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We perform an asymptotic analysis with respect to the parameter ε > 0 of the solution of the scalar advection–diffusion equation y t ε + M (x , t) y x ε − ε y x x ε = 0 , (x , t) ∈ (0 , 1) × (0 , T) , supplemented with Dirichlet boundary conditions. For small values of ε, the solution y ε exhibits a boundary layer of size O (ε) in the neighborhood of x = 1 (assuming M > 0) and an internal layer of size O (ε 1 / 2) in the neighborhood of the characteristic starting from the point (0 , 0). Assuming that these layers interact each other after a finite time T > 0 and using the method of matched asymptotic expansions, we construct an explicit approximation P ε satisfying ‖ y ε − P ε ‖ L ∞ (0 , T ; L 2 (0 , 1)) = O (ε 1 / 2). We emphasize the additional difficulties with respect to the case M constant considered recently by the authors. [ABSTRACT FROM AUTHOR]

A classical problem describing the collective motion of cells, is the movement driven by consumption/depletion of a nutrient. Here we analyze one of the simplest such model written as a coupled Partial Differential Equation/Ordinary Differential Equation system which we scale so as to get a limit describing the usually observed pattern. In this limit the cell density is concentrated as a moving Dirac mass and the nutrient undergoes a discontinuity. We first carry out the analysis without diffusion, getting a complete description of the unique limit. When diffusion is included, we prove several specific a priori estimates and interpret the system as a heterogeneous monostable equation. This allow us to obtain a limiting problem which shows the concentration effect of the limiting dynamics. [ABSTRACT FROM AUTHOR]

We study a pore-scale model where two mobile species with different diffusion coefficients react and precipitate in the form of immobile species (crystal) on the surface of the solid parts in a porous medium. The reverse may also happen, i.e. the crystals may dissolute to give mobile species. The mathematical modeling of these processes will give rise to a coupled system of ordinary and partial differential equations. We first prove the existence of a unique nonnegative global weak solution and then upscale the model from microscale to macroscale. [ABSTRACT FROM AUTHOR]

We propose a formal expansion of multiple relaxation times lattice Boltzmann schemes in terms of a single infinitesimal numerical variable. The result is a system of partial differential equations for the conserved moments of the lattice Boltzmann scheme. The expansion is presented in the nonlinear case up to fourth order accuracy. The asymptotic corrections of the nonconserved moments are developed in terms of equilibrium values and partial differentials of the conserved moments. Both expansions are coupled and conduct to explicit compact formulas. The new algebraic expressions are validated with previous results obtained with this framework. The example of isothermal D2Q9 lattice Boltzmann scheme illustrates the theoretical framework. [ABSTRACT FROM AUTHOR]

In this paper we study the asymptotic behaviour of a family of elliptic systems, as far as the existence of solutions is concerned. We give a special attention to the asymptotic behaviour of W and V as ε goes to zero in the system − ε 2 Δ u + W (x) u = Q u (u , v) in R N , − ε 2 Δ v + V (x) v = Q v (u , v) in R N , u , v ∈ H 1 (R N) , u (x) , v (x) > 0 for each x ∈ R N , where ε > 0 , W and V are positive potentials of C 2 class and Q is a p-homogeneous function with subcritical growth. We establish the existence of a positive solution by considering two classes of potentials W and V. Our arguments are based on penalization techniques, variational methods and the Moser iteration scheme. [ABSTRACT FROM AUTHOR]

In this article we obtain a 1-dimensional asymptotic model for a junction of thin hyperelastic rods as the thickness goes to zero. We show, under appropriate hypotheses on the loads, that the deformations that minimize the total energy weakly converge in a Sobolev space towards the minimum of a 1 D -dimensional energy for elastic strings by using techniques from Γ-convergence. [ABSTRACT FROM AUTHOR]

This work is concerned with the problem of identifying the shape, size and location of a small embedded tumor from measured temperature on the skin surface. The temperature distribution in the biological tissue is governed by the Pennes model equation. The proposed approach is based on the Kohn–Vogelius formulation and the topological sensitivity analysis method. The ill-posed geometric inverse problem is reformulated as a topology optimization. The temperature field perturbation, caused by the presence of a small anomaly, is analyzed and estimated. A topological asymptotic formula, describing the variation of the considered Kohn–Vogelius type functional with respect to the presence of a small anomaly is derived. [ABSTRACT FROM AUTHOR]

