bluuebunny/crossref_metadata_2025 huggingface dataset:
Weak finite‐dimensional approximations of semi‐linear elliptic PDEs with near‐critical exponents
Authors: Budd C.J. , Humphries A.R.
Published: December 2024

We make a formal study of the differential equation [Formula: see text] when posed as a variational problem over a finite‐dimensional subset $S_h$ of $H^1_0$ comprising piecewise‐linear functions defined on a mesh of size $h$ . We determine critical points $U_h \in S_h$ of the variational form of (1). Such functions are perturbations of $u$ when a solution of (1) exists, but we show that $U_h$ can also exist when (1) has no solution and we determine an asymptotic expression for the solution branch $(\lambda,U_h)$ when $\Vert U_h\Vert _{\infty}$ is large and $h \Vert U_h\Vert _{\infty}^2$ is small. If $\varepsilon = 0$ , then $u$ exists if $\lambda > \mbox{{\char"19}}^2/4$ , and we give a formula expressing $U_h$ as a perturbation of $u$ . If $\lambda \mathrel{\hbox{{\char"36}}}\mbox{{\char"19}}^2/4$ , then a solution of the differential equation does not exist, and $U_h$ grows as $h \rightarrow 0.$ We show that the rate of growth is proportional to $h^{-1/4}$ if $\lambda = \mbox{{\char"19}}^2/4$ , and $h^{-1/3}$ if $\lambda = 0$ . We compare these results with estimates for the solutions of (1) when $\varepsilon \rightarrow 0^-$ . Our results are obtained by using formal asymptotic methods – particulary the method of matched asymptotic expansions – and are supported by some numerical calculations.
Asymptotics of the circulation around a three‐dimensional thin wing
Authors: Hinder Rainer , Nazarov Sergueï A.
Published: December 2024

We are interested in finding the velocity distribution at the wings of an aeroplane. Within the scope of a three‐dimensional linear theory we analyse a model which is formulated as a mixed screen boundary value problem for the Helmholtz equation $(\Delta + k^{ 2}) \varPhi =0$ in $\mathbb{R}^{ 3}\backslash \overline{S}$ , where $\varPhi $ denotes the perturbation velocity potential, induced by the presence of the wings and $\overline{S} := \overline{L} \cup \overline{W}$ with the projection $L$ of the wings onto the $(y_{ 1},y_{ 2})$ ‐plane and the wake $W$ .

Not all Cauchy data are given explicitly on $L$ , respectively $W$ . These missing Cauchy data depend on the wing circulation $\varGamma $ . $\varGamma $ has to be fixed by the Kutta–Joukovskii condition: $\nabla \varPhi $ should be finite near the trailing edge $x_{\rm t}$ of $L$ . We reduce here this screen problem to an equivalent mixed boundary value problem in $\mathbb{R}^{ 3}_{ +}$ . The main problem is in both cases the calculation of $\varGamma $ . In order to find $\varPhi $ we use the method of matched asymptotics for some small geometrical parameter $\varepsilon$ and the ansatz $\varGamma = \varGamma _{ 0} + \varepsilon \varGamma _{ 1} + \cdots $ which makes it possible to split the problem into a sequence of problems for $\varGamma _{ 0},\varGamma _{ 1}, \ldots\,$ . Concretely, we calculate $\varGamma _{ 0}$ and $\varGamma _{ 1}$ explicitly by the demand of vanishing intensity factors of the solutions of the corresponding mixed problems at the borderline between $L$ and $W$ . Especially, we point out that $\varGamma _{ 0}$ can be obtained by solving a two‐dimensional problem for every cross‐section of $L$ while $\varGamma _{ 1}$ indicates the interaction of these cross‐sections.
On oscillations of an asymptotic equation of a nonlinear variational wave equation
Authors: Zhang Ping , Zheng Yuxi
Published: December 2024

The wave equation $\curpartial _t^2u - c\curpartial _x(c\curpartial _xu)=0$ , where $c = c(u)$ is a given function, arises in a number of different physical contexts and is the simplest example of an interesting class of nonlinear hyperbolic partial differential equations. For unidirectional weakly nonlinear waves, an asymptotic equation $\curpartial _x(\curpartial _tu + u\curpartial _xu) =(1/2)(\curpartial _xu)^2$ has been derived. It has been shown through concrete examples that oscillations in $v$ , $v=\curpartial _xu$ , in the initial data persist into positive time for the asymptotic equation. In the first part of this paper, we show by applying Young measure theory that no oscillations are generated if there are no oscillations (around a nonnegative state $v$ ) in the initial data, which implies in particular the global existence of weak solutions to the asymptotic equation with nonnegative $L^p(\mathbb{R})$ initial data $v$ with $p>2$ . In the second part, we obtain a regularity result for a large class of weak solutions to this equation by using its kinetic formulation. In particular this regularity result applies to both the conservative and dissipative weak solutions.
Energy decay of magnetoelastic waves in a bounded conductive medium
Authors: Perla Menzala G. , Zuazua Enrique
Published: December 2024

We study the decay of the energy of solutions of the system of magnetoelasticity in a bounded, three‐dimensional conductive medium. We prove that all solutions do decay as $t\to\infty$ in the energy‐space when the domain is simply connected. We also describe the large time behavior of solutions when the domain is not simply connected.
Decay of solutions of the wave equation with a local highly degenerate dissipation
Authors: Martinez Patrick
Published: December 2024

