bluuebunny/crossref_metadata_2025 huggingface dataset:
Controllability of the time discrete heat equation
Authors: Zheng Chuang
Published: November 2019

In this paper we study the controllability of an Euler Implicit time discrete heat equation in a bounded domain with a local internal controller. We prove that, based on Lebeau–Robbiano's time iteration method, the projection in appropriate filtered space is null controllable with uniformly bounded control. In this way, the well-known null-controllability property of the heat equation can be proven as the limit, as ▵t→0, of the controllability of projections of the time-discrete one. Consequently we prove the uniform approximate controllability after filtering with bounded control. A further study is made and analogous results are obtained for other discrete schemes, i.e. Euler Explicit schemes, θ-method schemes. We also discuss the null controllability of the Euler Implicit time discrete parabolic equation of fractional order.
Quenching for semidiscretizations of a heat equation with a singular boundary condition
Authors: Nabongo Diabate , Boni Théodore K.
Published: November 2019

We obtain some conditions under which the positive solution of the numerical approximation for the heat equation ut(x, t)=uxx(x, t), x∈(0, 1), t>0, with the singular boundary condition ux(1, t)=−u−β(1, t), where β>0 quenches in a finite time and estimate its semidiscrete quenching time. We also establish the convergence of the semidiscrete quenching time and obtain some results on numerical quenching rate and set. Finally we give some numerical results to illustrate our analysis.
Random integrals and correctors in homogenization
Authors: Bal Guillaume , Garnier Josselin , Motsch Sébastien , Perrier Vincent
Published: November 2019

This paper concerns the homogenization of a one-dimensional elliptic equation with oscillatory random coefficients. It is well known that the random solution to the elliptic equation converges to the solution of an effective medium elliptic equation in the limit of a vanishing correlation length in the random medium. It is also well known that the corrector to homogenization, i.e., the difference between the random solution and the homogenized solution, converges in distribution to a Gaussian process when the correlations in the random medium are sufficiently short-range. Moreover, the limiting process may be written as a stochastic integral with respect to standard Brownian motion. We generalize the result to a large class of processes with long-range correlations. In this setting, the corrector also converges to a Gaussian random process, which has an interpretation as a stochastic integral with respect to fractional Brownian motion. Moreover, we show that the longer the range of the correlations, the larger is the amplitude of the corrector. Derivations are based on a careful analysis of random oscillatory integrals of processes with long-range correlations. We also make use of the explicit expressions for the solutions to the one-dimensional elliptic equation.
Asymptotics for Gegenbauer–Sobolev orthogonal polynomials associated with non-coherent pairs of measures
Authors: de Andrade E.X.L. , Bracciali C.F. , Sri Ranga A.
Published: November 2019

Inner products of the type 〈f, g〉S=〈f, g〉ψ0+〈f′, g′〉ψ1, where one of the measures ψ0 or ψ1 is the measure associated with the Gegenbauer polynomials, are usually referred to as Gegenbauer–Sobolev inner products. This paper deals with some asymptotic relations for the orthogonal polynomials with respect to a class of Gegenbauer–Sobolev inner products. The inner products are such that the associated pairs of symmetric measures (ψ0, ψ1) are not within the concept of symmetrically coherent pairs of measures.
Smoothing effect for regularized Schrödinger equation on bounded domains
Authors: Aloui L.
Published: November 2019

We prove, under the geometric control condition, a smoothing effect for solutions of a regularized Schrödinger equation on bounded domains with boundary. This result is derived from the resolvent properties.
Spectral estimates for matrix-valued periodic Dirac operators
Authors: Korotyaev Evgeny
Published: November 2019

We consider the first order periodic systems perturbed by a 2N×2N matrix-valued periodic potential on the real line. The spectrum of this operator is absolutely continuous and consists of intervals separated by gaps. We define the Lyapunov function, which is analytic on an associated N-sheeted Riemann surface. On each sheet the Lyapunov function has the standard properties of the Lyapunov function for the scalar case. The Lyapunov function has branch points, which we call resonances. We prove the existence of real or complex resonances. We determine the asymptotics of the periodic, anti-periodic spectrum and of the resonances at high energy (in terms of the Fourier coefficients of the potential). We show that there exist two types of gaps: (i) stable gaps, i.e., the endpoints are periodic and anti-periodic eigenvalues, (ii) unstable (resonance) gaps, i.e., the endpoints are resonances (real branch points). Moreover, we determine various new trace formulae for potentials and the Lyapunov exponent.
Reaction–diffusion equations with nonlinear boundary conditions in narrow domains
Authors: Freidlin Mark , Spiliopoulos Konstantinos
Published: November 2019

Second initial boundary problem in narrow domains of width ε�1 for linear second-order differential equations with nonlinear boundary conditions is considered in this paper. Using probabilistic methods we show that the solution of such a problem converges as ε↓0 to the solution of a standard reaction–diffusion equation in a domain of reduced dimension. This reduction allows to obtain some results concerning wave front propagation in narrow domains. In particular, we describe conditions leading to jumps of the wave front.
Stabilization of the linear Kawahara equation with localized damping
Authors: Vasconcellos Carlos F. , da Silva Patricia N.
Published: November 2019

