bluuebunny/crossref_metadata_2025 huggingface dataset:
Summability in a direction of formal power series in several variables
Authors: Sanz Javier
Published: December 2024

In this paper a theory of summability in a direction, extending that of J.P. Ramis, is developed for formal power series of several variables. To this end, generalized Laplace and Borel transforms are studied, as well as their action on functions admitting Gevrey strongly asymptotic expansion as defined by H. Majima. The definition we give of summability in a direction turns out to be, in a sense, equivalent to an iterative classical summation procedure. As an application we provide a new proof of a well‐known result of R. Gérard and Y. Sibuya stating the convergence of the formal power series solution to a certain completely integrable Pfaffian system.
On initiation of subsonic detonation in porous media combustion
Authors: Gordon P.V. , Kamin S. , Sivashinsky G.I.
Published: December 2024

The long time behavior of the system γΘt−(γ−1)Πt=Ω(Ψ,Θ), Ψt=−Ω(Ψ,Θ), Πt−Θt=Πxx is explored. The model describes gaseous detonation subjected to strong hydraulic resistance, e.g., gaseous explosion in inert porous media. It is shown that successful initiation of a self‐sustaining detonation wave is not always attainable and depends on the initial conditions. The current work is an extension of the previous study of the problem based on a simplified formulation assuming a similarity relation between the temperature and concentration fields.
A method for asymptotic integration of almost diagonal systems
Authors: Elias Uri , Gingold Harry
Published: December 2024

A theorem of asymptotic integration is proven for linear systems of differential equations. The theorem is designed to fit a specialized family of differential systems which occur frequently in quantum mechanics. It is shown to be best possible in a certain sense. The method provided differs from an established trend that transforms the differential system, via a preparation theorem, to a differential system, where the coefficient matrix is the sum of a diagonal matrix and a remainder matrix that must be absolutely integrable at infinity. In this work the fundamental matrix solution is given as a product of a diagonal matrix and a perturbation of the identity matrix. The perturbation of the identity matrix being on the right in the product rather than on the left as is common in the literature.
Spectrum of a two‐dimensional Harper's operator
Authors: Assel Rachid
Published: December 2024

We give a semiclassical study of the spectrum of a two‐dimensional model of Harper's operator. We localize the part of the spectrum near the maximum in exponentially small intervals and give estimate on the splitting between them. Then we prove that in the rational case the Lebesgue measure of the spectrum remains strictly positive even in the semiclassical limit.
Asymptotic analysis of a metallostatic pressure type boundary condition modelled by a fictitious domain method in an aluminium casting
Authors: Barral P. , Quintela P.
Published: December 2024

In this paper we present an asymptotic analysis of the treatment of a metallostatic pressure boundary condition arising from the thermomechanical modelling of aluminium castings by means of a fictitious domain method depending on a small real parameter. We introduce an asymptotic expansion for the solution of the extended problem and we prove that the leading term of such expansion is the solution of the initial problem.
Effect of symmetry to the structure of positive solutions in nonlinear elliptic problems, III
Authors: Byeon Jaeyoung
Published: December 2024

We consider the problem: Δu+up=0 in ΩR,u=0 on ∂ΩR,u>0 in ΩR, where ΩR≡{x∈RN|R−1<|x|<R+1}, N≥3, and 1
0.
Homogenization of a soft elastic material reinforced by fibers
Authors: Bouchitté Guy , Bellieud Michel
Published: December 2024

We study the effective properties of an elastic composite medium under the assumption of small deformations. This composite is made of a periodic possibly disconnected subset filled up with a strong material surrounded by another material whose elastic coefficients are very small. The effective macroscopic behaviour obtained by homogenization turns out to be nonlocal and depends highly on the geometry of the strong component. In particular, second order derivative of the displacement appear in the limit energy when disconnected fibers are considered. These results were announced in [8].
Convergence of bounded solutions to gradient‐like semilinear Cauchy problems with radial nonlinearity
Authors: Chill Ralph
Published: December 2024

We prove convergence to a steady state of bounded solutions of the abstract first order semilinear Cauchy problem ut+Lu+g(Ψ(u))Cu=0, t∈R+, and of the second order semilinear Cauchy problem utt+αut+Lu+g(Ψ(u))Cu=0, t∈R+.