We consider the system of the wave equation with the Dirichlet boundary condition damped with a local linear dissipation of the type $a(x)u'$ . In order to simplify, we assume that the domain is the open ball of $\mathbf{R}^N$ centered in $O$ and of radius $R$ and that the function $a$ is radial near the boundary. When $a$ goes quickly to zero at the boundary, we obtain a precise decay rate estimate of the energy of the solutions. This extends some results of Zuazua and Nakao. The proof is based on a new nonlinear integral inequality and on the asymptotic behavior of the function $a$ at the boundary.
Angle singularities of solutions to the Neumann problem for the two‐dimensional Riccati’s equation
Authors: Kozlov Vladimir , Maz’ya Vladimir
Published: December 2024

A two‐dimensional Riccati’s equation with Neumann boundary data is considered in a domain with an angular point. Asymptotic formulas for an arbitrary solution near the vertex are obtained.
Perturbations of the Laplacian with variable coefficients in exterior domains and differentiability of the resolvent
Authors: Kerler Charlotte
Published: December 2024

We consider self‐adjoint, strictly elliptic, short‐range perturbations of the Laplacian with variable coefficients, that operate on square integrable functions defined in an exterior domain. According to the limiting absorption principle, the resolvent is extended to the positive real line, which is the continuous spectrum of these operators. In suitable weighted spaces, where the weight also depends on the grade of differentiation, we show that the extended resolvent is strongly differentiable w.r.t. the spectral parameter. Moreover, we obtain estimates for the derivatives of the extended resolvent in these weighted spaces, which are locally uniform in the spectral parameter.
The asymptotics of inhomogeneous factored Cauchy problems
Authors: Goldstein Gisèle Ruiz , Goldstein Jerome A. , Obrecht Enrico
Published: December 2024

Scattering theory tells how solutions of one abstract Schrödinger equation (of the form ${\rm i}({\rm d}u/{\rm d}t)=Hu$ with $H=H^*)$ are asymptotic to solutions of another (in principle simpler) abstract Schrödinger equation. We extend this theory to inhomogeneous problems of the form ${\rm i}({\rm d}u)/({\rm d}t)=Hu+h(t)$ , with special emphasis on factored equations of the form $\prod_{j=1}^N (({\rm d}/{\rm d}t)-{\rm i}A_j)u(t) = h(t),$ where $A_1,\ldots, A_N$ are commuting selfadjoint operators. As a special case, corresponding to $N=4$ and two‐space scattering, we conclude that every solution $u(\cdot, t)$ of the inhomogeneous elastic wave equation in the exterior of a bounded star shaped obstacle is of the form $u=v+w+z,$ where $v(\cdot, t)$ solves the free (homogeneous) elastic wave equation with no obstacle, $w(\cdot,t)$ is determined by the (rather general) inhomogeneity, and $z(\cdot,t)={\rm o}(1)$ as $t\to \pm \infty.$ Some of the results are presented in a more general Banach space context.
Some remarks on the off‐diagonal asymptotics in quantum ergodicity
Authors: Tate Tatsuya
Published: December 2024

Our purpose of this paper is to estimate the rate of the decay of the off‐diagonal asymptotics in Sunada (preprint) in case where the Hamilton flow is of Anosov type. For the sake of this, we use the central limit theorem for transitive Anosov flows (Sinai, Soviet Math. Dokl. 1 (1960), 983–987; Ratner, Israel J. Math. 16 (1973), 181–197; Zelditch, Comm. Math. Phys. 160 (1994), 81–92). Also it is shown that if the Hamilton flow has homogeneous Lebesgue spectrum, then the measure ${\rm d}m_{A}$ associated with a pseudodifferential operator $A$ , which is introduced by Zelditch (J. Funct. Anal. 140 (1996), 68–86), is absolutely continuous with respect to Lebesgue measure.
Semiclassical asymptotics of the residues of the scattering matrix for shape resonances
Authors: Lahmar‐Benbernou Amina , Martinez André
Published: December 2024

The aim of this study is to give complete semiclassical asymptotics of the residues ${\rm Res}[S(\lambda,\omega,\omega^\prime),\rho]$ at some pole $\rho$ of the distributional kernel of the scattering matrix $S(\lambda)$ corresponding to a semiclassical two‐body Schrödinger operator $P=-h^2\Delta + V$ , and considered as a meromorphic operator‐valued function with respect to the energy $\lambda$ . We do it in the case where the pole $\rho$ considered is a shape resonance of $P$ . This is a continuation of A. Benbernou, Estimation des residus de la matrice de diffusion associés a dès résonances de forme I (to appear in Ann. Inst. H. Poincaré), where an extra geometrical condition was assumed (namely the absence of caustics near the energy level ${\rm Re}\ \rho$ ). Here we drop this assumption by using an FBI transform which permits to work in the complexified phase space. Then we show that some semiclassical WKB expansions are global, and this allows us to find out estimates for the residue of the type ${\rm O}(h^N{\rm e}^{-2S_0/h})$ , where $S_0$ is the Agmon width of the potential barrier, and $N$ may be arbitrarily large depending on an explicit geometrical location of the incoming and outgoing waves $\omega$ and $\omega'$ one consider. Full asymptotic expansions are obtained under some additional generic geometric assumption on the potential $V$ .
Some asymptotic analysis for hyperbolic relaxed Stefan problems with memory
Authors: Bonetti Elena
Published: December 2024

The aim of this paper is to establish existence and uniqueness of the solution to a diffusive phase transition problem for an integrodifferential energy balance equation of hyperbolic type. We also examine some asymptotic relations with the related phase‐field problem and the limiting case of the hyperbolic Stefan problem with memory.
Ergodic type Bellman equations of risk‐sensitive control with large parameters and their singular limits
Authors: Kaise Hidehiro , Nagai Hideo
Published: December 2024

We consider existence of solutions of ergodic type Bellman equations of risk‐sensitive control with large parameters. Then we deduce as the singular limits first‐order nonlinear partial differential equations, which are considered to relate to $H_{\infty}$ control. This is a generalization of our previous work that appeared in Asymptotic Analysis 16 (1998), 347–362.
$H^0$‐convergence for the linearized elasticity system
Authors: El Hajji Mohamed , Donato Patrizia
Published: December 2024

In Nonlinear Partial Differential Equations and Their Applications, Vol. 13 (to appear), Briane, Damlamian and Donato extend to the case of perforated domains the $H$ ‐covergence introduced by Murat and Tartar (Sém. Anal. Fonct. Numér., 1977/1978; Topics in the Mathematical Modelling of Composite Materials, 1997) for the diffusion equation.