We study the stabilization of global solutions of the linear Kawahara (K) equation in a bounded interval under the effect of a localized damping mechanism. The Kawahara equation is a model for small amplitude long waves. Using multiplier techniques and compactness arguments we prove the exponential decay of the solutions of the (K) model. We also prove that the decay of solutions, in absence of damping, fails for some critical values of the length L and we define precisely this countable set. Finally, we include some remarks about nonlinear problem and we analyze the exact boundary control for linear Kawahara system.
Low-frequency dispersive estimates for the wave equation in higher dimensions
Authors: Moulin Simon
Published: November 2019

We prove dispersive estimates at low frequency in dimensions n≥4 for the wave equation for a very large class of real-valued potentials, provided zero is neither an eigenvalue nor a resonance. This class includes potentials V∈L∞(Rn) satisfying V(x)=O(〈x〉−(n+1)/2−ε), ε>0.
Gradient theory of phase transitions with a rapidly oscillating forcing term
Authors: Dirr Nicolas , Lucia Marcello , Novaga Matteo
Published: November 2019

We consider the Γ-limit of a family of functionals which model the interaction of a material that undergoes phase transition with a rapidly oscillating conservative vector field. These functionals consist of a gradient term, a double-well potential and a vector field. The scaling is such that all three terms scale in the same way and the frequency of the vector field is equal to the interface thickness. Difficulties arise from the fact that the two global minimizers of the functionals are nonconstant and converge only in the weak L2-topology.
Complete low frequency asymptotics for time-harmonic generalized Maxwell equations in nonsmooth exterior domains
Authors: Pauly Dirk
Published: November 2019

We continue the study of the operator of generalized Maxwell equations and completely discover the behavior of the solutions of the time-harmonic equations as the frequency tends to zero. Thereby we identify degenerate operators in terms of special ‘polynomially growing’ solutions of a corresponding static problem, which must be added to the ‘usual’ Neumann series in order to describe the low frequency asymptotic adequately.
Numerical verification of regularity in the three-dimensional Navier–Stokes equations for bounded sets of initial data
Authors: Robinson James C. , Sadowski Witold
Published: November 2019

Current theoretical results for the three-dimensional Navier–Stokes equations only guarantee that solutions remain regular for all time when the initial enstrophy (‖Du0‖2:=∫|curl u0|2) is sufficiently small, ‖Du0‖2≤χ0. In fact, this smallness condition is such that the enstrophy is always non-increasing. In this paper we provide a numerical procedure that will verify regularity of solutions for any bounded set of initial conditions, ‖Du0‖2≤χ1. Under the assumption that the equations are in fact regular we show that this procedure can be guaranteed to terminate after a finite time.
Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit
Authors: Glass O. , Guerrero S.
Published: November 2019

In this paper, we deal with controllability properties of linear and nonlinear Korteweg–de Vries equations in a bounded interval. The main part of this paper is a result of uniform controllability of a linear KdV equation in the limit of zero-dispersion. Moreover, we establish a result of null controllability for the linear equation via the left Dirichlet boundary condition, and of exact controllability via both Dirichlet boundary conditions. As a consequence, we obtain some local exact controllability results for the nonlinear KdV equation.
Plane Kolmogorov flows and Takens–Bogdanov bifurcation without parameters: The doubly reversible case
Authors: Afendikov Andrei , Fiedler Bernold , Liebscher Stefan
Published: November 2019

We consider the Kolmogorov problem of viscous incompressible planar fluid flow under external spatially periodic forcing. Looking for time-independent bounded solutions near the critical Reynolds number, we use the Kirchgässner reduction to obtain a spatial dynamical system on a 6-dimensional center manifold. The dynamics is generated by translations in the unbounded spatial direction. Reduction by first integrals yields a 3-dimensional reversible system with a line of equilibria. This line of equilibria is neither induced by symmetries, nor by first integrals. At isolated points, normal hyperbolicity of the line fails due to a transverse double eigenvalue zero. In particular we describe the complete set ℬ of all small bounded solutions. In the classical Kolmogorov case, ℬ consists of periodic profiles, homoclinic pulses and a heteroclinic front–back pair. This is a consequence of the symmetry of the external force.
Global existence and asymptotic behavior of weak solutions to the 1D compressible Navier–Stokes equations with degenerate viscosity coefficient
Authors: Guo Zhenhua , Jiang Song , Xie Feng
Published: November 2019

This paper is concerned with the existence of global weak solutions to the 1D compressible Navier–Stokes equations with density-dependent viscosity and initial density that is connected to vacuum with discontinuities. When the viscosity coefficient is proportional to ρθ with 0<θ<max{3−γ, 3/2} where ρ is the density, we prove a global existence theorem, improving thus the result in Meth. Appl. Anal. 12 (2005), 239–252, where 0<θ<1 is required. Moreover, we show that the domain occupied by the fluid expands into vacuum at an algebraic rate as time grows up due to the dispersion effect of the total pressure. It is worth pointing out that our result covers the interesting case of the Saint-Venant model for shallow water (i.e., θ=1, γ=2).
Multi-materials with strong interface: Variational modelings
Authors: Bessoud Anne Laure , Krasucki Françoise , Michaille Gérard
Published: November 2019