We apply the abstract results to semilinear parabolic and hyperbolic partial differential equations including the heat equation, the wave equation, a Kuramoto–Sivashinsky model and the Kirchhoff–Carrier equation.
Density of finite gap potentials for the Zakharov–Shabat system
Authors: Grébert B. , Kappeler T.
Published: December 2024

For various spaces of potentials we prove that the set of finite gap potentials of the Zakharov–Shabat system is dense. In particular our result holds for Sobolev spaces and for spaces of analytic potentials of a given type.
Wigner functions versus WKB‐methods in multivalued geometrical optics
Authors: Sparber Christof , Markowich Peter A. , Mauser Norbert J.
Published: December 2024

We consider the Cauchy problem for a class of scalar linear dispersive equations with rapidly oscillating initial data. The problem of the high‐frequency asymptotics of such models is reviewed, in particular we highlight the difficulties in crossing caustics when using (time‐dependent) WKB‐methods. Using Wigner measures we present an alternative approach to such asymptotic problems. We first discuss the connection of the naive WKB solutions to transport equations of Liouville type (with mono‐kinetic solutions) in the pre‐breaking regime. Further we show how the Wigner measure approach can be used to analyze high‐frequency limits in the post‐breaking regime, in comparison with the traditional Fourier integral operator method. Finally we present some illustrating examples.
Asymptotic methods for the eigenvalues of the Rayleigh equation for the linearized Rayleigh–Taylor instability
Authors: Helffer Bernard , Lafitte Olivier
Published: December 2024

In this paper, we study the spectrum of the Rayleigh equation, which models the Rayleigh–Taylor instability in a fluid of variable density ρ(x), where ρ(x) goes to ρ− at −∞ and ρ+ at +∞: [Formula: see text] The behavior of the smallest value δ(h) of δ>0 for which there exists a nontrivial solution u is investigated in terms of h for the two regimes h→0 (called the semi‐classical regime) and h→+∞. The so‐called growth rate of the Rayleigh–Taylor instability is $\sqrt{h/\delta(h)}$ .

When ρ−ρ−∈L2( $\mathbb{R} $ −) and ρ−ρ+∈L2( $\mathbb{R} $ +), we prove that δ(h)/h→(ρ++ρ−)/(ρ−−ρ+), as h→+∞, generalizing the result of Lord Rayleigh [23].

We then investigate the expansion of δ(h)/h in terms of 1/h and properties of ρ. In particular, we identify the number of terms n of the expansion in function of the behavior of ρ(x)−ρ± at ±∞, extending the results of Cherfils, Lafitte and Raviart [6].
Rigorous justification of the Kirchhoff law for junction of thin pipes filled with viscous fluid
Authors: Marušić‐Paloka Eduard
Published: December 2024

We study the junction of m pipes that are either thin or long (i.e., they have small ratio between the cross‐section and the length, denoted by ε). Pipes are filled with incompressible Newtonian fluid and the values of the pressure pi at end of each pipe are prescribed. By rigorous asymptotic analysis, as ε→0, we justify the analog of the Kirchhoff law for computing the junction pressure. In interior of each pipe the effective flow is the Poiseuille flow governed by the pressure drop between the end of the pipe and the junction point. The pressure at the junction point is equal to a weighted mean value of the prescribed pi‐s (Kirchhoff law). In the vicinity of the junction there is an interior layer, with thickness εlog (1/ε). To get a better approximation and to control the velocity gradient in vicinity of the junction, first order asymptotic approximation has to be corrected by solving an appropriate Leray problem. We prove the asymptotic error estimate for the approximation.
The concentration‐compactness principle and the asymptotic behavior for some degenerate parabolic equation
Authors: Tan Zhong
Published: December 2024

In this paper we study the existence and asymptotic behavior of the global solutions of some degenerate parabolic equation with critical Sobolev exponent. In particular, we apply the concentration‐compactness principle of P.‐L. Lions to the study of the asymptotic behavior of global solutions with the initial value in “stable set”.
Approximation of a quasilinear Schrödinger equation by a Klein–Gordon equation in space dimension 2
Authors: Colin Mathieu
Published: December 2024

