Robust exponential attractors for a phase-field system with memory

H.G. Rotstein et al. proposed a nonconserved phase-field system characterized by the presence of memory terms both in the heat conduction and in the order parameter dynamics. These hereditary effects are represented by time convolution integrals whose relaxation kernels k and h are nonnegative, smooth and decreasing. Rescaling k and h properly, we obtain a system of coupled partial integrodifferential equations depending on two relaxation times ɛ and σ. When ɛ and σ tend to 0, the formal limiting system is the well-known nonconserved phase-field model proposed by G. Caginalp. Assuming the exponential decay of the relaxation kernels, the rescaled system, endowed with homogeneous Neumann boundary conditions, generates a dissipative strongly continuous semigroup S_{ɛ, σ}(t) on a suitable phase space, which accounts for the past histories of the temperature as well as of the order parameter. Our main result consists in proving the existence of a family of exponential attractors for S_{ɛ, σ}(t), with ɛ, σ ∈ [0, 1], whose symmetric Hausdorff distance from tends to 0 in an explicitly controlled way.     

Energy decay for Timoshenko systems of memory type

Linear systems of Timoshenko type equations for beams including a memory term are studied. The exponential decay is proved for exponential kernels, while polynomial kernels are shown to lead to a polynomial decay. The optimality of the results is also investigated.     

Gibbs–Markov–Young structures with (stretched) exponential tail for partially hyperbolic attractors

In this work we extend the results obtained by Gouëzel in [12] to partially hyperbolic attractors. We study a forward invariant set K on a Riemannian manifold M   whose tangent space splits as dominated decomposition _TKM=E^cu⊕E^s$T_{K}M=E^{cu}\oplus E^s$, for which the center-unstable direction E^cu$E^{cu}$ is non-uniformly expanding on some local unstable disk. We prove that the (stretched) exponential decay of recurrence times for an induced scheme can be deduced under the assumption of (stretched) exponential decay of the time that typical points need to achieve some uniform expanding in the center-unstable direction. As an application of our results we obtain exponential decay of correlations and exponential large deviations for a class of partially hyperbolic diffeomorphisms considered in [1].     

Stability of global and exponential attractors for a three‐dimensional conserved phase‐field system with memory

We consider a conserved phase‐field system on a tri‐dimensional bounded domain. The heat conduction is characterized by memory effects depending on the past history of the (relative) temperature ?, which is represented through a convolution integral whose relaxation kernel k is a summable and decreasing function. Therefore, the system consists of a linear integrodifferential equation for ?, which is coupled with a viscous Cahn–Hilliard type equation governing the order parameter χ. The latter equation contains a nonmonotone nonlinearity ? and the viscosity effects are taken into account by a term ?αΔ?_tχ, for some α?0. Rescaling the kernel k with a relaxation time ε>0, we formulate a Cauchy–Neumann problem depending on ε and α. Assuming a suitable decay of k, we prove the existence of a family of exponential attractors {?_α,ε} for our problem, whose basin of attraction can be extended to the whole phase–space in the viscous case (i.e. when α>0). Moreover, we prove that the symmetric Hausdorff distance of ?_α,ε from a proper lifting of ?_α,0 tends to 0 in an explicitly controlled way, for any fixed α?0. In addition, the upper semicontinuity of the family of global attractors {??_α,ε} as ε→0 is achieved for any fixed α>0. Copyright © 2009 John Wiley & Sons, Ltd.     

Existence and general decay of a transmission problem for the plate equation with a memory condition on the boundary

In this paper, we study the existence and decay of a transmission problem for the plate equation with a memory condition on part of the boundary. First, we prove the global existence of weak solution by using Faedo–Galerkin’s method and compactness arguments. Then, without imposing \({u_0 = \frac{\partial u_0}{\partial\nu} = 0}\) on \({\Gamma_2}\), two explicit decay rate results are established under two different assumptions of the resolvent kernels. Both of these decay results allow a wider class of relaxation functions and initial data and thus generalize some previous results existing in the literature.     

End effects in three-phase-lag heat conduction

In this article we study the spatial behaviour of solutions for the three-phase-lag heat equation on a semi-infinite cylinder. First, we investigate the spatial evolution of solutions of an initial boundary-value problem with zero boundary conditions on the lateral surface of the cylinder. Under null initial data, a Phragmen–Lindelof alternative is obtained. An upper bound for the amplitude term in terms of the boundary data is also established. For the case of decay solutions, an improvement is obtained. We prove that the decay can be controlled by the exponential of a second-degree polynomial in the distance from the finite end of the cylinder. A class of non-standard problems is also considered for which the temperature and its first two time derivatives at a fixed time T ₀ are assumed proportional to their initial values. These results are relevant in the context of the Saint-Venant Principle for heat conduction problems.     

On the asymptotic behavior of the quasi-static problem for a linear viscoelastic fluid

In this paper we study the quasi-static problem for a viscoelastic fluid by means of the concept of minimal state. This implies the use of a different free energy defined in a wider space of data. The existence and uniqueness is proved in this new space and the asymptotic decay for the problem with non vanishing supplies is obtained for a large class of memory kernels, including those presenting an exponential or polynomial decay.     

