On the Number of Blocks in a Generalized Steiner System

We considert-designs withλ=1 (generalized Steiner systems) for which the block size is not necessarily constant. An inequality for the number of blocks is derived. Fort=2, this inequality is the well known De Bruijn–Erd?s inequality. Fort>2 it has the same order of magnitude as the Wilson–Petrenjuk inequality for Steiner systems with constant block size. The point of this note is that the inequality is very easy to derive and does not seem to be known. A stronger inequality was derived in 1969 by Woodall (J. London Math. Soc.(2)1, 509–519), but it requires Lagrange multipliers in the proof.     

A New Approach to Linearly Perturbed Riccati Equations Arising in Stochastic Control

In this paper a linearly perturbed version of the well-known matrix Riccati equations which arise in certain stochastic optimal control problems is studied. Via the concepts of mean square stabilizability and mean square detectability we improve previous results on both the convergence properties of the linearly perturbed Riccati differential equation and the solutions of the linearly perturbed algebraic Riccati equation. Furthermore, our approach unifies, in some way, the study for this class of Riccati equations with the one for classical theory, by eliminating a certain inconvenient assumption used in previous works (e.g., [10] and [26]). The results are derived under relatively weaker assumptions and include, inter alia, the following: (a) An extension of Theorem 4.1 of [26] to handle systems not necessarily observable. (b) The existence of a strong solution, subject only to the mean square stabilizability assumption. (c) Conditions for the existence and uniqueness of stabilizing solutions for systems not necessarily detectable. (d) Conditions for the existence and uniqueness of mean square stabilizing solutions instead of just stabilizing. (e) Relaxing the assumptions for convergence of the solution of the linearly perturbed Riccati differential equation and deriving new convergence results for systems not necessarily observable. Accepted 30 July 1996     

On martingale diffusions describing the ‘smile‐effect’ for implied volatilities

This paper discusses diffusion models describing the ‘smile‐effect’ of implied volatilities for option prices partly following the new approach of Bruno Dupire. If one restricts to the time homogeneous case, a careful study of this approach shows that the call option prices considered as a function of the price x of the underlying security, remaining time to maturity T–t and strike price K have necessarily to satisfy a certain functional equation, in order to fit into a coherent model. It is shown that for certain examples of empirically observed option prices which are reported in the literature, this functional equation does not hold. © 2000 John Wiley & Sons, Ltd.     

Elliptic equations,principal eigenvalue and dependence on the domain

We consider a general second order uniformly elliptic differential operator L and also the set θ of all open sets (not necessarily smooth) in the unit ball of ?ⁿ. We define a metric d in this space (up to an equivalence relation ～) that makes the space (θ/ ～,d) a complete metric space. We show that the principal eigenvalue and eigenfunction of L are continuous with the metric d. Similar results are obtained for the solutions of the equation Lv = ?.     

Behavior near hyperbolic stationary solutions for partial functional differential equations with infinite delay

The aim of this work is to investigate the asymptotic behavior of solutions near hyperbolic stationary solutions for partial functional differential equations with infinite delay. We suppose that the linear part satisfies the Hille–Yosida condition on a Banach space and it is not necessarily densely defined. Firstly, we establish a new variation of constants formula for the nonhomogeneous linear equations. Secondly, we use this formula and the spectral decomposition of the phase space to show the existence of stable and unstable manifolds. The estimations of solutions on these manifolds are obtained. For illustration, we propose to study the stability of stationary solutions for the Lotka–Volterra model with diffusion.     

Numerical approximation of Turing patterns in electrodeposition by ADI methods

In this paper we study the numerical approximation of Turing patterns corresponding to steady state solutions of a PDE system of reaction–diffusion equations modeling an electrodeposition process. We apply the Method of Lines (MOL) and describe the semi-discretization by high order finite differences in space given by the Extended Central Difference Formulas (ECDFs) that approximate Neumann boundary conditions (BCs) with the same accuracy. We introduce a test equation to describe the interplay between the diffusion and the reaction time scales. We present a stability analysis of a selection of time-integrators (IMEX 2-SBDF method, Crank–Nicolson (CN), Alternating Direction Implicit (ADI) method) for the test equation as well as for the Schnakenberg model, prototype of nonlinear reaction–diffusion systems with Turing patterns. Eventually, we apply the ADI-ECDF schemes to solve the electrodeposition model until the stationary patterns (spots & worms and only spots) are reached. We validate the model by comparison with experiments on Cu film growth by electrodeposition.     