In this paper, we define in the same spirit a notion of $H^0$ ‐convergence for the linearized elasticity system in perforated domains. This definition allows to obtain compactness, locality and corrector results.
On the existence of symmetric solutions to some singularly perturbed semilinear elliptic problems
Authors: Pistoia Angela
Published: December 2024

We prove the existence of symmetric solutions to the singularly perturbed semilinear Dirichlet or Neumann problem [Formula: see text] under certain assumptions on the nonlinearity $f$ .
Uniqueness of solutions of the Cauchy problem for parabolic equations degenerating at infinity
Authors: Eidelman S. , Kamin S. , Porper F.
Published: December 2024

This paper is devoted to the investigation of classes of uniqueness for classical solutions of the Cauchy problem [Formula: see text] We study how the behaviour of the function $\rho(t,x)$ as $\vert x\vert \to \infty$ influences the classes of uniqueness of the Cauchy problem.

In particular the dependence of such classes on the number of the space coordinates is clarified. We also discuss the sharpness of the obtained results.
On some nonlinear partial differential equations involving the $p$‐Laplacian and critical Sobolev trace maps
Authors: Demengel Françoise , Nazaret Bruno
Published: December 2024

In this article, we study some partial differential equations with Neumann nonlinear boundary conditions, related to the problem of the best constant from $W^{1,p}(\varOmega)$ to $L^{p^\star}(\curpartial\varOmega)$ , where $p^\star$ is the critical Sobolev exponent for the embedding of $W^{1,p}(\varOmega)$ into $L^{q}(\curpartial\varOmega)$ ( $\varOmega $ is a bounded $\mathcal{C}^1$ open set and $p<N$ ).
An upper bound for the temperature for some combustion equations in the limit of large activation energy
Authors: Langlois A. , Marion M.
Published: December 2024

This paper is concerned with the thermal diffusional model borrowed from the theory of combustion. The analytic methods currently used to investigate this model require a near‐equidiffusion condition, the inverse of the activation energy being the small parameter. The first step in these methods consists in assuming a suitable bound on the temperature. The aim of this paper is to derive rigorously such a bound. We show that the temperature is bounded independently of the activation energy β uniformly in time, at least for sufficiently large values of β.

The proof relies on the use of a Lyapunov function first introduced by Barabanova [1] and energy type estimates.
Some refinements of Wigner’s semi‐circle law for Gaussian random matrices using superanalysis
Authors: Inoue Atsushi , Nomura Yuji
Published: December 2024

In Random Matrix Theory (R.M.T.), Wigner’s semi‐circle law gives a well‐known corner‐stone. We report mathematical refinements of this law as an application of superanalysis. That is, using Efetov’s idea, we rewrite the average of the empirical measure of the eigenvalue distribution of the Hermitian Gaussian matrices (GUE) in a compact form. Careful calculations give not only the precise convergence rate of that law, but also the precise rate of the edge mobility.
Homogenization of a neutronic multigroup evolution model
Authors: Capdeboscq Yves
Published: December 2024

In this paper is studied the homogenization of an evolution problem for a cooperative system of weakly coupled elliptic partial differential equations, called neutronic multigroup diffusion model, in a periodic heterogenous domain. Such a model is used for studying the evolution of the neutron flux in nuclear reactor core. In this paper, we show that under a symmetry assumption, the oscillatory behavior of the solutions is controled by the first eigenvector of a multigroup eigenvalue problem posed in the periodicity cell, whereas the global trend is asymptotically given by a homogenized evolution problem. We then turn to cases when the symmetry condition is not fulfilled. In domains without boundaries, the limit equation for the global trend is then a homogenized transport equation. Alternatively, we show that in bounded domains and with well prepared initial data, the microscopic scale does not only control the oscillatory behavior of the solutions, but also induces an exponential drift.
Transformation operators for class of damped hyperbolic equations
Authors: Shubov Marianna A.
Published: December 2024

We extend the classical concept of transformation operators to the one‐dimensional wave equation with spatially nonhomogeneous coefficients containing the first order damping term. The equation governs the vibrations of a damped string. Our results hold in the cases of an infinite, semi‐infinite or a finite string. Transformation operators were introduced in the fifties by I.M. Gelfand, B.M. Levitan and V.A. Marchenko in connection with the inverse scattering problem for the one‐dimensional Schrödinger equation. In the classical case, the transformation operator maps the exponential function (stationary wave function of a free particle) into the so‐called Jost solution of the perturbed Schrödinger equation. In our case, it is natural to introduce two transformation operators, which we call outgoing and incoming transformation operators respectively. (The terminology is motivated by an analog with the Lax–Phillips scattering theory.) The first of them is related to the nonselfadjoint quadratic operator pencil generated by the original problem, and the second one is related to the adjoint pencil. We introduce a pair of asymptotically exponential solutions for each of the pencils and show that our transformation operators map certain exponential type functions to these solutions. Our main results are the proof of the existence of transformation operators (which have the forms of the identity operator plus certain Volterra integral operators) and estimates for their kernels. To obtain these results, we derive a pair of integral equations for the kernels of the transformation operators. These equations are the generalizations of the corresponding classical equation which is valid in the case of a wave equation without damping term. One of possible applications of the method developed in this paper is given in our forthcoming work. In that work, we use the transformation operators to prove the fact that the dynamics generator of a finite string with damping both in the equation and in the boundary conditions is a Riesz spectral operator. The latter result provides a class of nontrivial examples of nonselfadjoint operators which admit an analog of the spectral decomposition. The result also has significant applications in the control theory of distributed parameter systems.
Modelling of piezoelectric static thin plates
Authors: Sene Abdou
Published: December 2024