In this article, we study the nonlinear plasma wave equation [Formula: see text] with initial data $u_{\varepsilon}(\cdot,0)=u_{0}^{\varepsilon}(\cdot)\in H^{8}(\mathbb{R}^{2}),\ \curpartial_{t}u_{\varepsilon}(\cdot,0)=u_{1}^{\varepsilon}(\cdot)\in H^{7}(\mathbb{R}^{2})$ . We show that the Cauchy problem is locally well‐posed on an interval [0,T] where the time T is independent of ε if u1ε is small enough. Then, we demonstrate the strong convergence of uε towards the solution u of a nonlinear relativistic Schrödinger equation as ε goes to 0.
On the double porosity model of a single phase flow in random media
Authors: Bourgeat A. , Mikelić A. , Piatnitski A.
Published: December 2024

We consider the linearized equations of slightly compressible single fluid flow through a highly heterogeneous random porous medium, consisting of two types of material. Due to the high heterogeneity of the two materials the ratio of their permeability coefficients is of order ε2, where ε is the characteristic scale of heterogeneities. Supposing that the matrix blocks set of the porous medium consists of random stationary inclusions, and assuming positive definiteness of the effective permeability tensor associated to the corresponding Neumann problem for the random fractures system, we obtain the homogenized problem for a random version of the double porosity model used in geohydrology. It includes as a particular case the periodic setting, already studied by homogenization theory methods (see, for example, [1,7]). The homogenized problem is obtained by using the stochastic two scale convergence in the mean, and by means of convergence results specially adapted to our a priori estimates and to the random geometry, which do not require extension of solutions to the matrix part.
Rigorous Navier–Stokes limit of the lattice Boltzmann equation
Authors: Junk Michael , Yong Wen‐An
Published: December 2024

In this article, we rigorously investigate the diffusive limit of a velocity‐discrete Boltzmann equation which is used in the lattice Boltzmann method (LBM) to construct approximate solutions of the incompressible Navier–Stokes equation. Our results apply to LBM collision operators with multiple collision frequencies (generalized lattice Boltzmann) which include the widely used BGK operators.
Singular perturbations for certain partial differential equations without boundary‐layers
Authors: Akhmetov Denis R. , Lavrentiev, Jr. Mikhail M. , Spigler Renato
Published: December 2024

Linear parabolic partial differential equations with a small parameter multiplying some of the higher space derivatives are considered, in the limiting case when such parameter vanishes. A number of different boundary‐value problems for singularly perturbed equations are examined. Such problems are unified by the rather unexpected property that no boundary‐layers are required, despite of the presence of the small vanishing parameter. The high points are the following. First, solvability theorems for some classes of ultraparabolic problems have been established. Second, the boundary conditions to be imposed to obtain well‐posed problems do not depend on the sign of the coefficient multiplying the time‐like derivative. Third, all coefficients are allowed to depend on all space and time variables. These results, in part, have been established by imposing suitably generalized compatibility conditions on coefficients and data.
On some nonlinear equations involving the 1‐Laplacian with critical Sobolev growth and perturbation terms
Authors: Wigniolle Jérôme
Published: December 2024

Let Ω be a smooth bounded domain in RN, N≥2; let a, f, h be smooth functions on $\overline{\varOmega }$ , f being positive on $\overline{\varOmega }$ and a satisfying the following condition:

∫Ω|∇u|+∫Ωa|u|≥C∫Ω|u|, ∀u∈W1,10(Ω), where C is some positive constant.