We introduce a simplified model for a multi-material made up of two elastic bodies connected by a strong thin material layer whose stiffness grows as 1/ε. The model is obtained by identifying the Γ-limit of the stored strain energy functional of the physical problem when the thickness ε of the intermediate layer tends to zero. The intermediate layer behaves as a stiffening elastic membrane. Furthermore, in the linear anisotropic case, we establish the strong convergence of the exact solution toward the solution of the limit problem.
Asymptotic behavior of least energy solutions for a biharmonic problem with nearly critical growth
Authors: Takahashi Futoshi
Published: November 2019

We study the asymptotic behavior of least energy solutions to the equation Δ2u=c0K(x)upε with the Navier boundary condition as ε→+0, where Ω is a smooth bounded domain in RN (N≥5), c0=(N−4)(N−2)N(N+2) and pε=(N+4)/(N−4)−ε, ε>0. Under some assumptions on the coefficient function K, we obtain fairly precise asymptotics of the L∞-norm of least energy solutions.
A reduced model for spatially structured predator–prey systems with fast spatial migrations and slow demographic evolutions
Authors: Castella François , Hoffbeck Jean-Philippe , Lagadeuc Yvan
Published: November 2019

We consider a spatially structured predator–prey model where fast migrations occur inside a given spatial domain, while slow predator–prey interactions prescribe the demographic evolution. The unknowns of our model are the numbers of predators and prey at each time t and each site x of the domain. In the idealized limit where migrations are infinitely fast, we show one can approximate the global dynamics using the mere two unknowns corresponding to the total number of preys and predators, irrespective of their respective spatial repartition. Besides, the error term induced by this approximation can be made exponentially small with respect to the natural asymptotic parameter.

In doing so, we completely characterize how migrations do modify both the qualitative and quantitative properties of the global demography.

Our analysis relies on a convenient version of the central manifold theorem, in conjunction with a spectral gap estimate on the involved migration operator.
Large time behaviour for degenerate parabolic equations with convection
Authors: Andreucci Daniele , Tedeev Anatoli F.
Published: November 2019

We consider the Cauchy problem for a doubly nonlinear parabolic equation, obtaining optimal estimates both for the sup norm of nonnegative solutions, and for their support for large times.
Stability property of the two-dimensional Keller–Segel model
Authors: Raczyński Andrzej
Published: November 2019

The paper contains results concerning the stability properties of solutions to a simplified Keller–Segel problem modelling chemotaxis in the whole space R2. First, the existence of solutions to considered system is proved, as well as for this system with the elliptic equation replaced by its parabolic version with time derivative multiplied by a positive parameter ε. Secondly, the stability of the model is shown, namely the convergence of solutions of parabolic–parabolic system to a solution of parabolic–elliptic one, when a suitable parameter in the system under consideration converges towards zero.
The Vlasov–Poisson system with strong external magnetic field. Finite Larmor radius regime
Authors: Bostan Mihai
Published: November 2019

We study here the finite Larmor radius regime for the Vlasov–Poisson equations with strong external magnetic field. The derivation of the limit model follows by formal expansion in power series with respect to a small parameter. If we replace the particle distribution by the center distribution of the Larmor circles the limit of these densities satisfies a transport equation, whose velocity is given by the gyro-average of the electric field. We justify rigorously the convergence towards the above model and we investigate the well-posedness of it.
Homogenization for a nonlinear ferroelectric model
Authors: Timofte Aida
Published: November 2019

We prove homogenization results for a class of rate-independent, nonlinear ferroelectric models. The PDE system defining the model is restated in terms of a stability condition and an energy balance law using an energy-storage functional and a dissipation functional. By the method of weak and strong two-scale convergence via periodic unfolding, we show that the solutions of the problem with periodicity converge to the energetic solution of the homogenized problem associated with the corresponding Γ-limits of the functionals. The main difficulties are the nonlinearity of the model, as well as the general form considered for the stored energy, which is neither convex nor quadratic.
Uniqueness and multiplicity results for N-Laplace equation with critical and singular nonlinearity in a ball
Authors: Giacomoni J. , Prashanth S. , Sreenadh K.
Published: November 2019

Let B1 be the unit open ball with center at the origin in RN, N≥2. We consider the following quasilinear problem depending on a real parameter λ>0:

−ΔNu=λ f(u), u>0 in Ω,

u=0 on ∂Ω,   (Pλ)

where f(t) is a nonlinearity that grows like etN/N−1 as t→∞ and behaves like tα, for some α∈(−∞, 0), as t→0+. More precisely, we require f to satisfy assumptions (A1) and (A2) listed in Section 1. For such a general nonlinearity we show that if λ>0 is small enough, (Pλ) admits at least one weak solution (in the sense of distributions). We further study the question of uniqueness and multiplicity of solutions to (Pλ) when Ω=B1 under additional structural conditions on f (see assumptions (A3)–(A8) in Section 2). Using shooting methods and asymptotic analysis of ODEs, under the additional assumptions (A3)–(A5), we prove uniqueness of solution to (Pλ) for all λ>0 small whereas under (A6), (A7) or (A8), we show multiplicity of solutions to (Pλ) for all λ>0 in a maximal interval. These results clearly show that the borderline between uniqueness and multiplicity is given by the growth condition lim inf t→∞h(t)teεt1/(N−1)=∞ ∀ε>0.
Bloch waves homogenization of a Dirichlet problem in a periodically perforated domain
Authors: Dupuy Delphine , Orive Rafael , Smaranda Loredana
Published: November 2019