We study the behavior of a piezoelectric plate when its thickness tends to 0. Our analysis regorously justifies the classical a priori assumption that the electric potential is a second order polynomial with respect to the thickness variable. We prove that the electric potential appears in the limit bidimensional equation only. Moreover, we obtain the very contribution of the piezoelectric constants in the bending operator.
On singular perturbation problems for the nonlinear Poisson equation
Authors: Ambroso A. , Méhats F. , Raviart P.A.
Published: December 2024

We study a singular perturbation problem for the nonlinear Poisson equation and we classify its solutions. This classification is based on their monotonicity properties, which mainly depend on the stationary points of the so‐called Sagdeev potential. For each class of solution, we give necessary and sufficient conditions for the resolution of the problem and provide a boundary layer analysis.
Adiabatic limits of closed orbits for some Newtonian systems in ${\mathbb{R}}$n
Authors: Malchiodi Andrea
Published: December 2024

We deal with a Newtonian system like $\ddot{x}$ +V′(x)=0. We suppose that V : ${\mathbb{R}}$ n→ ${\mathbb{R}}$ possesses an (n−1)‐dimensional compact manifold M of critical points, and we prove the existence of arbitrarily slow periodic orbits. When the period tends to infinity these orbits, rescaled in time, converge to some closed geodesics on M.
Estimates on periodic and Dirichlet eigenvalues for the Zakharov–Shabat system
Authors: Grébert B. , Kappeler T.
Published: December 2024

Consider the 2×2 first order system due to Zakharov–Shabat, $LY:={\rm i}\left(\matrix{1&0\cr 0&-1}\right)Y'+\left(\matrix{0&\psi_{1}\cr \psi _{2}&0}\right)Y=\lambda Y$ with ψ1,ψ2 being complex valued functions of period one in the weighted Sobolev space $H^{w}\equiv H^{w}_{\mathbb{C}}.$ Denote by spec(ψ1,ψ2) the set of periodic eigenvalues of L(ψ1,ψ2) with respect to the interval [0,2] and by specDir(ψ1,ψ2) the set of Dirichlet eigenvalues of L(ψ1,ψ2) when considered on the interval [0,1]. It is well known that spec(ψ1,ψ2) and specDir(ψ1,ψ2) are discrete.

Theorem. Assume that w is a weight such that, for some δ>0, w−δ(k)=(1+|k|)−δw(k) is a weight as well. Then for any bounded subset ${\mathbb{B}}$ of 1‐periodic elements in Hw×Hw there exist N≥1 and M≥1 so that for any |k|≥N, and $(\psi_{1},\psi_{2})\in{\mathbb{B}} $ , the set $\mathit{spec}(\psi_{1},\psi_{2})\cap \{\lambda \in{\mathbb{C}} \mid |\lambda -k\pi | < \pi/2\}$ contains exactly one isolated pair of eigenvalues {λ+k,λ−k} and $\mathit{spec}_{\rm Dir}(\psi_{1},\psi_{2})\cap \{\lambda \in{\mathbb{C}} \mid |\lambda -k\pi |<{\pi}/{2}\}$ contains a single Dirichlet eigenvalue μk. These eigenvalues satisfy the following estimates

(i) Σ|k|≥Nw(2k)2|λ+k−λ−k|2≤M;

(ii) $\sum _{|k|\geq N}w(2k)^{2}|\frac{(\lambda ^{+}_{k}+\lambda ^{-}_{k})} {2}-\mu _{k}|^{2}\leq M.$

Furthermore spec(ψ1,ψ2)\{λ±k,|k|≥N} and $\mathit{spec}_{\rm Dir}(\psi_{1},\psi_{2})\backslash \{\mu _{k}\mid |k|\geq N\}$ are contained in $\{\lambda \in{\mathbb{C}} \mid |\lambda | < N\pi -\pi /2\}$ and its cardinality is 4N−2, respectively 2N−1.

When $\psi_{2}=\overline{\psi } _{1}$ (respectively $\psi_{2}=-\overline{\psi } _{1}),L(\psi_{1},\psi_{2})$ is one of the operators in the Lax pair for the defocusing (resp. focusing) nonlinear Schrödinger equation.
Sensitivity analysis of stress intensity factors in 2D coupled elastic structures
Authors: Bochniak M. , Sändig A.‐M.
Published: December 2024

Material discontinuities and geometrical peculiarities like cracks, notches, corners and edges lead to stress singularities in linear elastic structures. Their strength can be characterised by stress intensity or notch factors. In this paper we investigate the influence of the shape and of the material parameters on these factors. The sensitivity analysis is performed by means of asymptotic expansions.
Direct and inverse three‐point Sturm–Liouville problem with parameter‐dependent boundary conditions
Authors: Pivovarchik Vyacheslav
Published: December 2024

