denoising

We present a variational model for image denoising and/or texture identification. Noise and textures may be modelled as oscillating components of images. The model involves a L ²-data fitting term and a Tychonov-like regularization term. We choose the BV ² norm instead of the classical BV norm. Here BV ² is the bounded hessian function space that we define and describe. The main improvement is that we do not observe staircasing effects any longer, during denoising process. Moreover, texture extraction can be performed with the same method. We give existence results and present a discretized problem. An algorithm close to the one set by Chambolle (J Math Imaging Vis 20:89–97, 2004) is used: we prove convergence and present numerical tests.     

Error estimates for a finite element-finite volume discretization of convection-diffusion equations

We consider a time-dependent linear convection-diffusion equation. This equation is approximated by a combined finite element-finite volume method: the diffusion term is discretized by Crouzeix-Raviart piecewise linear finite elements, and the convection term by upwind barycentric finite volumes on a triangular grid. An implicit Euler approach is used for time discretization. It is shown that the error associated with this scheme, measured by a discrete L^∞-L²- and L²-H¹-norm, respectively, decays linearly with the mesh size and the time step. This result holds without any link between mesh size and time step. The dependence of the corresponding error bound on the diffusion coefficient is completely explicit.     

Curve denoising by multiscale singularity detection and geometric shrinkage

We propose a method for denoising piecewise smooth curves, given a number of noisy sample points. Using geometric variants of wavelet shrinkage methods, our algorithm preserves corners while enforcing that the smoothed arcs lie in an L² Sobolev space Hα of order α chosen by the operator. The reconstruction is scale-invariant when using the Sobolev space H3/2, adapts to the local noise level, and is essentially free of tuning parameters. In particular, our noise-adaptivity ensures that there is no arbitrarily-chosen “diffusion time” parameter in the denoising. Further, in cases where the distinction between signal and noise is unclear, we show how statistics gathered from the curve can be used to identify a finite number of “good” choices for the denoising.     

Minimum-energy wavelet frame on the interval with arbitrary integer dilation factor

In this paper, we study minimum-energy frame Ψ={ψ¹,ψ²,…,ψ^M} on the interval with arbitrary factor d for L²[0,1], Ψ corresponding to some refinable functions with compact support. We give the constructive proof as well as the necessary and sufficient conditions of minimum-energy frames for L²[0,1], present the decomposition and reconstruction formulas of minimum-energy frame on the interval [0,1], and some examples. The experimental results show that the proposed minimum-energy frame on the interval improves the performance in the application of image denoising significantly.     

Local and 2-local derivations on noncommutative Arens algebras

The paper is devoted to so-called local and 2-local derivations on the noncommutative Arens algebra L ^ω(M,τ) associated with a von Neumann algebra M and a faithful normal semi-finite trace τ. We prove that every 2-local derivation on L ^ω(M,τ) is a spatial derivation, and if M is a finite von Neumann algebra, then each local derivation on L ^ω(M,τ) is also a spatial derivation and every 2-local derivation on M is in fact an inner derivation.     

Diffusion vanishing limit of the nonlinear pipe Magnetohydrodynamic flow with fixed viscosity

We establish magnetic diffusion vanishing limit of the nonlinear pipe Magnetohydrodynamic flow by the mathematical validity of the Prandtl boundary layer theory with fixed viscosity. The convergence is verified under various Sobolev norms, including the L~∞(L~2)and L~∞(H~1) norm.     

Varieties of Lattices with Geometric Descriptions

A lattice L is spatial if every element of L is a join of completely join-irreducible elements of L (points), and strongly spatial if it is spatial and the minimal coverings of completely join-irreducible elements are well-behaved. Herrmann et al. proved in 1994 that every modular lattice can be embedded, within its variety, into an algebraic and spatial lattice. We extend this result to n-distributive lattices, for fixed n. We deduce that the variety of all n-distributive lattices is generated by its finite members, thus it has a decidable word problem for free lattices. This solves two problems stated by Huhn in 1985. We prove that every modular (resp., n-distributive) lattice embeds within its variety into some strongly spatial lattice. Every lattice which is either algebraic modular spatial or bi-algebraic is strongly spatial. We also construct a lattice that cannot be embedded, within its variety, into any algebraic and spatial lattice. This lattice has a least and a largest element, and it generates a locally finite variety of join-semidistributive lattices.     

Harmonic two-forms on manifolds with non-negative isotropic curvature

We prove that L ² harmonic two-forms are parallel if a complete manifold (M, g) has the non-negative isotropic curvature. Furthermore, if (M, g) has positive isotropic curvature at some point, then there is no non-trivial L ² harmonic two-form. We obtain that an almost K?hler manifold of non-negative isotropic curvature is K?hler and a symplectic manifold can not admit any almost K?hler structure of positive isotropic curvature.     

Systems of p-Laplacian differential inclusions with large diffusion

In this paper we consider coupled systems of p-Laplacian differential inclusions and we prove, under suitable conditions, that a homogenization process occurs when diffusion parameters become arbitrarily large. In fact we obtain that the attractors are continuous at infinity on L²(Ω)×L²(Ω) topology, with respect to the diffusion coefficients, and the limit set is the attractor of an ordinary differential problem.     

Dual mixed volumes and the slicing problem

We develop a technique using dual mixed-volumes to study the isotropic constants of some classes of spaces. In particular, we recover, strengthen and generalize results of Ball and Junge concerning the isotropic constants of subspaces and quotients of Lp and related spaces. An extension of these results to negative values of p is also obtained, using generalized intersection-bodies. In particular, we show that the isotropic constant of a convex body which is contained in an intersection-body is bounded (up to a constant) by the ratio between the latter's mean-radius and the former's volume-radius. We also show how type or cotype 2 may be used to easily prove inequalities on any isotropic measure.     