A recent three-dimensional (3D) model that revisited earlier theoretical work for longitudinal electromagnetic stirring in the continuous casting of steel blooms is analyzed further to explore how the bloom width interacts with the pole pitch of the stirrer to affect the magnetic flux density. Whereas the first work indicated the presence of a boundary layer in the steel near the interface with the stirrer, with all three components of the magnetic flux density vector being coupled to each other, in the analysis presented here we find that the component along the direction of the travelling wave decouples from those in the other two directions and can even be determined analytically in the form of a series solution. Moreover, it is found that the remaining two components can be found via a two-dimensional computation, but that it is not possible in general to determine these components without taking into account the surrounding air. The validity of the asymptotically reduced model solution is confirmed by comparing it with the results of 3D numerical computations. Moreover, the asymptotic approach provides a way to compute the time-averaged Lorentz force components that requires two orders of magnitude less computational time than the fully 3D approach. [ABSTRACT FROM AUTHOR]

This work addresses the mathematical analysis – by means of asymptotic analysis – of a penalisation strategy for the full discretisation of elastic wave propagation problems in quasi-incompressible media that has been recently developed by the authors. We provide a convergence analysis of the solution to the continuous version of the penalised problem towards its formal limit when the penalisation parameter tends to infinity. Moreover, as a fundamental intermediate step we provide an asymptotic analysis of the convergence of solutions to quasi-incompressible problems towards solutions to purely incompressible problems when the incompressibility parameter tends to infinity. Finally, we further detail the regularity assumptions required to guarantee that the mentioned convergence holds. [ABSTRACT FROM AUTHOR]

The Petrowsky type equation y t t ε + ε y x x x x ε − y x x ε = 0 , ε > 0 encountered in linear beams theory is null controllable through Neumann boundary controls. Due to the boundary layer of size of order ε occurring at the extremities, these boundary controls get singular as ε goes to 0. Using the matched asymptotic method, we describe the boundary layer of the solution y ε and derive a rigorous second order asymptotic expansion of the control of minimal weighted L 2 -norm, with respect to the parameter ε. The weight in the norm is chosen to guarantee the smoothness of the control. In particular, we recover and enrich earlier results due to J.-L. Lions in the eighties showing that the leading term of the expansion is a null Dirichlet control for the limit hyperbolic wave equation. The asymptotic analysis also provides a robust discrete approximation of the control for any ε small enough. Numerical experiments support our study. [ABSTRACT FROM AUTHOR]

We consider diffusivity of random walks with transition probabilities depending on the number of consecutive traversals of the last traversed edge, the so called senile reinforced random walk (SeRW). In one dimension, the walk is known to be sub-diffusive with identity reinforcement function. We perturb the model by introducing a small probability δ of escaping the last traversed edge at each step. The perturbed SeRW model is diffusive for any δ > 0 , with enhanced diffusivity (≫ O (δ 2)) in the small δ regime. We further study stochastically perturbed SeRW models by having the last edge escape probability of the form δ ξ n with ξ n 's being independent random variables. Enhanced diffusivity in such models are logarithmically close to the so called residual diffusivity (positive in the zero δ limit), with diffusivity between O ( 1 | log δ | ) and O ( 1 log | log δ | ). Finally, we generalize our results to higher dimensions where the unperturbed model is already diffusive. The enhanced diffusivity can be as much as O (log − 2 δ). [ABSTRACT FROM AUTHOR]

We give sufficient conditions for the existence of weak solutions to quasilinear elliptic Dirichlet problem driven by the A-Laplace operator in a bounded domain Ω. The techniques, based on a variant of the symmetric mountain pass theorem, exploit variational methods. We also provide information about the asymptotic behavior of the solutions as a suitable parameter goes to 0 + . In this case, we point out the existence of a blow-up phenomenon. The analysis developed in this paper extends and complements various qualitative and asymptotic properties for some cases described by homogeneous differential operators. [ABSTRACT FROM AUTHOR]