We look for some u∈BV(Ω), u not identically 0, which satisfies:

[Formula: see text] where 1*=N/(N−1) denotes the critical Sobolev exponent for the embedding of W1,1(Ω) and BV(Ω) into Lk(Ω), q is a real in ]1,1*[ and sign(u) is some L∞ function such that sign(u) u=|u|.
An example in the homogenization of a degenerate elliptic equation
Authors: Nandakumaran A.K. , Rajesh M. , Mallikarjuna Rao K.S.
Published: December 2024

In this article we study the homogenization, of a particular example, of degenerate elliptic equations of second order in the setup of viscosity solutions. These results are an attempt to extend the corresponding results of Evans [8] to degenerate situations.
On the limit of some penalized problems involving increasing powers
Authors: Benkirane A. , Elmahi A. , Meskine D.
Published: December 2024

We prove an existence and uniqueness result for solutions of some bilateral problems with right‐hand side in L1 by using an approach involving increasing powers.
A parabolic singular perturbation problem with an internal layer
Authors: Grasman Johan , Shih Shagi‐Di
Published: December 2024

A method is presented to approximate with singular perturbation methods a parabolic differential equation for the quarter plane with a discontinuity at the corner. This discontinuity gives rise to an internal layer. It is necessary to match the local solution in this layer with the one in a corner layer as otherwise terms in the internal layer solution remain unnoticed. The problem is explained using the exact solution of a special case. The asymptotic solution is proved to approximate the exact solution in the general case using the maximum principle for parabolic differential equations.
The Boltzmann equation as limit of a Schrödinger equation with random potential
Authors: Brassart Matthieu
Published: December 2024

We study the transport of particles in time‐dependent random media in the so‐called weak coupling limit. We show how a limiting irreversible dynamic of Boltzmann's type can be derived from a reversible dynamic of Schrödinger's type. The homogenization is performed at the level of Wigner transforms for a mixed states formulation. We obtain a simple modelization of the influence of microscopic chaos on physical observables by means of a space–time Fourier transform of a covariance associated with the random variations.
Semiclassical states for a class of nonlinear elliptic field equations
Authors: D'Aprile Teresa
Published: December 2024

In this paper we are concerned with the problem of finding solutions for the following nonlinear field equation

−ħ2Δv+V(x)v−ħpΔpv+W′(v)=0,

where v : $\mathbb{R} $ N→ $\mathbb{R} $ N+1, N≥3, p>N, ħ>0, the potential V is positive and W is an appropriate singular function. Assuming that V has a mountain pass geometry, determined by the existence of two strict local minima, then for small values of the parameter ħ we are able to find a solution which concentrates at the related mountain pass point of V in the semiclassical limit (i.e., as ħ→0+). The proof of our results uses a variational approach and the solutions are captured as critical points for the associated energy functional.
Nonlocal front propagation problems in bounded domains with Neumann‐type boundary conditions and applications
Authors: Da Lio Francesca , Inwon Kim Christina , Slepčev Dejan
Published: December 2024

This paper is concerned with the asymptotic behavior as ε→0 of the solutions of nonlocal reaction–diffusion equations of the form ut−Δu+ε−2f(u,ε∫0u)=0 in O×(0,T) associated with nonlinear oblique derivative boundary conditions. We show that such behavior is described in terms of an interface evolving with normal velocity depending not only on its curvature but also on the measure of the set it encloses. To this purpose we introduce a weak notion of motion of hypersurfaces with nonlocal normal velocities depending on the volume they enclose, which extends the geometric definition of generalized motion of hypersurfaces in bounded domains introduced by G. Barles and the first author to solve a similar problem with local normal velocities depending on the normal direction and the curvature of the front. We also establish comparison and existence theorems of viscosity solutions to initial‐boundary value problems for some singular degenerate nonlocal parabolic pde's with nonlinear Neumann‐type boundary conditions.
Study of the double porosity model versus the fissures thickness
Authors: Bourgeat A. , Pankratov L. , Panfilov M.
Published: December 2024

We study by homogenization the various macroscopic models associated to the diffusion through a fissured media, i.e., a set of low conductivity blocks crossed by a net of highly conducting fissures. According to the fissure thickness, δε, range we obtain different models. For this, first we homogenize by seeking the limit when ε, the small parameter associated to the blocks size, goes to zero and then we study the limit when δ goes to zero. In each situation we prove the convergence to the corresponding macroscopic model.
Uniform rates of decay of solutions for a nonlinear lattice with memory
Authors: Perla Menzala G. , Muñoz Rivera Jaime E. , Konotop V.V.
Published: December 2024