In this paper we use the spectral method of Bloch waves to study the homogenization process of the Poisson equation in a periodically perforated domain, under homogeneous Dirichlet conditions on both exterior and interior boundaries, as the hole size goes to zero more rapidly than the micro-structure size. Using this method, we find the exact value of the critical hole size, which separates the different behaviors, where the classical strange term may or may not appear in the homogenized equation. This strange term is related to the asymptotic behavior of the first eigenvalue with respect to the hole radius.
An averaging principle for moment equations of branching diffusion processes with fast transformation rate
Authors: Chen Zhiwei , Molchanov Stanislav
Published: November 2019

A multi-type branching diffusion process with fast transmutations between particles are considered in this paper. The first moment equations for the process are reaction–diffusion equations. As the inverse transmutation rate approaches 0, we derive an averaging principle for these reaction–diffusion equations.
The two-scale Fourier transform approach to homogenization; periodic homogenization in Fourier space
Authors: Wellander Niklas
Published: November 2019

A two-scale Fourier transform for periodic homogenization in Fourier space is introduced. The transform connects the various existing techniques for periodic homogenization, i.e., two-scale convergence, periodic unfolding and the Floquet–Bloch expansion approach to homogenization. It turns out that the two-scale compactness results are easily obtained by the use of the two-scale Fourier transform. Moreover, the Floquet–Bloch eigenvalue problems for differential operators is recovered in a natural and straight forward way by the use of this transform. The transform is generalized to the (N+1)-scale case.
Asymptotics of solutions of the Neumann problem in a domain with closely posed components of the boundary
Authors: Cardone G. , Nazarov S.A. , Sokolowski J.
Published: November 2019

The Neumann problem for the Poisson equation is considered in a domain Ωε⊂Rn with boundary components posed at a small distance ε>0 so that in the limit, as ε→0+, the components touch each other at the point 𝒪 with the tangency exponent 2m≥2. Asymptotics of the solution uε and the Dirichlet integral ‖∇xuε; L2(Ωε)‖2 are evaluated and it is shown that main asymptotic term of uε and the existence of the finite limit of the integral depend on the relation between the spatial dimension n and the exponent 2m. For example, in the case n<2m−1 the main asymptotic term becomes of the boundary layer type and the Dirichlet integral has no finite limit. Some generalizations are discussed and certain unsolved problems are formulated, in particular, non-integer exponents 2m and tangency of the boundary components along smooth curves.
On one method of improving weakly converging sequence of gradients
Authors: Kałamajska Agnieszka
Published: November 2019

Let Ω⊆Rn be a bounded open domain, F⊆Omega¯ be a closed subset and let {uk}k∈N be a bounded sequence in Sobolew space where 1≤p<∞, converging weakly to u∈W1,p(Ω, Rm). Let ℛ be a given complete separable ring of continuous functions on Rm×n and assume that for each f∈ℛ the sequence of compositions {f(∇uk)(1+|∇uk|p) dx}k∈N embedded into the space of measures on Ω¯ converges weakly * to some measure μf. We discuss the possibility to modify the sequence {uk}k∈N in such a way that the new sequence {wk}k∈N is still bounded in W1,p(Ω, Rm), converges weakly to u, each sequence of measures {f(∇wk)(1+|∇wk|p) dx}k∈N also converges weakly * to μf, where f∈ℛ, but additionally the new sequence satisfies the condition “wk=u” on F. Our results are applied to the minimization problems in the Calculus of Variations.
Stabilization of solutions of a semilinear parabolic equation modelling quorum sensing in a biofilm
Authors: Fila Marek , Winkler Michael
Published: November 2019

We analyze a mathematical model introduced by Anguige, Ward and King (J. Math. Biol. 51 (2005), 557–594) to describe a quorum sensing mechanism in spatially-structured populations of the bacteria Pseudomonas aeruginosa. In the biologically relevant limit case when the spatial distribution of bacteria is constant in time, this model reduces to a single semilinear parabolic equation for the concentration of the signal substance N-(3-oxododecanoyl)-homoserine lactone (AHL). We show that under some mild technical assumptions on the nonlinear AHL production rate, the AHL concentration approaches a uniquely determined steady state, and that this convergence takes place at an exponential rate.
Semiclassical reduction for magnetic Schrödinger operator with periodic zero-range potentials and applications
Authors: Helffer Bernard , Pankrashkin Konstantin
Published: November 2019

The two-dimensional Schrödinger operator with a uniform magnetic field and a periodic zero-range potential is considered. For weak magnetic fields and a weak coupling we reduce the spectral problem to the semiclassical analysis of one-dimensional Harper-like operators. This shows the existence of parts of Cantor structure in the spectrum for special values of the magnetic flux.
Analytic smoothing effect for the Schrödinger equation relative to the harmonic oscillator equation
Authors: Baraket Amel Atallah , Mechergui Chokri
Published: November 2019

We study the microlocal analytic smoothing effect for the solutions of the Schrödinger equation associated with the harmonic oscillator using the method of Martinez, Nakamura and Sordoni (Comm. Pure Appl. Math. 59 (2006), 1330–1351) of microlocal weights estimates.
Stochastic homogenization of quasilinear PDEs with a spatial degeneracy
Authors: Delarue François , Rhodes Rémi
Published: November 2019

We investigate stochastic homogenization for some degenerate quasilinear parabolic PDEs. The underlying nonlinear operator degenerates along the space variable, uniformly in the nonlinear term: the degeneracy points correspond to the degeneracy points of a reference diffusion operator on the random medium.