In previous work the authors found the asymptotic expansion of the L 2 -norm of the solution u (t , x) of the strongly damped wave equation u t t − Δ u t − Δ u = 0 and also of the L 2 -norm of the difference between u (t , x) and its asymptotic approximation ν (t , x). This was done in space dimension N ⩾ 3. In the present work results are extended to the exceptional cases N = 1 and N = 2. This extension is achieved by deriving new lemmas on the asymptotic expansion of some parameter dependent integrals. [ABSTRACT FROM AUTHOR]

This paper studies the wave-breaking mechanism and dynamical behavior of solutions near the explicit self-similar singularity for the two component Camassa–Holm equations, which can be regarded as a model for shallow water dynamics and arising from the approximation of the Hamiltonian for Euler's equation in the shallow water regime. [ABSTRACT FROM AUTHOR]

This paper deals with the relation of the dynamic elastic Cosserat rod model and the Kirchhoff beam equations. We show that the Kirchhoff beam without angular inertia is the asymptotic limit of the Cosserat rod, as the slenderness parameter (ratio between rod diameter and length) and the Mach number (ratio between rod velocity and typical speed of sound) approach zero, i.e., low-Mach-number–slenderness limit. The asymptotic framework is exact up to fourth order in the small parameter and reveals a mathematical structure that allows a uniform handling of the transition regime between the models. To investigate this regime numerically, we apply a scheme that is based on a Gauss–Legendre collocation in space and an α-method in time. [ABSTRACT FROM AUTHOR]

In this paper, we study a system of three evolutionary operator equations involving fractional powers of selfadjoint, monotone, unbounded, linear operators having compact resolvents. This system constitutes a generalized and relaxed version of a phase field system of Cahn–Hilliard type modelling tumor growth that has originally been proposed in Hawkins-Daarud et al. (Int. J. Numer. Meth. Biomed. Eng.28 (2012), 3–24). The original phase field system and certain relaxed versions thereof have been studied in recent papers co-authored by the present authors and E. Rocca. The model consists of a Cahn–Hilliard equation for the tumor cell fraction φ, coupled to a reaction–diffusion equation for a function S representing the nutrient-rich extracellular water volume fraction. Effects due to fluid motion are neglected. Motivated by the possibility that the diffusional regimes governing the evolution of the different constituents of the model may be of different (e.g., fractional) type, the present authors studied in a recent note a generalization of the systems investigated in the abovementioned works. Under rather general assumptions, well-posedness and regularity results have been shown. In particular, by writing the equation governing the evolution of the chemical potential in the form of a general variational inequality, also singular or nonsmooth contributions of logarithmic or of double obstacle type to the energy density could be admitted. In this note, we perform an asymptotic analysis of the governing system as two (small) relaxation parameters approach zero separately and simultaneously. Corresponding well-posedness and regularity results are established for the respective cases; in particular, we give a detailed discussion which assumptions on the admissible nonlinearities have to be postulated in each of the occurring cases. [ABSTRACT FROM AUTHOR]

We provide a comprehensive study on the asymptotic solutions of an interface problem corresponding to an elliptic partial differential equation with Dirichlet boundary condition and transmission condition, subject to the small geometric perturbation and/or the high contrast ratio of the conductivity. All asymptotic terms can be solved in the unperturbed reference domains, which significantly reduces computations in practice, especially for random perturbations. Our setting is quite general and allows two types of elliptic problems: the perturbation of the domain boundary where the Dirchlet condition is imposed and the perturbation of the interface where the transmission condition is imposed. As the perturbation size and the ratio of the conductivities tends to zero, the two-parameter asymptotic expansions on the reference domain are derived to any order after the single parameter expansions are solved beforehand. The results suggest the emergence of the Neumann or Robin boundary condition, depending on the relation of the two asymptotic parameters. Our method is the classic asymptotic analysis techniques but in a new unified approach to both problems. [ABSTRACT FROM AUTHOR]

We study an optimal control problem for the mixed Dirichlet–Neumann boundary value problem for the strongly non-linear elliptic equation with exponential nonlinearity in a domain with rugous boundary. A density of surface traction u acting on a part of rugous boundary is taken as a control. The optimal control problem is to minimize the discrepancy between a given distribution and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the state equation for a given control. After having defined a suitable class of the weak solutions, we provide asymptotic analysis of the above mentioned optimal control problem posed in a family of perturbed domains and give the characterization of the limiting behavior of its optimal solutions. [ABSTRACT FROM AUTHOR]