The problem of small vibrations of a smooth inhomogeneous string damped at an interior point and fixed at the endpoints is reduced to a three‐point Sturm–Liouville boundary problem. This problem is considered as an eigenvalue problem for a nonmonic quadratic operator pencil of a special type with the spectrum located in the upper half‐plane of the spectral parameter. Concerning the corresponding inverse problem it is shown that the spectrum does not determine the potential of the Sturm–Liouville problem (and consequently the density of the string) uniquely. The conditions are given sufficient for a sequence of complex numbers to be the spectrum of the considered Sturm–Liouville problem with real‐valued potential which belongs to L2(0,a). In order to recover the potential uniquely the spectrum of the corresponding so to say truncated Sturm–Liouville problem is chosen as an additional information. Then the problem of recovering of the potential turns out to be overdetermined. The self‐consistency of the two spectra is discussed.
Analysis of a Brownian particle moving in a time‐dependent drift field
Authors: Knessl Charles , Yang Yongzhi (Peter)
Published: December 2024

We consider a Brownian particle that is subject to (1) a time‐dependent convection (or drift) field and (2) a reflecting barrier. We let Y(T) be the particle's position at time T. There is a standard reflecting barrier that constrains the particle to the non‐negative real axis (i.e., Y(T)≥0). We assume that Y(T0)=X0≥0 with probability one, and that the drift field is linearly dependent upon time. Specifically, we assume that the drift changes sign at T=0 and becomes positive for T>0. Such models arise naturally in several areas, including convection–diffusion problems in mathematical physics and the study of time dependent queues. We obtain an exact expression for the probability density Q, with Q(X,T) dX=Prob[Y(T)∈(X,X+ dX)|Y(T0)=X0≥0], in terms of Airy functions. We then obtain detailed asymptotic results, that apply for X0 and/or T0→∞, and various ranges of the space–time (X,T) plane. We interpret our results in terms of semi‐classical mechanics.
Regular and singular solutions of a quasilinear equation with weights
Authors: Bidaut‐Véron Marie‐Françoise , García‐Huidobro Marta
Published: December 2024

In this article we study the behavior near 0 of the nonnegative solutions of the equation −div(a(x)|∇u|p−2∇u)=b(x)|u|δ−1u, x∈Ω\{0}, where Ω is a domain of RN containing 0, and δ>p−1>0, a, b are nonnegative weight functions. We give a complete classification of the solutions in the radial case, and punctual estimates in the nonradial one. We also consider the Dirichlet problem in Ω.
Asymptotic expansion of the solution of a mixed Dirichlet–Ventcel problem with a small parameter
Authors: Bourlard Maryse , Maghnouji Abderrahman , Nicaise Serge , Paquet Luc
Published: December 2024

A mixed Dirichlet–Ventcel problem with a small parameter in a polygonal domain of the plane is considered. Since the limit problem is a mixed Dirichlet–Neumann problem, we are in presence of a singular perturbed problem. We give the full asymptotic of the solution of the Dirichlet–Ventcel problem with the help of outer terms and boundary layer terms. Here the main difficulty relies on the singular behaviour of the solutions near the corner points where the boundary conditions change, so that the boundary layer terms are corner layer terms.
Variational limit of a one dimensional discrete and statistically homogeneous system of material points
Authors: Iosifescu Oana , Licht Christian , Michaille Gérard
Published: December 2024

We show that the energy of a discrete system of material points on a line and subjected to random nearest‐neighbour interactions, almost surely converges in a variational sense to a deterministic energy defined on spaces of functions with bounded variation.

Résumé. Nous montrons que l'énergie discrète en dimension 1 d'un système de points matériels soumis à des intéractions aléatoires locales, converge presque surement, en un sens variationnel, vers une énergie déterministe définie dans les espaces BV ou SBV.
Diffusion approximation in heterogeneous media
Authors: Goudon Thierry , Mellet Antoine
Published: December 2024

We investigate the behaviour of the solutions of kinetic equations as a parameter related to both the mean free path and a characteristic length of heterogeneities goes to 0. We obtain in the limit a diffusion equation for the macroscopic density which combines the homogenization effects to the diffusion approximation. In particular the limit equation contains drift terms related to the behaviour of the kernel of the collision operator.
Ondes guidées dans une bande élastique et existence d'une onde de Rayleigh asymptotique
Authors: Bouhennache Tark
Published: December 2024

Nous étudions le comportement asymptotique à haute fréquence des ondes guidées dans une bande élastique isotrope et hétérogène Ω={(x,y); y∈(0,L)} ⊂ R2, avec les conditions de surface libre en y=0 et de Dirichlet en y=L. Nous montrons en particulier l'existence d'une “onde de Rayleigh asymptotique”.

Abstract. We study the asymptotic behavior at high frequency of the guided waves in an isotropic and heterogeneous elastic strip Ω={(x,y); y∈(0,L)} ⊂ R2, with the free surface condition on y=0 and Dirichlet condition on y=L. We prove, in particular, the existence of an “asymptotic Rayleigh wave”.
Asymptotic behaviour of nonlinear problems in periodically perforated domains. The case p=N
Authors: Casado‐Díaz Juan
Published: December 2024

We present an extension of the two‐scale convergence method to the homogenization of monotone Dirichlet problems in periodically perforated domains, with critical size. The case of operators of order p strictly less than the dimension N of the space was considered in a previous paper. The case p=N presents several differences because it is a limit case for the Sobolev's imbedding theorem. Its study is the aim of this paper.
On the diffusion theory of neutron transport on the torus
Authors: Mokhtar‐Kharroubi M. , Thevenot L.
Published: December 2024