Assuming that this reference diffusion operator is ergodic, we can prove the homogenization property for the quasilinear PDEs, by means of the first-order approximation method. The (nonlinear) limit operator need not be nondegenerate. Concrete examples are provided.
Asymptotic expansions for Taylor coefficients of the composition of two functions
Authors: Hilberdink Titus
Published: November 2019

We give an asymptotic expansion for the Taylor coefficients of L(P(z)) where L(z) is analytic in the open unit disc whose Taylor coefficients vary ‘smoothly’ and P(z) is a probability generating function. We show how this result applies to a variety of problems, amongst them obtaining the asymptotics of Bernoulli transforms and weighted renewal sequences.
Two-parameter asymptotics in the Cauchy problem for a quasi-linear parabolic equation
Authors: Zakharov Sergei V.
Published: November 2019

The Cauchy problem for a quasi-linear parabolic equation with a small parameter ε at the higher derivative is considered. The derivative of the initial function is of order O(1/ρ), where ρ is another small parameter. We study the asymptotics of the solution of the problem in parameters ε and ρ tending to zero.
Asymptotic behavior of a thermoviscoelastic plate with memory effects
Authors: Grasselli Maurizio , Muñoz Rivera Jaime E. , Squassina Marco
Published: November 2019

We consider a coupled linear system describing a thermoviscoelastic plate with hereditary effects. The system consists of a hyperbolic integrodifferential equation, governing the temperature, which is linearly coupled with the partial differential equation ruling the evolution of the vertical deflection, presenting a convolution term accounting for memory effects. It is also assumed that the thermal power contains a memory term characterized by a relaxation kernel. We prove that the system is exponentially stable and we obtain a closeness estimate between the system with memory effects and the corresponding memory-free limiting system, as the kernels fade in a suitable sense.
On the Weyl law for Toeplitz operators
Authors: Paoletti Roberto
Published: November 2019

A Weyl law for Toeplitz operators was proved by Boutet de Monvel and Guillemin for general Toeplitz structures. In the setting of positive line bundles, we revisit this theme in light of local asymptotic techniques based on the microlocal theory of the Szego kernel. By pairing this approach with classical arguments used to estimate the spectral function of a pseudodifferential operator, we first establish a local Weyl law (that is, a pointwise estimate on the spectral function of the Toeplitz operator). The global Weyl law follows by integration.
Hyperasymptotic expansions of the modified Bessel function of the third kind of purely imaginary order
Authors: Shi Wei , Wong R.
Published: November 2019

In this paper, we use a new method to derive hyperasymptotic expansions for the modified Bessel function of the third kind of purely imaginary order. Our approach is based on the method of Hadamard-series expansion developed by Paris (Proc. Roy. Soc. Lond. A 457 (2001), 2855–2869). Modifications are made to deal with the presence of an infinite number of saddle points, and numerical computation is used to illustrate the accuracy of our results.
Modulation analysis for a stochastic NLS equation arising in Bose–Einstein condensation
Authors: de Bouard Anne , Fukuizumi Reika
Published: November 2019

We study the asymptotic behavior of the solution of a model equation for Bose–Einstein condensation, in the case where the trapping potential varies randomly in time. The model is the so called Gross–Pitaevskii equation, with a quadratic potential with white noise fluctuations in time whose amplitude ε tends to zero. The initial condition of the solution is a standing wave solution of the unperturbed equation. We prove that up to times of the order of ε−2, the solution decomposes into the sum of a randomly modulated standing wave and a small remainder, and we derive the equations for the modulation parameters. In addition, we show that the first order of the remainder, as ε goes to zero, converges to a Gaussian process, whose expected mode amplitudes concentrate on the third eigenmode generated by the Hermite functions, on a certain time scale.
Global asymptotics for polynomials orthogonal with exponential quartic weight
Authors: Wong R. , Zhang L.
Published: November 2019

In this paper, we study the asymptotics of polynomials orthogonal with respect to the varying quartic weight ω(x)=e−nV(x), where V(x)=Vt(x)=x4/4+x2t/2. We focus on the critical case t=−2, in the sense that for t≥−2, the support of the associated equilibrium measure is a single interval, while for t<−2, the support consists of two intervals. Globally uniform asymptotic expansions are obtained for z in three unbounded regions. These regions together cover the whole complex z-plane. In particular, in the region containing the origin, the expansion involves the Ψ function affiliated with the Hastings–McLeod solution of the second Painlevé equation. Our approach is based on a modified version of the steepest-descent method for Riemann–Hilbert problems introduced by Deift and Zhou (Ann. Math. 137 (1993), 295–370).
Dispersive estimates for the Schrödinger equation in dimensions four and five
Authors: Cardoso Fernando , Cuevas Claudio , Vodev Georgi
Published: November 2019

We prove optimal (that is, without loss of derivatives) dispersive estimates for the Schrödinger group eit(−Δ+V) for a class of real-valued potentials V∈Ck(Rn), V(x)=O(〈x〉−δ), where n=4, 5, k>(n−3)/2, δ>3 if n=4 and δ>5 if n=5.
On continuous dependence of solution of parabolic equations on coefficients
Authors: Kim Hyejin
Published: November 2019

It is well known that solutions of classical initial-boundary problems for second-order parabolic equations depend continuously on the coefficients if the coefficients converge to their limits in a strong enough topology.