We consider the propagation of surface water waves in a straight planar channel perturbed at the bottom by several thin curved tunnels and wells. We propose a method to construct non reflecting underwater topographies of this type at an arbitrary prescribed wave number. To proceed, we compute asymptotic expansions of the diffraction solutions with respect to the small parameter of the geometry taking into account the existence of boundary layer phenomena. We establish error estimates to validate the expansions using advances techniques of weighted spaces with detached asymptotics. In the process, we show the absence of trapped surface waves for perturbations small enough. This analysis furnishes asymptotic formulas for the scattering matrix and we use them to determine underwater topographies which are non-reflecting. Theoretical and numerical examples are given. [ABSTRACT FROM AUTHOR]

A distributed optimal control problem for a phase field system which physical context is that of tumor growth is discussed. The system we are going to take into account consists of a Cahn–Hilliard equation for the phase variable (relative concentration of the tumor) coupled with a reaction-diffusion equation for the nutrient. The cost functional is of standard tracking-type and the control variable models the intensity at which it is possible to dispense medication. The model we deal with presents two small and positive parameters which are introduced in previous contributions as relaxation terms. Here, starting from the already investigated optimal control problem for the relaxed model, we aim at confirming the existence of optimal control and characterizing the first-order necessary optimality condition, via asymptotic schemes, when one of the two occurring parameters goes to zero. [ABSTRACT FROM AUTHOR]

We perform the asymptotic analysis of the scalar advection-diffusion equation y t ε − ε y x x ε + M y x ε = 0 , (x , t) ∈ (0 , 1) × (0 , T) , M > 0 , with respect to the diffusion coefficient ε. We use the matched asymptotic expansion method which allows to describe the boundary layers of the solution. We then use the asymptotics to discuss the controllability property of the solution for T ⩾ 1 / M. [ABSTRACT FROM AUTHOR]

We consider a family of linear viscoelastic shells with thickness 2 ε (where ε is a small parameter), clamped along a portion of their lateral face, all having the same middle surface S. We formulate the three-dimensional mechanical problem in curvilinear coordinates and provide existence and uniqueness of (weak) solution of the corresponding three-dimensional variational problem. We are interested in studying the limit behavior of both the three-dimensional problems and their solutions when ε tends to zero. To do that, we use asymptotic analysis methods. First, we formulate the variational problem in a fixed domain independent of ε. Then we assume an asymptotic expansion of the scaled displacements field, u ( ε ) = ( u i ( ε ) ) , and we characterize the zeroth order term as the solution of a two-dimensional scaled limit problem. Moreover, we find that, depending on the order of the applied forces, the limit of the field u ( ε ) is the solution of one of the two sets of two-dimensional variational equations derived, which can be described as viscoelastic membrane shell and viscoelastic flexural shell problems. In both cases, we find a model which presents a long-term memory that takes into account the deformations at previous times. We finally comment on the existence and uniqueness of solution for the two-dimensional variational problems found and announce convergence results. [ABSTRACT FROM AUTHOR]

We consider the two-dimensional steady-state heat conduction on a bounded domain Ω containing a heat source at an unknown location ω ⊂ Ω. We are interested in determining the locations ω allowing a suitable thermal environment. The resulting shape optimization problem consists of a geometric control of either the maximum of the temperature or its L² mean oscillations in Ω. We derive the topological asymptotic expansion of the considered shape functionals with respect to the insertion of small circular coated inclusion characterized by a discontinuous thermal conductivity and their radius. We propose a reconstruction algorithm based on this topological gradient to identify the locations. Some numerical simulations are presented to show the efficiency of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamics of a liquid film flowing over a periodic wavy wall. This study is based on a long-wave model that is valid at near-critical Reynolds number. For the periodic wall surface, we prove the existence of a periodic steady-state solution to the model by the method of abstract contraction mapping in a particular functional space. Using the Floquet--Bloch theory and asymptotic method, we establish several analytic results on the stability of the periodic steady-state solution in a weighted functional space. [ABSTRACT FROM AUTHOR]