A posteriori error estimation of residual type for anisotropic diffusion-convection-reaction problems

This paper presents a robust a posteriori residual error estimator for diffusion-convection-reaction problems with anisotropic diffusion, approximated by a SUPG finite element method on isotropic or anisotropic meshes in R^d, d=2 or 3. The equivalence between the energy norm of the error and the residual error estimator is proved. Numerical tests confirm the theoretical results.     

Surface diffusion flow near spheres

We consider closed immersed hypersurfaces evolving by surface diffusion flow, and perform an analysis based on local and global integral estimates. First we show that a properly immersed stationary (ΔH ≡ 0) hypersurface in ${\mathbb{R}^3}$ or ${\mathbb{R}^4}$ with restricted growth of the curvature at infinity and small total tracefree curvature must be an embedded union of umbilic hypersurfaces. Then we prove for surfaces that if the L ² norm of the tracefree curvature is globally initially small it is monotonic nonincreasing along the flow. We also derive pointwise estimates for all derivatives of the curvature assuming that its L ² norm is locally small. Using these results we show that if a singularity develops the curvature must concentrate in a definite manner, and prove that a blowup under suitable conditions converges to a nonumbilic embedded stationary surface. We obtain our main result as a consequence: the surface diffusion flow of a surface initially close to a sphere in L ² is a family of embeddings, exists for all time, and exponentially converges to a round sphere.     

Lifespan Theorem for simple constrained surface diffusion flows

We consider closed immersed hypersurfaces in R³ and R⁴ evolving by a special class of constrained surface diffusion flows. This class of constrained flows includes the classical surface diffusion flow. In this paper we present a Lifespan Theorem for these flows, which gives a positive lower bound on the time for which a smooth solution exists, and a small upper bound on the total curvature during this time. The hypothesis of the theorem is that the surface is not already singular in terms of concentration of curvature. This turns out to be a deep property of the initial manifold, as the lower bound on maximal time obtained depends precisely upon the concentration of curvature of the initial manifold in L² for M² immersed in R³ and additionally on the concentration in L³ for M³ immersed in R⁴. This is stronger than a previous result on a different class of constrained surface diffusion flows, as here we obtain an improved lower bound on maximal time, a better estimate during this period, and eliminate any assumption on the area of the evolving hypersurface.     

Some convergence results on quadratic forms for random fields and application to empirical covariances

Limit theorems are proved for quadratic forms of Gaussian random fields in presence of long memory. We obtain a non central limit theorem under a minimal integrability condition, which allows isotropic and anisotropic models. We apply our limit theorems and those of Ginovian (J. Contemp. Math. Anal. 34(2):1?C15) to obtain the asymptotic behavior of the empirical covariances of Gaussian fields, which is a particular example of quadratic forms. We show that it is possible to obtain a Gaussian limit when the spectral density is not in L ². Therefore the dichotomy observed in dimension d?=?1 between central and non central limit theorems cannot be stated so easily due to possible anisotropic strong dependence in d?>?1.     

A null series with small anti-analytic part

We show that it is possible for an L² function on the circle, which is a sum of an almost everywhere convergent series of exponentials with positive frequencies, to not belong to the Hardy space H². A consequence in the uniqueness theory is obtained. To cite this article: G. Kozma, A. Olevskiǐ, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Convergence and superconvergence analyses of HDG methods for time fractional diffusion problems

We study the hybridizable discontinuous Galerkin (HDG) method for the spatial discretization of time fractional diffusion models with Caputo derivative of order 0 < α < 1. For each time t ∈ [0, T], when the HDG approximations are taken to be piecewise polynomials of degree k ≥ 0 on the spatial domain Ω, the approximations to the exact solution u in the L _∞(0, T; L ₂(Ω))-norm and to ?u in the \(L_{\infty }(0, \textit {T}; \mathbf {L}_{2}({\Omega }))\)-norm are proven to converge with the rate h ^k+1 provided that u is sufficiently regular, where h is the maximum diameter of the elements of the mesh. Moreover, for k ≥ 1, we obtain a superconvergence result which allows us to compute, in an elementwise manner, a new approximation for u converging with a rate h ^k+2 (ignoring the logarithmic factor), for quasi-uniform spatial meshes. Numerical experiments validating the theoretical results are displayed.     

Approximations of the resolvent for a non-self-adjoint diffusion operator with rapidly oscillating coefficients

A strongly inhomogeneous diffusion operatorwith drift depending on a small parameter ? is studied in the space L ²(? ⁿ ). The strong inhomogeneity consists in that the coefficients of the operator are ?-periodic and, in addition, the drift vector is of the order of ? ^?1. As ? → 0, approximations in the operator L ²-norm of order ? and ? ² are constructed for the resolvent of the operator. For each of these orders of approximation, an averaged diffusion operator is obtained. A spectral method based on the Bloch representation for an operator with periodic coefficients is used.     

Rigorous derivation of nonlinear plate theory and geometric rigidityDérivation rigoureuse de la théorie non linéaire des plaques et rigidité géométrique

We show that nonlinear plate theory arises as a Γ-limit of three-dimensional nonlinear elasticity. A key ingredient in the proof is a sharp rigidity estimate for maps $\text{v:(0,1)}{}^3\text{→}\text{R}{}^3$. We show that the L² distance of ?v from a single rotation is bounded by a multiple of the L² distance from the set SO(3) of all rotations. To cite this article: G. Friesecke et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 173–178