In case of one spatial variable, we consider the question of the weakest possible topology providing convergence of the solutions. The problem is closely related to weak convergence of corresponding diffusion processes. Continuous Markov processes corresponding to the Feller operators DvDu arise, in general, as limiting processes. Solutions of the parabolic equations converge, in general, to the solutions of corresponding initial-boundary problems for the limiting operator DvDu.
Double scale analysis of a Schrödinger–Poisson system with quantum wells and macroscopic nonlinearities in dimension 2 and 3
Authors: Faraj A. , Mantile A. , Nier F.
Published: November 2019

We consider the stationary Schrödinger–Poisson model with a background potential describing a quantum well. The Hamiltonian of this system composes of contributions – the background potential well plus a nonlinear repulsive term – which extends on different length scales with ratio parametrized by the small parameter h. With a partition function which forces the particles to remain in the quantum well, the limit h→0 in the nonlinear system leads to different asymptotic behaviours, including spectral renormalization, depending on the dimensions 1, 2 or 3.
Boundary layer solutions to singularly perturbed problems via the implicit function theorem
Authors: Omel'chenko Oleh , Recke Lutz
Published: November 2019

We prove existence, local uniqueness and asymptotic estimates for boundary layer solutions to singularly perturbed problems of the type ε2u″=f(x, u, εu′, ε), 0<x<1, with Dirichlet and Neumann boundary conditions. For that we assume that there is given a family of approximate solutions which satisfy the differential equation and the boundary conditions with certain low accuracy. Moreover, we show that, if this accuracy is high, then the closeness of the approximate solution to the exact solution is correspondingly high. The main tool of the proofs is a generalized implicit function theorem which is close to those of Fife and Greenlee (Uspechi Mat. Nauk 24 (1974), 103–130) and of Magnus (Proc. Royal Soc. Edinburgh 136A (2006), 559–583). Finally we show how to construct approximate solutions under certain natural conditions.
Singular perturbation problems in a general smooth domain
Authors: Gie Gung-Min
Published: November 2019

The goal of this article is to study the boundary layer of a reaction-diffusion equation with a small viscosity in a general (curved), bounded and smooth domain in Rn, n≥2. To the best of our knowledge, the classical expansion in the case of a bounded interval or of a channel is not valid for a general domain. Using the techniques of differential geometry, a new asymptotic expansion proposed in this article recovers the optimal convergence rate of the remainder at all orders.
Decay estimates for the solutions of the system of crystal acoustics for cubic crystals
Authors: Liess Otto
Published: November 2019

The aim of this paper is to study decay estimates for global solutions of the system of linear crystal elasticity in three space dimensions for cubic crystals in the generic nearly isotropic case. Our main result is that for large time, the solutions to the system decay with a decay rate of order t−1/2−1/κ for some κ∈N which is strictly positive (cf. Remark 1.2). We should observe that in the isotropic case, the optimal decay rate is t−1 and that the decay rate for the wave equation in n space variables is t−(n−1)/2. On the other hand, the decay rate for solutions of the Maxwell system for optically biaxial crystals is of order t−1/2. (See O. Liess, Asymptot. Anal. 4 (1991), 61–95.)
Local H1-regularity and H1/3−δ-regularity up to the boundary in time dependent viscoplasticity
Authors: Alber Hans-Dieter , Nesenenko Sergiy
Published: November 2019

Local and boundary regularity for quasistatic initial-boundary value problems from viscoplasticity is studied. The problems considered belong to a general class with monotone constitutive equations modelling materials showing kinematic hardening. A standard example is the Melan–Prager model. It is shown that the strain/stress/internal variable fields have the regularity H4/3−δ/H1/3−δ/H1/3−δ up to the boundary. The proof uses perturbation estimates for monotone operator equations.
A Moser-type inequality in Lorentz–Sobolev spaces for unbounded domains in RN
Authors: Cassani Daniele , Tarsi Cristina
Published: November 2019

We derive a Pohožaev–Trudinger type embedding for the Lorentz–Sobolev space W10LN,q(Ω), for general domains Ω⊆RN and in particular for Ω=RN. Precisely, we first prove that the corresponding inequality is domain independent and then, by constructing explicit concentrating sequences à la Moser, we establish that the embedding inequality is sharp and we exhibit the best constant.
Emergence of exponentially small reflected waves
Authors: Betz Volker , Joye Alain , Teufel Stefan
Published: November 2019

We study the time-dependent scattering of a quantum mechanical wave packet at a barrier for energies larger than the barrier height, in the semi-classical regime. More precisely, we are interested in the leading order of the exponentially small scattered part of the wave packet in the semiclassical parameter when the energy density of the incident wave is sharply peaked around some value. We prove that this reflected part has, to leading order, a Gaussian shape centered on the classical trajectory for all times soon after its birth time. We give explicit formulas and rigorous error bounds for the reflected wave for all of these times.
The limiting absorption principle for an acoustic equation with a refraction coefficient vanishing at infinity
Authors: Eidus D.
Published: November 2019

We prove the limiting absorption principle for an equation of forced acoustic stationary oscillations outside a reflecting surface in a case where the refraction coefficient ρ(x) tends to zero at infinity.
Asymptotic behaviour of solutions to coercive models of monotone type for inelastic deformation with damping
Authors: Bartczak Leszek
Published: November 2019