Approximate eigenpairs (quasimodes) of axisymmetric thin elastic domains with laterally clamped boundary conditions (Lamé system) are determined by an asymptotic analysis as the thickness (2ε) tends to zero. The departing point is the Koiter shell model that we reduce by asymptotic analysis to a scalar model that depends on two parameters: the angular frequency k and the half-thickness ε. Optimizing k for each chosen ε, we find power laws for k in function of e that provide the smallest eigenvalues of the scalar reductions. Corresponding eigenpairs generate quasimodes for the 3D Lamé system by means of several reconstruction operators, including boundary layer terms. Numerical experiments demonstrate that in many cases the constructed eigenpair corresponds to the first eigenpair of the Lamé system. Geometrical conditions are necessary to this approach: The Gaussian curvature has to be nonnegative and the azimuthal curvature has to dominate the meridian curvature in any point of the midsurface. In this case, the first eigenvector admits progressively larger oscillation in the angular variable as e tends to 0. Its angular frequency exhibits a power law relation of the form k = γ ε -β with β = ¼ in the parabolic case (cylinders and trimmed cones), and the various βs 2/5, 3/7, and 1/3 in the elliptic case. For these cases where the mathematical analysis is applicable, numerical examples that illustrate the theoretical results are presented. [ABSTRACT FROM AUTHOR]

In this paper we properly justify the modeling of a thin piezoelectric shallow shell in unilateral contact with a rigid plane. Starting from the three-dimensional nonlinear Signorini problem, we establish the convergence of the displacement field and of the electric potential as the thickness of the shell goes to zero. More precisely we obtain that the transverse mechanical displacement field coupled with the in-plane components solve an obstacle problem described new piezoelectric characteristics. We also investigate the very popular case of cubic crystals and show that, for two-dimensional shallow shells, the coupling piezoelectric effect disappears. [ABSTRACT FROM AUTHOR]

A rigorous proof is given for the convergence of the solutions of a viscous Cahn-Hilliard system to the solution of the regularized version of the forward-backward parabolic equation, as the coefficient of the diffusive term goes to 0. Nonhomogenous Neumann boundary conditions are handled for the chemical potential and the subdifferential of a possible nonsmooth double-well functional is considered in the equation. An error estimate for the difference of solutions is also proved in a suitable norm and with a specified rate of convergence. [ABSTRACT FROM AUTHOR]

The present work deals with the resolution of the Poisson equation in a bounded domain made of a thin and periodic layer of finite length placed into a homogeneous medium.We provide and justify a high order asymptotic expansion which takes into account the boundary layer effect occurring in the vicinity of the periodic layer as well as the corner singularities appearing in the neighborhood of the extremities of the layer. Our approach combines the method of matched asymptotic expansions and the method of periodic surface homogenization. [ABSTRACT FROM AUTHOR]

This paper aims at properly justifying the modeling of a thin piezoelectric plate in unilateral contact with a rigid plane. In order to do that we start from the three-dimensional non-linear Signorini problem which couples the elastic and the electric effects. By an asymptotic analysis we study the convergence of the displacement field and of the electric potential as the thickness of the plate goes to zero. We establish that, at the limit, the in-plane elastic components and the electric potential are coupled and solve a bilateral linear piezoelectric problem. However the transversemechanical displacement field, is independent of the electric effect and solves a two-dimensional elastic obstacle problem. We also investigate the very popular case of cubic crystals and show that, for thin plates, the piezoelectric coupling effect disappears. [ABSTRACT FROM AUTHOR]

We study the existence, uniqueness and asymptotic expansions to perturbed Poisson-Boltzmann equations on an unbounded domain in R2 or R3. First, a shooting method is applied to prove the existence and uniqueness of the exact solution. For the approximation to the regularly perturbed Poisson-Boltzmann equation, the solution via the classical method fails. We develop a novel approximate solution in terms of generalized asymptotic expansions. For the singularly perturbed problem, we show that a formula of asymptotic expansions with a boundary layer near the left end point provides a valid approximation. All our results are proved rigorously. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an innovative and systematic technique to study delay differential equations via polynomials. First, we review an intrinsic relation between delay differential equations and polynomials. From this relation, we obtain long time behaviors of the solutions to delay differential equations via asymptotic analysis of the corresponding polynomials. Moreover, we derive asymptotic formulas and upper bounds for the intrinsic growth rate of delay differential equations, as well as a Gronwall-type inequality for delay differential inequalities. [ABSTRACT FROM AUTHOR]