We give a spectral approach to the diffusion approximation for general neutron transport equations on the torus with a particular emphasis on the initial layer problem. By Fourier analysis, we diagonalize the transport operator and deal with diffusive limit for each Fourier mode by a Dunford functional calculus.
Time global solutions to a parabolic‐elliptic system modelling chemotaxis
Authors: Senba Takasi , Suzuki Takashi
Published: December 2024

This paper is concerning with asymptotic behavior of infinite time blowup solutions to a parabolic‐elliptic system to chemotaxis. In this paper, we show that the solutions form a delta function singularity at each blowup point in infinite time and that the weight of each delta function singularity is equal to 8π and 4π if the blowup point is in the domain and on the boundary, respectively. We refer to the former and the latter phenomenon as chemotactic collapse and quantization, respectively. Next, we investigate the behavior of some norms of the solutions. Finally, we show that the location of the collapse moves continuously if total mass of the solution is between 4π and 8π.
A unilateral hyperbolic problem with small parameters
Authors: Hajouj Brahim , Madaune‐Tort Monique
Published: December 2024

In this paper some variational inequalities of second order used in the elasto‐plasticity theory are compared with first order inequalities. Convergence properties in Sobolev spaces are given thanks to a priori estimates and a uniqueness result for first order inequalities.
Longterm dynamics of a conserved phase‐field system with memory
Authors: Grasselli Maurizio , Pata Vittorino , Vegni Federico Mario
Published: December 2024

We consider a phase‐field model based on hereditary constitutive equations for the internal energy and the heat flux and on the assumption that the spatial average of the order parameter χ is conserved. This model consists of a parabolic integrodifferential equation for the (relative) temperature ϑ coupled with a nonlinear fourth‐order evolution equation for χ. We first show that the obtained system is indeed a nonautonomous dynamical system, provided that the phase space accounts for the past history of ϑ and appropriate boundary conditions are given. Then we establish the existence of an absorbing set, uniformly with respect to a certain class of heat source terms. Finally, we prove that, under suitable assumptions, our dissipative dynamical system possesses a uniform attractor of finite Hausdorff and fractal dimensions.
A multiscale colloid transport model with anisotropic degenerate diffusion
Authors: Belhadj M. , Gerbeau J.‐F. , Perthame B.
Published: December 2024

We consider a weakly coupled semilinear parabolic‐hyperbolic system with a degenerate and anisotropic diffusion. It arises to model the evolution of a chemical or biological tracer in a porous medium. We study the well‐posedness of the system using a L1 theory. Then, we establish the relaxation limit as the reaction constant becomes large. We prove that the system converges to a nonlinear parabolic‐hyperbolic equation that generalizes the Stefan problem. Two specificities of this paper are (i) to deal with ill‐prepared initial data and (ii) with unique entropy solutions based on a precise entropy inequality.
Asymptotic behavior of a two fluid flow in a thin domain: From Stokes equations to Buckley–Leverett equation and Reynolds law
Authors: Paoli Laetitia
Published: December 2024

We consider the flow of two Newtonian, incompressible and nonmiscible fluids in a 2D thin domain. Starting from the Stokes equations, we derive a generalized Buckley–Leverett equation for the first fluid saturation. We study the asymptotic behavior of the flow when the thickness of the gap tends to zero. Assuming that the fluids interface, which is a free boundary, is described by curves of uniformly bounded variation, we prove that the limit problem obeys a generalized Reynolds law. Moreover, when the two sides of the gap are fixed, the saturation of the limit problem is solution of the classical Buckley–Leverett equation.
Quantitative homogenization of global attractors for reaction–diffusion systems with rapidly oscillating terms
Authors: Fiedler Bernold , Vishik Mark I.
Published: December 2024

For rapidly spatially oscillating nonlinearities f and inhomogeneities g we compare solutions uε of reaction–diffusion systems

∂tuε=aΔuε−f(ε,x,x/ε,u)+g(ε,x,x/ε)

with solutions u0 of the formally homogenized, spatially averaged system

∂tu0=aΔu0−f0(x,u0)+g0(x,u0).

We consider a smooth bounded domain x∈Ω⊆ $\mathbb{R}^{n}$ , n≤3, with Dirichlet boundary conditions. We also impose sufficient regularity and dissipation conditions, such that solutions exist globally in time and, in fact, converge to their compact global attractors 𝒜ε and 𝒜0, respectively, in L2(Ω).

Based on ε‐independent a priori estimates we prove

‖uε(·,t)−u0(·,t)‖L2(Ω)≤Cε eρt,

uniformly for all t≥0 and 0<ε≤ε0. Here the solutions uε and u0 start at the same initial condition u=u0(x)∈H1(Ω) for t=0, and C=C(‖u0‖H1).

Based on an ε‐independent H2‐bound on the global attractors 𝒜ε as well as an exponential attraction rate ν of the homogenized attractor 𝒜0 in L2(Ω), we also prove fractional order upper semicontinuity of the global attractors for $\varepsilon\searrow 0$ ,

distL2(Ω)(𝒜ε,𝒜0)≤Cεγ′

for γ′=(1+ρ/ν)−1. This result requires the homogenized nonlinearity f0(x,w) to be near a potential vector function f1(x,w)=∇wF(x,w) with scalar potential F.

Both quantitative homogenization estimates are proved only for quasiperiodic dependence of f,g on the spatially rapidly oscillating variable x/ε. Moreover, the finitely many frequencies describing this quasiperiodic dependence are assumed to satisfy Diophantine conditions, as are familiar from small divisor problems in Kolmogorov–Arnold–Moser theory. Alternatively, the results hold if f, g admit a sufficiently regular divergence representation which describes their explicit spatial dependence. All results apply to, and are illustrated for, the case of FitzHugh–Nagumo systems with spatially rapidly oscillating quasiperiodic coefficients and inhomogeneities.