We study the asymptotic behaviour of the solution to coercive models of inelastic deformation with damping. We look for conditions on the boundary data, the external force and the nonlinearity such that the solution of our problem converges to a stationary solution.
Stationary solutions for the one-dimensional Nordström–Vlasov system
Authors: Bostan Mihai
Published: November 2019

The Nordström–Vlasov system describes the evolution of self-gravitating collisionless particles. We prove the existence of stationary solutions in one dimension. We show also the propagation of impulsion moments and perform an asymptotic analysis.
Viscoelastic fluids in thin domains: A mathematical proof
Authors: Bayada Guy , Chupin Laurent , Grec Bérénice
Published: November 2019

The present paper deals with non-Newtonian viscoelastic flows of Oldroyd-B type in thin domains. Such geometries arise for example in the context of lubrication. More precisely, we justify rigorously the asymptotic model obtained heuristically by proving the mathematical convergence of the Navier–Stokes/Oldroyd-B system towards the asymptotic model.
Asymptotical form of Possio integral equation in theoretical aeroelasticity
Authors: Shubov Marianna A.
Published: November 2019

The paper is the second in a series of works devoted to the solvability of the Possio singular integral equation. This equation relates the pressure distribution over a typical section of a slender wing in subsonic compressible air flow to the normal velocity of the points of a wing (downwash). In spite of the importance of the Possio equation, the question of the existence of its solution has not been settled yet. In the first paper “Reduction of the boundary-value problem to Possio integral equation in theoretical aeroelasticity”, we reformulated the initial boundary-value problem, involving a partial differential equation for the velocity potential and highly nonstandard boundary conditions, in the form of a singular integral equation, the Possio equation. In our derivation, we have used the ideas and techniques totally different from the ones used by C. Possio in his original work. In the present paper, we show that the integral equation can be split up into two parts in such a way that the first part is not small with respect to a complex parameter entering the equation, while the second part is asymptotically small and tends to zero as the above parameter tends to infinity. In this paper, we justify such a splitting and in the next paper, we will prove solvability of the nonvanishing part of the Possio equation and then prove that asymptotically small terms cannot destroy the aforementioned unique solvability.
Diffusive limits for the Knudsen gas in a thin channel with accommodation on the boundary
Authors: Bernard Yann
Published: November 2019

We consider a model for a free molecular flow in a thin channel bounded by two parallel plates on which Maxwellian boundary conditions with (fixed) accommodation and a generic scattering kernel apply. Using functional analytic tools, we show that as the width of the channel vanishes, and on a suitable temporal scale, the evolution of the density is described by a diffusion problem. We distinguish two classes of temporal scalings (normal and anomalous) and we show that an infinitesimal amount of grazing collisions with the walls of the channel is responsible for the anomalous diffusion. The method employed is adapted from the original work of F. Golse in Asymptot. Anal. 17 (1998), 1–12.
H-convergence of singular solutions of some Dirichlet problems with terms of order one
Authors: Boccardo Lucio , Casado-Diaz Juan
Published: November 2019

We study the H-convergence of boundary value problems of the type

 −div(Mε(x)∇uε)=−div(uεEε(x))+f(x), inΩ,

 uε=0 on∂Ω

if Eε∈(L2(Ω))N.
Singular perturbations of weakly coupled systems of Hamilton–Jacobi equations
Authors: Camilli Fabio , Cesaroni Annalisa
Published: November 2019

In this paper we study singular perturbations of weakly coupled systems of elliptic equations. A model problem is given by small random perturbations of random evolution processes. In this setting we give a PDE proof of large deviation results, analogous to those studied in Ann. Probab. 15 (1987), 646–658, Stochastics Stochastics Rep. 33 (1990), 111–148, Probab. Theory Related Fields 94 (1993), 335–374. An essential tool in our approach is the weak KAM theory introduced in Calc. Var. Partial Differential Equations 22 (2005), 185–228.
Weighted energy decay for 3D wave equation
Authors: Kopylova E.A.
Published: November 2019

We obtain a dispersive long-time decay in weighted energy norms for solutions to the 3D wave equation with generic potential. The decay extends the results obtained by Jensen and Kato for the 3D Schrödinger equation.
Semiclassical resonances for a two-level Schrödinger operator with a conical intersection
Authors: Fujiié Setsuro , Lasser Caroline , Nédélec Laurence
Published: November 2019

We study the resonant set of a two-level Schrödinger operator with a linear conical intersection. This model operator can be decomposed into a direct sum of first order systems on the real half-line. For these ordinary differential systems we locally construct exact WKB solutions, which are connected to global solutions, amongst which are resonant states. The main results are a generalized Bohr–Sommerfeld quantization condition and an asymptotic description of the set of resonances as a distorted lattice.
Explicit observation of Schrödinger equation with potential
Authors: Maddouri Feten
Published: November 2019

We study an explicit interior observability estimate for the Schrödinger equation with potential in a bounded domain of RN (N≥1), and with zero Dirichlet boundary condition. The method combines Carleman inequality and multiplier techniques. Moreover, by microlocal analysis tools, we prove a precised observability estimate in high frequencies.
On convergence to equilibria for a chemotaxis model with volume-filling effect
Authors: Jiang Jie , Zhang Yanyan
Published: November 2019