We provide an asymptotic analysis of linear transport problems in the diffusion limit under minimal regularity assumptions on the domain, the coefficients, and the data. The weak form of the limit equation is derived and the convergence of the solution in the L2 norm is established without artificial regularity requirements. This is important to be able to deal with problems involving realistic geometries and heterogeneous media. In a second step we prove the usual O(ε) convergence rates under very mild additional assumptions. The generalization of the results to convergence in Lp with p≠2 and some limitations are discussed. [ABSTRACT FROM AUTHOR]

A viscous incompressible fluid with a large number of small axially symmetric solid particles is considered. It is assumed that the particles are identically oriented and under the influence of the fluid move translationally or rotate around a symmetry axis with the direction of their symmetry axes unchanged. The asymptotic behavior of oscillations of the system is studied, when the diameters of particles and distances between the nearest particles are decreased. The equations, describing the homogenized model of the system, are derived. It is shown that the homogenized equations correspond to a non-standard hydrodynamics. Namely, the homogenized stress tensor linearly depends not only on the strain tensor but also on the rotation tensor. [ABSTRACT FROM AUTHOR]

We consider a three-dimensional body whose energy couples mechanic, electric and thermic effects. By using an asymptotic approach we establish the associated bi-dimensional dynamic model for a thermopiezoelectric plate. The computation of the new constitutive laws shows that pyroelectric effect (coupling of the electric field and the temperature) can appear in the limit 2D model even though the 3D dielectric displacement field does not present any coupling with the temperature. Pyrokinetic effect (coupling of elastic and temperature fields) is exhibited by the same way. [ABSTRACT FROM AUTHOR]

In this paper we introduce a PDE system which aims at describing the dynamics of a dispersed phase of particles moving into an incompressible perfect fluid, in two space dimensions. The system couples a Vlasov-type equation and an Euler-type equation: the fluid acts on the dispersed phase through a gyroscopic force whereas the latter contributes to the vorticity of the former. First we give a Dobrushin-type derivation of the system as a mean-field limit of a PDE system which describes the dynamics of a finite number of massive pointwise particles moving into an incompressible perfect fluid. This last system is itself inferred from the paper 'On the motion of a small body immersed in a two-dimensional incompressible perfect fluid', accepted for publication in Bulletin de la SMF where the system for one massive pointwise particle was derived as the limit of the motion of a solid body when the body shrinks to a point with fixed mass and circulation. Then we deal with the well-posedness issues including the existence of weak solutions. Next we exhibit the Hamiltonian structure of the system and finally, we study the behavior of the system in the limit where the mass of the particles vanishes. [ABSTRACT FROM AUTHOR]

Polarization matrices (or tensors) are generalizations of mathematical objects like the harmonic capacity or the virtual mass tensor. They participate in many asymptotic formulae with broad applications to problems of structural mechanics. In the present paper polarization matrices for anisotropic heterogeneous elastic inclusions are investigated, the ambient anisotropic elastic space is allowed to be inhomogeneous near the inclusion as well. By variational arguments the existence of unique solutions to the corresponding transmission problems is proved. Using results about elliptic problems in domains with a compact complement, polarization matrices can be properly defined in terms of certain coefficients in the asymptotic expansion at infinity of the solution to the homogeneous transmission problem. Representation formulae are derived from which properties like positivity or negativity can be read of directly. Further the behavior of the polarization matrix is investigated under small changes of the interface. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new technique to obtain the slow-motion dynamics in nonequilibrium and singularly perturbed problems characterized by multiple scales. Our method is based on a straightforward asymptotic reduction of the order of the governing differential equation and leads to amplitude equations that describe the slowly-varying envelope variation of a uniformly valid asymptotic expansion. This may constitute a simpler and in certain cases a more general approach toward the derivation of asymptotic expansions, compared to other mainstream methods such as the method of Multiple Scales or Matched Asymptotic expansions because of its relation with the Renormalization Group. We illustrate our method with a number of singularly perturbed problems for ordinary and partial differential equations and recover certain results from the literature as special cases. [ABSTRACT FROM AUTHOR]