For an earlier preprint version of the present paper see [6]. In the companion paper [7], based on analytic semigroup methods, similar results are obtained for the quantitative homogenization of solutions and invariant manifolds. Examples include homogenization of the Navier–Stokes system under periodic boundary conditions and for spatially rapidly oscillating quasiperiodic forces. For a recent extension to strongly continuous semigroups with an application to damped hyperbolic wave equations see [8].
Behaviour of the energy density associated to a Kelvin–Voight model in viscoelasticity
Authors: Petrini Milena
Published: December 2024

We study an initial values problem related to a viscoelasticity model in a periodic medium, with oscillating data at t=0, looking for the homogenization limit of the associated energy density.

This limit is determined in a specific case in which there is a constant proportionality between the elastic coefficient and the one in the viscosity part, where it follows an exponential decay in time of the energy density.
Carleman estimates and distribution of resonances for the transparent obstacle and application to the stabilization
Authors: Bellassoued Mourad
Published: December 2024

The purpose of this paper is to study the resonances associated to the transparent obstacles. We prove the existence of an exponentially small neighborhood of the real axis free from the poles of the resolvent. As an application of these results, we get information about decay‐speed of the local energy. We prove that for regular data, the energy decays at least as fast as the inverse of the logarithm of the time.
Elliptic regularization for the heat equation with nonlinear boundary conditions
Authors: Apreutesei N.C. , Volpert V.A.
Published: December 2024

The paper is devoted to an elliptic perturbation of a parabolic problem. The solution v of the heat equation with a nonlinear boundary condition is compared with the solution vε of an elliptic regularization. An asymptotic expansion of vε is constructed and the order of accuracy for the difference vε−v is obtained. The boundary function method by Vishik and Lyusternik is used.
Non‐convergent radial solution of a semilinear elliptic equation in $\mathbb{R}$N
Authors: Iaia Joseph A.
Published: December 2024

We examine the bounded radial solutions of Δu+f(u)=0 in $\mathbb{R}$ N with N>1. We show for many nonlinearities f that if lim r→∞u(r)= does not exist then f≡0 on some nonempty open interval and N=2.
Dependence of the blow‐up time with respect to parameters and numerical approximations for a parabolic problem
Authors: Groisman Pablo , Rossi Julio D.
Published: December 2024

We find a bound for the modulus of continuity of the blow‐up time for the problem ut=λΔu+up, with initial datum u(x,0)=ϕ(x)+hf(x) respect to the parameters λ, p and h. We also find an estimate for the rate of convergence of the blow‐up times for a semi‐discrete numerical scheme.
On solutions of the wave equation with homogeneous Cauchy data
Authors: Kato Jun , Ozawa Tohru
Published: December 2024

In this article, the behavior of solutions to the free wave equation with homogeneous Cauchy data are considered. In particular, the propagation of singularities are observed explicitly. Such Cauchy data are of special interest in view of applications to self‐similar solutions to nonlinear wave equations.
Inverse resonance scattering on the half line
Authors: Korotyaev Evgeni
Published: December 2024

For the Schrödinger operator on the half line we prove the following results: the mapping from real compactly supported potentials to the associated Jost functions (in some class of entire functions) is one‐to‐one and onto. Moreover, we show that the resonances in any bounded domain in $\mathbb{C}$ ‐ are so‐called free parameters.
Landau–Zener transitions for eigenvalue avoided crossings in the adiabatic and Born–Oppenheimer approximations
Authors: Rousse Vidian
Published: December 2024

In the Born–Oppenheimer approximation context, we study the propagation of Gaussian wave packets through the simplest type of eigenvalue avoided crossings of an electronic Hamiltonian 𝒞4 in the nuclear position variable. It yields a two‐parameter problem: the mass ratio ε4 between electrons and nuclei and the minimum gap δ between the two eigenvalues. We prove that, up to first order, the Landau–Zener formula correctly predicts the transition probability from a level to another when the wave packet propagates through the avoided crossing in the two different regimes: δ being asymptotically either smaller or greater than ε when both go to 0.
Nonlinear diffractive optics with curved phases: beam dispersion and transition between light and shadow
Authors: Dumas E.
Published: December 2024

We give asymptotic descriptions of smooth oscillating solutions of hyperbolic systems with variable coefficients, in the weakly nonlinear diffractive optics regime. The dependence of the coefficients of the system in the space–time variable (corresponding to propagation in a non‐homogeneous medium) implies that the rays are not parallel lines – the same occurs with non‐planar initial phases. Approximations are given by WKB asymptotics with 3‐scales profiles and curved phases. The fastest scale concerns oscillations, while the slowest one describes the modulation of the envelope, which is along rays for the oscillatory components. We consider two kinds of behaviors at the intermediate scale: ‘weakly decaying’ (Sobolev), giving the transverse evolution of a ‘ray packet’, and ‘shock‐type’ profiles describing a region of rapid transition for the amplitude.
Classical solvability and asymptotic behavior of combustion free boundary problem in multidimensional case
Authors: Yi Fahuai , Wang Lijie
Published: December 2024

A free boundary problem for a parabolic system arising from the mathematical theory of combustion in multidimensional case will be considered in this paper. The existence and uniqueness of classical solution locally in time will be obtained by the use of contraction mapping principle. A convergence result for parameter λ→0 is also obtained.
Homogenization of some problems with gradient constraints
Authors: Cardone G. , Corbo Esposito A. , Yosifian G.A. , Zhikov V.V.
Published: December 2024