Existence and uniqueness of solutions to nonlinear evolution equations with locally monotone operators

In this paper we establish the existence and uniqueness of solutions for nonlinear evolution equations on a Banach space with locally monotone operators, which is a generalization of the classical result for monotone operators. In particular, we show that local monotonicity implies pseudo-monotonicity. The main results are applied to PDE of various types such as porous medium equations, reaction–diffusion equations, the generalized Burgers equation, the Navier–Stokes equation, the 3D Leray-α model and the p-Laplace equation with non-monotone perturbations.     

Stability and convergence for a complete model of mass diffusion

We propose a fully discrete scheme for approximating a three-dimensional, strongly nonlinear model of mass diffusion, also called the complete Kazhikhov–Smagulov model. The scheme uses a C⁰ finite-element approximation for all unknowns (density, velocity and pressure), even though the density limit, solution of the continuous problem, belongs to H². A first-order time discretization is used such that, at each time step, one only needs to solve two decoupled linear problems for the discrete density and the velocity–pressure, separately.We extend to the complete model, some stability and convergence results already obtained by the last two authors for a simplified model where λ²-terms are not considered, λ being the mass diffusion coefficient. Now, different arguments must be introduced, based mainly on an induction process with respect to the time step, obtaining at the same time the three main properties of the scheme: an approximate discrete maximum principle for the density, weak estimates for the velocity and strong ones for the density. Furthermore, the convergence towards a weak solution of the density-dependent Navier–Stokes problem is also obtained as λ→0 (jointly with the space and time parameters).Finally, some numerical computations prove the practical usefulness of the scheme.     

Two‐Timing Hypothesis,Distinguished Limits,Drifts, and Pseudo‐Diffusion for Oscillating Flows

The aim of this paper is: using the two‐timing method to study and classify the multiplicity of distinguished limits and asymptotic solutions for the advection equation with a general oscillating velocity field. Our results are:

• (i) the dimensionless advection equation that contains two independent small parameters, which represent the ratio of two characteristic time‐scales and the spatial amplitude of oscillations; the related scaling of the variables and parameters uses the Strouhal number;
• (ii) an infinite sequence of distinguished limits has been identified; this sequence corresponds to the successive degenerations of a drift velocity;
• (iii) we have derived the averaged equations and the oscillatory equations for the first four distinguished limits; derivations are performed up to the fourth orders in small parameters;
• (v) we have shown, that each distinguished limit generates an infinite number of parametric solutions; these solutions differ from each other by the slow time‐scale and the amplitude of the prescribed velocity;
• (vi) we have discovered the inevitable presence of pseudo‐diffusion terms in the averaged equations, pseudo‐diffusion appears as a Lie derivative of the averaged tensor of quadratic displacements; we have analyzed the matrix of pseudo‐diffusion coefficients and have established its degenerated form and hyperbolic character; however, for one‐dimensional cases, the pseudo‐diffusion can appear as ordinary diffusion;
• (vii) the averaged equations for four different types of oscillating velocity fields have been considered as the examples of different drifts and pseudo‐diffusion;
• (viii) our main methodological result is the introduction of a logical order into the area and classification of an infinite number of asymptotic solutions; we hope that it can help in the study of the similar problems for more complex systems;
• (ix) our study can be used as a test for the validity of the two‐timing hypothesis, because in our calculations we do not employ any additional assumptions.

     

Existence of periodic travelling wave solutions of non-autonomous reaction–diffusion equations with lambda–omega type

Periodic travelling wave solutions of reaction–diffusion equations were studied by many authors. The λ–ω$\lambda \text{–}\omega$ type reaction–diffusion system is a notable special model that admits explicit periodic travelling wave solutions and was introduced by Kopell and Howard in 1973. There are now similar systems which are investigated by means of autonomous dynamics. In contrast, there are few papers which are concerned with non-autonomous cases. For this reason, we apply Mawhin’s continuation theorem to derive the existence of periodic travelling wave solutions for non-autonomous λ–ω$\lambda \text{–}\omega$ systems, and we describe the ‘disappearance’ of periodic travelling wave solutions under special situations. Our main result is also illustrated by examples.     

Global well posedness for the Maxwell–Navier–Stokes system in 2D

We prove global existence of regular solutions to the full MHD system (or more precisely the Maxwell–Navier–Stokes system) in 2D. We also provide an exponential growth estimate for the Hs norm of the solution when the time goes to infinity.     

