The Limit of the Stefan Problem with Surface Tension and Kinetic Undercooling on the Free Boundary

In this paper we consider the Stefan problem with surface tension and kinetic undercooling effects, that is with the temperature u satisfying the condition u = -σK - εV_n on the interface Γ_t, σ, ε = const. ≥ 0 where K and V_n are the mean curvature and the normal velocity of Γ_t, respectively. In any of the following situations: (1) σ > 0 fixed, ε > 0, (2) σ = ε → 0; (3) σ → 0, ε = 0, we shall prove the convergence of the corresponding local (in time) classical solution of the Stefan problem.     

Well‐posedness and asymptotic analysis for a Penrose–Fife type phase field system

In this paper, an asymptotic analysis of the (non‐conserved) Penrose–Fife phase field system for two vanishing time relaxation parameters ε and δ is developed, in analogy with the similar analyses for the phase field model proposed by G. Caginalp (Arch. Rational Mech. Anal. 1986; 92 :205–245), which were carried out by Rossi and Stoth (Adv. Math. Sci. Appl. 2003; 13 :249–271; Quart. Appl. Math. 1995; 53 :695–700). Although formally the singular limits for ε ↓ 0 and for ε and δ ↓ 0 are, respectively, the viscous Cahn–Hilliard equation and the Cahn–Hilliard equation, it turns out that the Penrose–Fife system is indeed a bad approximation for these equations. Therefore, we consider an alternative approximating phase field system, which could be viewed as a generalization of the classical Penrose–Fife phase field system, featuring a double non‐linearity given by two maximal monotone graphs. A well‐posedness result is proved for such a system, and it is shown that the solutions converge to the unique solution of the viscous Cahn–Hilliard equation as ε ↓ 0, and of the Cahn–Hilliard equation as ε ↓ 0 and δ ↓ 0. Copyright © 2004 John Wiley & Sons, Ltd.     

Asymptotic properties of the moment convergence for NA sequences

It is well-known that the complete convergence theorem for i.i.d. random variables has been an active topic since the famous work done by Hsu and Robbins [6]. Chow [4] obtained a moment version of Hsu and Robbins series. However, the series tends to infinity whenever ε goes to zero, so it is of interest to investigate the asymptotic behavior of the series as ε goes to zero. This note gives some limit theorems of the series generated by moments for NA random variables.     

Convergence of trajectories in fractal interpolation of stochastic processes

The notion of fractal interpolation functions (FIFs) can be applied to stochastic processes. Such construction is especially useful for the class of α-self-similar processes with stationary increments and for the class of α-fractional Brownian motions. For these classes, convergence of the Minkowski dimension of the graphs in fractal interpolation of the Hausdorff dimension of the graph of original process was studied in [Herburt I, Małysz R. On convergence of box dimensions of fractal interpolation stochastic processes. Demonstratio Math 2000;4:873–88. [11]], [Małysz R. A generalization of fractal interpolation stochastic processes to higher dimension. Fractals 2001;9:415–28. [15]], and [Herburt I. Box dimension of interpolations of self-similar processes with stationary increments. Probab Math Statist 2001;21:171–8. [10]].We prove that trajectories of fractal interpolation stochastic processes converge to the trajectory of the original process. We also show that convergence of the trajectories in fractal interpolation of stochastic processes is equivalent to the convergence of trajectories in linear interpolation.     

3‐D viscous Cahn–Hilliard equation with memory

We deal with the memory relaxation of the viscous Cahn–Hilliard equation in 3‐D, covering the well‐known hyperbolic version of the model. We study the long‐term dynamic of the system in dependence of the scaling parameter of the memory kernel ε and of the viscosity coefficient δ. In particular we construct a family of exponential attractors, which is robust as both ε and δ go to zero, provided that ε is linearly controlled by δ. Copyright © 2008 John Wiley & Sons, Ltd.     