Segal–Bargmann and Weyl transforms on compact Lie groups

We present an elementary derivation of the reproducing kernel for invariant Fock spaces associated with compact Lie groups which, as ólafsson and ?rsted showed in (Lie Theory and its Applicaitons in Physics. World Scientific, 1996), yields a simple proof of the unitarity of Hall’s Segal–Bargmann transform for compact Lie groups K. Further, we prove certain Hermite and character expansions for the heat and reproducing kernels on K and K_{\mathbb C}{K_{\mathbb C}} . Finally, we introduce a Toeplitz (or Wick) calculus as an attempt to have a quantization of the functions on K_{\mathbb C}{K_{\mathbb C}} as operators on the Hilbert space L ²(K).     

The covering number in learning theory

The covering number of a ball of a reproducing kernel Hilbert space as a subset of the continuous function space plays an important role in Learning Theory. We give estimates for this covering number by means of the regularity of the Mercer kernel K. For convolution type kernels K(x,t)=k(x−t) on [0,1]ⁿ, we provide estimates depending on the decay of k̂, the Fourier transform of k. In particular, when k̂ decays exponentially, our estimate for this covering number is better than all the previous results and covers many important Mercer kernels. A counter example is presented to show that the eigenfunctions of the Hilbert–Schmidt operator Lm_K associated with a Mercer kernel K may not be uniformly bounded. Hence some previous methods used for estimating the covering number in Learning Theory are not valid. We also provide an example of a Mercer kernel to show that L_K^1/2 may not be generated by a Mercer kernel.     

Heat Kernels, Smoothness Estimates, and Exponential Decay

In this article, we characterize functions whose Fourier transforms have exponential decay. We characterize such functions by showing that they satisfy a family of estimates that we call quantitative smoothness estimates (QSE). Using the QSE, we establish Gaussian decay in the “bad direction” for the □ _b -heat kernel on polynomial models in ? ⁿ⁺¹. On the transform side, the problem becomes establishing QSE on a heat kernel associated to the weighted $\bar{\partial}$ -operator on L ²(?). The bounds are established with Duhamel’s formula and careful estimation. In ?², we can prove both on and off-diagonal decay for the □ _b -heat kernel on polynomial models.     

A general method for proving sharp energy decay rates for memory-dissipative evolution equations

This Note is concerned with stabilization of hyperbolic systems by a distributed memory feedback. We present here a general method which gives energy decay rates in terms of the asymptotic behavior of the kernel at infinity. This method, which allows us to recover in a natural way the known cases (exponential, polynomial, …), applies to a large quasi-optimal class of kernels. It also provides sharp energy decay rates compared to the ones that are available in the literature. We give a general condition under which the energy of solutions is shown to decay at least as fast as the kernel at infinity. To cite this article: F. Alabau-Boussouira, P. Cannarsa, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Stability of a transmission problem for Kirchhoff‐type wave equations with memory on the boundary

In this paper, we investigate the influence of boundary dissipation on the decay property of solutions for a transmission problem of Kirchhoff‐type wave equations with a memory condition on one part of the boundary. Without the condition u₀ = 0 on Γ₀, we establish a general decay of energy depending on the behavior of relaxation function by introducing suitable energy and Lyapunov functionals. This result allows a wider class of relaxation functions. Copyright © 2016 John Wiley & Sons, Ltd.     

Approach to self‐similarity in Smoluchowski's coagulation equations

We consider the approach to self‐similarity (or dynamical scaling) in Smoluchowski's equations of coagulation for the solvable kernels K(x, y) = 2, x + y and xy. In addition to the known self‐similar solutions with exponential tails, there are one‐parameter families of solutions with algebraic decay, whose form is related to heavy‐tailed distributions well‐known in probability theory. For K = 2 the size distribution is Mittag‐Leffler, and for K = x + y and K = xy it is a power‐law rescaling of a maximally skewed α‐stable Lévy distribution. We characterize completely the domains of attraction of all self‐similar solutions under weak convergence of measures. Our results are analogous to the classical characterization of stable distributions in probability theory. The proofs are simple, relying on the Laplace transform and a fundamental rigidity lemma for scaling limits. © 2003 Wiley Periodicals, Inc.     

Polyharmonic and Related Kernels on Manifolds: Interpolation and Approximation

This article is devoted to developing a theory for effective kernel interpolation and approximation in a general setting. For a wide class of compact, connected C ^∞ Riemannian manifolds, including the important cases of spheres and SO(3), and using techniques involving differential geometry and Lie groups, we establish that the kernels obtained as fundamental solutions of certain partial differential operators generate Lagrange functions that are uniformly bounded and decay away from their center at an algebraic rate, and in certain cases, an exponential rate. An immediate corollary is that the corresponding Lebesgue constants for interpolation as well as for L ₂ minimization are uniformly bounded with a constant whose only dependence on the set of data sites is reflected in the mesh ratio, which measures the uniformity of the data. The kernels considered here include the restricted surface splines on spheres, as well as surface splines for SO(3), both of which have elementary closed-form representations that are computationally implementable. In addition to obtaining bounded Lebesgue constants in this setting, we also establish a “zeros lemma” for domains on compact Riemannian manifolds—one that holds in as much generality as the corresponding Euclidean zeros lemma (on Lipschitz domains satisfying interior cone conditions) with constants that clearly demonstrate the influence of the geometry of the boundary (via cone parameters) as well as that of the Riemannian metric.     