On the numerical solution of diffusion–reaction equations with singular source terms

A numerical study is presented of reaction–diffusion problems having singular reaction source terms, singular in the sense that within the spatial domain the source is defined by a Dirac delta function expression on a lower dimensional surface. A consequence is that solutions will be continuous, but not continuously differentiable. This lack of smoothness and the lower dimensional surface form an obstacle for numerical discretization, including amongst others order reduction. In this paper the standard finite volume approach is studied for which reduction from order two to order one occurs. A local grid refinement technique is discussed which overcomes the reduction.     

Large time behavior for some nonlinear degenerate parabolic equations

We study the asymptotic behavior of Lipschitz continuous solutions of nonlinear degenerate parabolic equations in the periodic setting. Our results apply to a large class of Hamilton–Jacobi–Bellman equations. Defining Σ as the set where the diffusion vanishes, i.e., where the equation is totally degenerate, we obtain the convergence when the equation is uniformly parabolic outside Σ and, on Σ, the Hamiltonian is either strictly convex or satisfies an assumption similar of the one introduced by Barles–Souganidis (2000) for first-order Hamilton–Jacobi equations. This latter assumption allows to deal with equations with nonconvex Hamiltonians. We can also release the uniform parabolic requirement outside Σ. As a consequence, we prove the convergence of some everywhere degenerate second-order equations.     

Analyticity estimates for the Navier–Stokes equations

We study spatial analyticity properties of solutions of the three-dimensional Navier–Stokes equations and obtain new growth rate estimates for the analyticity radius. We also study stability properties of strong global solutions of the Navier–Stokes equations with data in Hr, r?1/2, and prove a stability result for the analyticity radius.     

The unique solvability of a complex 3D heat transfer problem

Conductive–convective–radiative heat transfer in a scattering and absorbing medium with reflecting and radiating boundaries is considered. The P₁${}_1$ approximation (diffusion model) is used for the simplification of the original problem. The existence of bounded states of the diffusion model is proved. The uniqueness of solutions is established under certain assumptions.     

Well-posedness and dynamics of fractional FitzHugh-Nagumo systems on ℝN driven by nonlinear noise

This article is concerned with the well-posedness as well as long-term dynamics of a wide class of non-autonomous, non-local, fractional, stochastic FitzHugh-Nagumo systems driven by nonlinear noise defined on the entire space?R^N. The well-posedness is proved for the systems with polynomial drift terms of arbitrary order as well as locally Lipschitz nonlinear diffusion terms by utilizing the pathwise and mean square uniform estimates. The mean random dynamical system generated by the solution operators is proved to possess a unique weak pullback mean random attractor in a Bochner space. The existence of invariant measures is also established for the autonomous systems with globally Lipschitz continuous diffusion terms. The idea of uniform tail-estimates of the solutions in the appropriate spaces is employed to derive the tightness of a family of probability distributions of the solutions in order to overcome the non-compactness of the standard Sobolev embeddings on ?^N as well as the lack of smoothing effect on one component of the solutions. The results of this paper are new even when the fractional Laplacian is replaced by the standard Laplacian.

     

A compact 9 point stencil based on integrated RBFs for the convection–diffusion equation

In this paper, a high-order compact stencil for solving the convection–diffusion equation in two dimensions is proposed. The convection and diffusion terms are both approximated by means of radial basis functions (RBFs) that are constructed over 3×3$3\times 3$ rectangular stencils. Salient features here are that (i) integration is employed to construct local RBF approximations; and (ii) through the constants of integration, values of the convection–diffusion equation at several selected nodes on the stencil are also enforced. Numerical results indicate that (i) the inclusion of the governing equation into the stencil leads to a significant improvement in accuracy; (ii) when the convection dominates, accurate solutions are obtained at a regime of the RBF width which makes the RBFs peaked; and (iii) high levels of accuracy are achieved using relatively coarse grids.     

On the Topology of Vacuum Spacetimes

We prove that there are no restrictions on the spatial topology of asymptotically flat solutions of the vacuum Einstein equations in (n + 1)-dimensions. We do this by gluing a solution of the vacuum constraint equations on an arbitrary compact manifold Sⁿ \Sigma^n to an asymptotically Euclidean solution of the constraints on \mathbbRⁿ \mathbb{R}^n . For any Sⁿ \Sigma^n which does not admit a metric of positive scalar curvature, this provides for the existence of asymptotically flat vacuum spacetimes with no maximal slices. Our main theorem is a special case of a more general gluing construction for nondegenerate solutions of the vacuum constraint equations which have some restrictions on the mean curvature, but for which the mean curvature is not necessarily constant. This generalizes the construction [16], which is restricted to constant mean curvature data.