Derivation of effective interface conditions for reaction-diffusion equations

We address the derivation of effective interface conditions for reaction-diffusion systems. The considered system is defined in a domain containing a thin layer that shrinks to the interface when its thickness ε tends to zero. The evolution of the system can be written in the form of an energy balance involving an energy and a dissipation functional. Using the Mosco convergence of the dual of the dissipation functional for ε → 0 it is possible to do a limit passage in the energy balance and obtain a limit system that describes the evolution on the interface. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Remarks on summability of series formed of deviation probabilities of sums of independent identically distributed random variables

We make some remarks leading to a refinement of the recent work of Klesov (1993) on the connection between the convergence of the series $\Sigma _{n = 1}^\infty \tau _n P(|S_n | \ge \varepsilon n^\alpha )$ for every ε > 0 and the convergence of $\Sigma _{n = 1}^\infty n\tau _n P(|X_1 | \ge \varepsilon n^\alpha )$ again for every ε > 0.     

The Extragradient Method for Convex Optimization in the Presence of Computational Errors

In this article, we study convergence of the extragradient method for constrained convex minimization problems in a Hilbert space. Our goal is to obtain an ε-approximate solution of the problem in the presence of computational errors, where ε is a given positive number. Most results known in the literature establish convergence of optimization algorithms, when computational errors are summable. In this article, the convergence of the extragradient method for solving convex minimization problems is established for nonsummable computational errors. We show that the the extragradient method generates a good approximate solution, if the sequence of computational errors is bounded from above by a constant.     

Stability theorems for 2×2 hypermatrices

In this paper, stability properties of 2×2 hypermatrices of the form $$\left[ {\begin{array}{*{20}c} { \in {\rm A} \in {\rm B}} \\ {CD} \\ \end{array} } \right]$$ are investigated, where ε is a small parameter. Stability theorems on these and similar matrices play an important role in the convergence analysis of certain numerical methods of mathematical programming and control theory, as pointed out in Refs. 1–3.     

The Projected Subgradient Method for Nonsmooth Convex Optimization in the Presence of Computational Errors

We study the convergence of the projected subgradient method for constrained convex optimization in a Hilbert space. Our goal is to obtain an ε-approximate solution of the problem in the presence of computational errors, where ε is a given positive number. The results that we obtain are important in practice because computations always introduce numerical errors.     

A convergence method for stochastic differential games with a small parameter

A stochastic differential game problem for wideband noise driven system is considered. Often in applications, one have a single realization, then expectation is not appropriate in the cost function. First we will consider the payoff structure in the pathwise but not necessarily in the expected value sense. For N-person noncooperative games, under very general conditions, it will be shown that the optimal equilibrium policies of the limit diffusion when applied to the physical processes, will be δ-equilibrium as the parameters ε > 0 and T→ ∞. A combination of direct averaging and perturbed test function techniques will be used in convergence analysis. Results are shown to hold when mathematical expectations are used in the payoff structure. Two person zero sum games can also be considered in this framework     

Long time behavior of a singular perturbation of the viscous Cahn–Hilliard–Gurtin equation

We consider a singular perturbation of the generalized viscous Cahn–Hilliard equation based on constitutive equations introduced by Gurtin. This equation rules the order parameter ρ, which represents the density of atoms, and it is given on a n‐rectangle (n?3) with periodic boundary conditions. We prove the existence of a family of exponential attractors that is robust with respect to the perturbation parameter ε>0, as ε goes to 0. In a similar spirit, we analyze the stability of the global attractor. If n=1, 2, then we also construct a family of inertial manifolds that is continuous with respect to ε. These results improve and generalize the ones contained in some previous papers. Finally, we establish the convergence of any trajectory to a single equilibrium via a suitable version of the ?ojasiewicz–Simon inequality, provided that the potential is real analytic. Copyright © 2007 John Wiley & Sons, Ltd.     

Dynamics of Hodgkin–Huxley systems revisited

We consider the singularly perturbed Hodgkin–Huxley system subject to Neumann boundary conditions. We construct a family of exponential attractors {?_ε} which is continuous at ε = 0, ε being the parameter of perturbation. Moreover, this continuity result is obtained with respect to a metric independent of ε, compared with all previous results where the metric always depends on ε. In the latter case, one needs to consider more regular function spaces and more smoother absorbing sets. Our results show that we can construct and analyse the stability of exponential attractors in a natural phase-space as it is known for the global attractor. Also, a new proof of the upper semicontinuity of the global attractor 𝒜_ε at ε = 0 is given.     