Exponential and polynomial stability of a wave equation for boundary memory damping with singular kernels

This work is concerned with stabilization of a wave equation stabilized by a boundary feedback. When the feedback is both frictional and with memory, we prove exponential stability of the solutions. In case of a boundary feedback which is only of memory type, uniform stability is not expected. We prove in this latter case, that the solutions decay polynomially. The method is new and uses the method of higher order energies (see [F. Alabau-Boussouira, J. Prüss, R. Zacher, Exponential and polynomial stabilization of wave equations subjected to boundary-memory dissipation with singular kernels, in preparation; F. Alabau, Stabilisation frontière indirecte de systèmes faiblement couplés, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999) 1015–1020; F. Alabau, P. Cannarsa, V. Komornik, Indirect internal damping of coupled systems, J. Evolution Equations 2 (2002) 127–150; F. Alabau, Indirect boundary stabilization of weakly coupled systems, SIAM J. Control Optim. 41 (2002) 511–541]), the multiplier method and the properties of a large class of singular kernels. Moreover, our method can be extended to include cases of nonsingular kernels (see [V. Vergara, R. Zacher, Lyapunov functions and convergence to steady state for differential equations of fractional order, Math. Z. 259 (2008) 287–309; R. Zacher, Convergence to equilibrium for second order differential equations with weak damping of memory type, preprint.]). To cite this article: F. Alabau-Boussouira et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Refinable Functions with Exponential Decay: An?Approach via Cascade Algorithms

Refinable functions with exponential decay arise from applications such as the Butterworth filters in signal processing. Refinable functions with exponential decay also play an important role in the study of Riesz bases of wavelets generated from multiresolution analysis. A fundamental problem is whether the standard solution of a refinement equation with an exponentially decaying mask has exponential decay. We investigate this fundamental problem by considering cascade algorithms in weighted L _p spaces (1≤p≤∞). We give some sufficient conditions for the cascade algorithm associated with an exponentially decaying mask to converge in weighted L _p spaces. Consequently, we prove that the refinable functions associated with the Butterworth filters are continuous functions with exponential decay. By analyzing spectral properties of the transition operator associated with an exponentially decaying mask, we find a characterization for the corresponding refinable function to lie in weighted L ₂ spaces. The general theory is applied to an interesting example of bivariate refinable functions with exponential decay, which can be viewed as an extension of the Butterworth filters.     

Décroissance non exponentielle du noyau de la chaleur

Summary In this paper, we give estimates on the heat kernel on the diagonal in terms of exp[–|Logt|], >1 when1 vanishes. Moreover, we compute, by means of the stochastic calculus of variations, the decay of the flatness of some heat kernels. On the other hand, we show that the computations in both cases are very similar.     

Diffusion processes on an interval under linear moment conditions

We discuss a class of linear and nonlinear diffusion-type partial differential equations on a bounded interval and discuss the possibility of replacing the boundary conditions by certain linear conditions on the moments of order 0 (the total mass) and of another arbitrarily chosen order n. Each choice of n induces the addition of a certain potential in the equation, the case of zero potential arising exactly in the special case of n=1 corresponding to a condition on the barycenter. In the linear case we exploit smoothing properties and perturbation theory of analytic semigroups to obtain well-posedness for the classical heat equation (with said conditions on the moments). Long time behavior is studied for both the linear heat equation with potential and certain nonlinear equations of porous medium or fast diffusion type. In particular, we prove polynomial decay in the porous medium range and exponential decay in the fast diffusion range, respectively.     

Energy decay result in a Timoshenko‐type system of thermoelasticity of type III with weak damping

We investigate in this paper a thermoelastic system where the oscillations are defined by the Timoshenko model and the heat conduction is given by Green and Naghdi theories. We introduce 2 new stability numbers κ₁ , κ₂, and we prove a general decay result, from which the exponential and polynomial decays are only special cases.     

Computational study of decaying annular vortex flow using the Rε/k − ε turbulence model

In this article, we present three dimensional CFD study of turbulent vortex flow in an annular passage using OpenFOAM 1.6. The vortex flow is generated by introducing the flow through a tangential entry to the passage. For the analysis presented in this article, turbulence was modeled using the R_ε/k − ε model, in addition, a comparison between such model with the standard k − ε model was conducted and discussed. The main characteristics of the flow such as vortex structure and recirculation zone were investigated. It was found that flow is subjected to Rankine vortex structure with three forced vortex regimes and a free vortex region near to the outer wall. The phenomenon of vortex decay was investigated by depicting the swirl number trend along the axial direction of the flow domain. It was found that the vortex decay is subjected to an exponential decay behavior. New coefficients for the exponential decay correlation were derived based on local values of velocity components in different radial planes.