In this paper, we study the asymptotic behavior of solutions to a chemotaxis model with volume-filling effect subject to the homogeneous Neumann boundary conditions. We establish a non-smooth version of Łojasiewicz–Simon inequality, and we prove that as time goes to infinity the solution to our system converges to an equilibrium in W1,p(Ω)×W1,p(Ω), p>max (n, 2), Ω⊂Rn. We also obtain an estimate of the decay rate to equilibrium.
Degenerated codimension 1 crossings and resolvent estimates
Authors: Duyckaerts Thomas , Fermanian Kammerer Clotilde , Jecko Thierry
Published: November 2019

In this article, we analyze the propagation of Wigner measures of a family of solutions to a system of semi-classical pseudodifferential equations presenting eigenvalues crossings on hypersurfaces. We prove the propagation along classical trajectories under a geometric condition which is satisfied, for example, as soon as the Hamiltonian vector fields are transverse or tangent at finite order to the crossing set. We derive resolvent estimates for semi-classical Schrödinger operator with matrix-valued potential under a geometric condition of the same type on the crossing set and we analyze examples of degenerate situations where one can prove transfers between the modes.
Asymptotic expansions of Legendre series coefficients for functions with endpoint singularities
Authors: Sidi Avram
Published: November 2019

Let Σ∞n=0en[f]Pn(x) be the Legendre expansion of a function f(x) on (−1, 1). In this work, we derive an asymptotic expansion as n→∞ for en[f], assuming that f∈C∞(−1, 1), but may have arbitrary algebraic-logarithmic singularities at one or both endpoints x=±1. Specifically, we assume that f(x) has asymptotic expansions of the forms

 f(x)~Σ∞s=0Usl(log(1−x))(1−x)αs as x→1−,

 f(x)~Σ∞s=0Vs(log(1+x))(1+x)βs as x→−1+,

where Us(y) and Vs(y) are some polynomials in y. Here, αs and βs are in general complex and Rαs, Rβs>−1. An important special case is that in which Us(y) and Vs(y) are constant polynomials; for this case, the asymptotic expansion of en[f] assumes the form

 en[f]~Σs=0αs∉Z+∞Σ∞i=0asihαs+i+1/2+(−1)n Σs=0βs∉Z+∞Σ∞i=0bsihβs+i+1/2 as n→∞,

where h=(n+1/2)−2, Z+={0, 1 , 2, …}, and asi and bsi are constants independent of n.
Exponential decay phenomenon of the principal eigenvalue of an elliptic operator with a large drift term of gradient type
Authors: Jimbo Shuichi , Kimura Masato , Notsu Hirofumi
Published: November 2019

We study an asymptotic behaviour of the principal eigenvalue for an elliptic operator with large advection which is given by a gradient of a potential function. It is shown that the principal eigenvalue decays exponentially under the velocity potential well condition as the parameter tends to infinity. We reveal that the depth of the potential well plays an important role in the estimate. Particularly, in one-dimensional case, we give a much more elaborate characterization for the eigenvalue. Some numerical examples are also shown.
Multipliers in the space of analytic functions with exponential mean growth
Authors: Dostanić Milutin R.
Published: November 2019

The multipliers in the space of analytic functions with exponential mean growth are described.
Asymptotic expansions of option price under regime-switching diffusions with a fast-varying switching process
Authors: Yin G.
Published: November 2019

This work aims to developing asymptotic expansions of solutions of a system of coupled differential equations with applications to option price under regime-switching diffusions. The main motivation stems from using switching diffusions to model stochastic volatility so as to obtain uniform asymptotic expansions of European-type options. By focusing on fast mean reversion, our effort is placed on finding the “effective volatility”. Under simple conditions, asymptotic expansions are developed with uniform asymptotic error bounds. The leading term in the asymptotic expansions satisfies a Black–Scholes equation in which the mean return rate and volatility are averaged out with respect to the stationary measure of the switching process. In addition, the full asymptotic series is developed, which will help us to gain insight on the behavior of the option price when the time approaches maturity. The asymptotic expansions obtained in this paper are interesting in their own right and can be used for other problems in control optimization of systems involving fast varying switching processes.
The Fujita phenomenon in exterior domains under dynamical boundary conditions
Authors: Rault Jean-François
Published: November 2019

The Fujita phenomenon for nonlinear parabolic problems ∂tu=Δu+up in an exterior domain of RN under dissipative dynamical boundary conditions σ ∂tu+∂νu=0 is investigated in the superlinear case. As in the case of Dirichlet boundary conditions (see Trans. Amer. Math. Soc. 316 (1989), 595–622 and Israel J. Math. 98 (1997), 141–156), it turns out that there exists a critical exponent p=1+2/N such that blow-up of positive solutions always occurs for subcritical exponents, whereas in the supercritical case global existence can occur for small non-negative initial data.
The three-dimensional finite Larmor radius approximation
Authors: Han-Kwan Daniel
Published: November 2019

Following Frénod and Sonnendrücker [SIAM J. Math. Anal. 32(6) (2001) 1227–1247], we consider the finite Larmor radius regime for a plasma submitted to a large magnetic field and take into account both the quasineutrality and the local thermodynamic equilibrium of the electrons. We then rigorously establish the asymptotic gyrokinetic limit of the rescaled and modified Vlasov–Poisson system in a three-dimensional setting with the help of an averaging lemma.