Zero Dissipation Limit to Rarefaction Waves for the 1-D Compressible Navier-Stokes Equations

The zero dissipation limit for the one-dimensional Navier-Stokes equations of compressible,isentropic gases in the case that the corresponding Euler equations have rarefaction wave solutions is investi...     

Free point processes and free extreme values

We continue here the study of free extreme values begun in Ben Arous and Voiculescu (Ann Probab 34:2037–2059, 2006). We study the convergence of the free point processes associated with free extreme values to a free Poisson random measure (Voiculescu in Lecture notes in mathematics. Springer, Heidelberg, pp. 279–349, 1998; Barndorff-Nielsen and Thorbjornsen in Probab Theory Relat Fields 131:197–228, 2005). We relate this convergence to the free extremal laws introduced in Ben Arous and Voiculescu (Ann Probab 34:2037–2059, 2006) and give the limit laws for free order statistics.     

Reduction principles for quantile and Bahadur–Kiefer processes of long-range dependent linear sequences

In this paper we consider quantile and Bahadur–Kiefer processes for long range dependent linear sequences. These processes, unlike in previous studies, are considered on the whole interval (0, 1). As it is well-known, quantile processes can have very erratic behavior on the tails. We overcome this problem by considering these processes with appropriate weight functions. In this way we conclude strong approximations that yield some remarkable phenomena that are not shared with i.i.d. sequences, including weak convergence of the Bahadur–Kiefer processes, a different pointwise behavior of the general and uniform Bahadur–Kiefer processes, and a somewhat “strange” behavior of the general quantile process.     

Numerical analysis of a model of cure process for composites

We consider a composite material composed of fibres included in a resin which becomes solid when it is heated up (reaction of reticulation). The mathematical modelling of the cure process is given by a kinetic equation describing the evolution of the reaction of reticulation coupled with the heat equation. In this paper, we are interested in the computation of approximate solutions. We propose a family of discretized problems depending on two parameters (β₁, β₂) ε [0, 1]² which split the linear and non‐ linear terms in implicit and explicit parts. We prove the stability and convergence of the discretization for any (β₁, β₂) ε [½, 1 ] × [0, 1]. We present also some numerical results. Copyright © 2006 John Wiley & Sons, Ltd.     

Homogenization of the Neumann problem in thick multi‐structures of type 3 : 2 : 2

Using some special extension operator, a convergence theorem is proved for the solution to the Neumann boundary value problem for the Ukawa equation in a junction Ω_ε, which is the union of a domain Ω₀ and a large number N of ε‐periodically situated thin annular disks with variable thickness of order ε=??(N^‐1), as ε → 0. Copyright © 2004 John Wiley & Sons, Ltd.     

Contrôlabilité de l'équation des ondes à densité rapidement oscillante à une dimension d'espace

We consider the one-dimensional wave equation with periodic density of period ε → 0. By a counterexample due to Avellaneda, Bardos, and Rauch, we know that the boundary controllability property does not hold uniformly as ε → 0. We prove that the control remains uniformly bounded if we control the projection of the solution over the subspace generated by the eigenfunctions associated with the eigenvalues λ ≤ C_ε⁻², C > 0 being small enough. This result is sharp in the sense that the control diverges when the projection over the eigenfunctions such that λ ~ C_ε⁻², with C large, is controlled. We use the classical WKB asymptotic development that provides sharp results on the convergence of the spectrum and the theory of non-harmonic Fourier series.     

Random attractor for the stochastic Cahn–Hilliard–Navier–Stokes system with small additive noise

The objective of this paper is to study the asymptotic behavior of solutions, in terms of the upper semi-continuous property of random attractor, of the Cahn–Hilliard–Navier–Stokes system with small additive noise. We prove the existence of a random attractor for the Cahn–Hilliard–Navier–Stokes system with small additive noise. Furthermore, we consider the stability of global attractor and prove the random attractor of the Cahn–Hilliard–Navier–Stokes system with small additive noise will convergent to the global attractor of the unperturbed Cahn–Hilliard–Navier–Stokes system when the parameter of the perturbation ε tends to zero.