A homogenization theorem is established for the problem of minimization of a quadratic integral functional on a set of admissible functions whose gradients are subjected to rapidly changing constraints imposed on a disperse periodic set (regarded as inclusions). At each point of the inclusion, the gradients must belong to a given closed convex set of an arbitrary structure which may vary from point to point within the inclusion. Our approach is based on two‐scale convergence and an explicit construction of a Γ‐realizing sequence. This homogenization method can be directly applied to variational problems for vector‐valued functions, which is demonstrated on problems of elasticity with convex constraints on the strain tensor at the points of disperse inclusions. We also consider some problems with constraints on periodic sets of zero Lebesgue measure and study homogenization problems for some cases of nondisperse inclusions.
Gevrey class regularity for the primitive equations in space dimension 2
Authors: Petcu M.
Published: December 2024

The aim of this article is to prove results of space and time regularity of solutions for the Primitive Equations of the ocean in space dimension two with periodic boundary conditions. It is shown that these solutions belong to a certain Gevrey class of functions which is a subset of real analytic functions.
Diffusion in a highly heterogeneous thin domain
Authors: Hoang Viet Ha
Published: December 2024

The paper considers a diffusion equation in a thin domain of thickness ε in $\mathbb{R}^{n}$ whose diffusivity varies periodically with a period cell of size εp (p>0) in the cross section. The domain is highly heterogeneous so that the diffusivity is of the scale εα (α>0) in a part of the period cell and is of order 1 in the rest. The asymptotic behaviour of the solution when ε→0 is studied for all the values of p and α. In some cases, the limiting equation is in $\mathbb{R}^{n-1}$ as in the previous works on diffusion in thin domains but in some others, an equation in $\mathbb{R}^{n}$ is obtained.
The nonlinear sieve problem and applications to thin films
Authors: Ansini Nadia
Published: December 2024

We consider variational problems defined on domains ‘weakly’ connected through a separation hyperplane (‘sieve plane’) by an ε‐periodically distributed ‘contact zone’. We study the asymptotic behaviour as ε tends to 0 of integral functionals in such domains in the nonlinear and vector‐valued case, showing that the asymptotic memory of the sieve is described by a nonlinear ‘capacitary‐type’ formula. In particular we treat the case when the integral energies on both sides of the sieve plane satisfy different growth conditions. We also study the case of thin films, with flat profile and thickness ε, connected by the same sieve plane.
Towards a rigorous derivation of the cubic NLSE in dimension one
Authors: Adami Riccardo , Bardos Claude , Golse François , Teta Alessandro
Published: December 2024

We consider a system of N particles in dimension one, interacting through a zero‐range repulsive potential whose strength is proportional to N−1. We construct the finite and the infinite Schrödinger hierarchies for the reduced density matrices of subsystems with n particles. We show that the solution of the finite hierarchy converges in a suitable sense to a solution of the infinite one. Besides, the infinite hierarchy is solved by a factorized state, built as a tensor product of many identical one‐particle wave functions which fulfil the cubic nonlinear Schrödinger equation. Therefore, choosing a factorized initial datum and assuming propagation of chaos, we provide a derivation for the cubic NLSE. The result, achieved with operator‐analysis techniques, can be considered as a first step towards a rigorous deduction of the Gross–Pitaevskii equation in dimension one. The problem of proving propagation of chaos is left untouched.
Homogenization of a parabolic operator with Signorini boundary conditions in perforated domains
Authors: Beliaev A.
Published: December 2024

The paper deals with homogenization of initial value problems for the standard parabolic equation in a periodically or randomly perforated domain. The boundary of the domain is assumed to be semi‐permeable, and this implies Signorini boundary conditions for the sought variable. Convergence of solutions to the solution of the homogenized problem is proved under the minimal number of assumptions with respect to regularity of initial data and geometry of the boundaries.
L1 and L∞ intermediate asymptotics for scalar conservation laws
Authors: Dolbeault Jean , Escobedo Miguel
Published: December 2024

In this paper, using entropy techniques, we study the rate of convergence of nonnegative solutions of a simple scalar conservation law to their asymptotic states in a weighted L1 norm. After an appropriate rescaling and for a well chosen weight, we obtain an exponential rate of convergence. Written in the original coordinates, this provides intermediate asymptotics estimates in L1, with an algebraic rate. We also prove a uniform convergence result on the support of the solutions, provided the initial data is compactly supported and has an appropriate behaviour on a neighborhood of the lower end of its support.
Convergence of weak solutions for the primitive system of the quasigeostrophic equations
Authors: Charve Frédéric
Published: December 2024

On s'intéresse ici au comportement des solutions faibles de Leray du système primitif lorsque le nombre de Rossby tend vers zéro. On sépare notamment la solution en deux parties chacune solution d'un système particulier, l'une appelée partie oscillante, l'autre partie quasigéostrophique. La première tend vers zéro lorsque la rotation devient forte et de la seconde on peut extraire une sous‐suite tendant vers une solution du système quasigéostrophique.
Infinitely many fine modes bifurcating from radially symmetric internal layers
Authors: Sakamoto Kunimochi
Published: December 2024

For a propagator–controller type system of reaction–diffusion equations on an N‐dimensional ball, it is shown that there exists a family of radially symmetric stationary transition layer solutions when the propagator diffuses slowly compared with the controller. A spectral analysis is also carried out for the linearization around the layered solutions, which reveals that these layer‐solutions are unstable. More importantly, it is found that for activator(propagator)–inhibitor(controller) systems there exist infinitely many static bifurcation points from the radially symmetric solutions as the diffusion rate of the activator goes to zero. This confirms the validity of a conjecture raised in Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), 359–401, in the special situation where the domain is a ball.