In this paper we study a nonlinear lattice with memory and show that the problem is globally well posed. Furthermore we find uniform rates of decay of the total energy. Our main result shows that the memory effect is strong enough to produce a uniform rate of decay. That is, if the relaxation function decays exponentially then, the corresponding solution also decays exponentially. When the relaxation kernel decays polynomially then, the solution also decays polynomially as time goes to infinity.
Resolvent estimates and applications to wave equations with potential
Authors: Zappacosta Stefano
Published: December 2024

We improve estimates for power resolvent of the free Laplacian −Δ on $\mathbb{R}^{n}$ , and we use them to get a control on the solution to the wave equation with a potential (∂2t−Δ)u+Vu=0 in $\mathbb{R}^{3}$ . We finally obtain a local energy decay for u.
Ventcel's boundary conditions for a dynamic nonlinear plate
Authors: Rahmani Leila
Published: December 2024

In this paper, we consider the full system of dynamic von Karman equations for an heterogeneous plate that comprises two parts: a thin rigid body inserted into an elastic plate. We show that Ventcel's boundary conditions may be obtained, as the thickness of the rigid body goes to zero.
Some analytical results on the Gray–Scott model
Authors: Dkhil F. , Logak E. , Nishiura Y.
Published: December 2024

We consider the Gray–Scott model for cubic autocatalysis. We prove nonexistence results on stationary and travelling pulse solutions for some domain of parameters. We obtain an explicit travelling front in the one dimensional case for equal diffusivities and equal transformation rates of the reactants. We prove that travelling fronts continue to exist near the equal diffusivities and equal transformation rates case. We also provide numerical simulations illustrating our results.
On the Sturm–Liouville problem for the complex cubic oscillator
Authors: Trinh Duc Tai
Published: December 2024

In this paper, solutions of the Sturm–Liouville problem associated with the one‐parameter family of non‐Hermitian Hamiltonians Hα=p2+i(q3+αq) are investigated. We also present some initial results on the distribution of zeros of eigenfunctions. This distribution seems to exhibit a complex version of the classical results on real oscillators.
On the fundamental state energy for a Schrödinger operator with magnetic field in domains with corners
Authors: Bonnaillie Virginie
Published: December 2024

The superconducting properties of a sample submitted to an external magnetic field are mathematically described by the minimizers of the Ginzburg–Landau's functional. The analysis of the Hessian of the functional leads to estimate the fundamental state for the Schrödinger operator with intense magnetic field for which the superconductivity appears. So we are interested in the asymptotic behavior of the energy for the Schrödinger operator with a magnetic field. A lot of papers have been devoted to this problem, we can quote the works of Bernoff–Sternberg, Lu–Pan, Helffer–Mohamed. These papers deal with estimates of the energy in a regular domain and our goal is to establish similar results in a domain with corners. Although this problem is often mentioned in the physical literature, there are very few mathematical papers. We only know the contributions by Pan and Jadallah which deal with very particular domains like a square or a quarter plane. The physicists Brosens, Devreese, Fomin, Moshchalkov, Schweigert and Peeters propose a non optimal upper bound for the energy. Here, we present a more rigorous analysis and give an asymptotics of the smallest eigenvalue of the operator in a sector Ωα of angle α when α is closed to 0, an estimate for the eigenfunctions and we use these results to study the fundamental state in the semi‐classical case.
Smoothness characterization and stability in nonlinear multiscale framework: theoretical results
Authors: Matei Basarab
Published: December 2024

The aim of this paper is to characterize the smoothness of a function belonging to a Hölder or to a Besov space, by the rate of decay of its nonlinear multiscale coefficients. The study is carried out in the general framework of nonlinear multiscale representations of data due to A. Harten. We also study the stability of these representations, a key issue in their applications to design adaptive algorithms.
Penalty approximation and analytical characterization of the problem of super‐replication under portfolio constraints
Authors: Bensoussan Alain , Touzi Nizar , Menaldi José Luis
Published: December 2024

In this paper, we consider the problem of super‐replication under portfolio constraints in a Markov framework. More specifically, we assume that the portfolio is restricted to lie in a convex subset, and we show that the super‐replication value is the smallest function which lies above the Black–Scholes price function and which is stable for the so‐called face lifting operator. A natural approach to this problem is the penalty approximation, which not only provides a constructive smooth approximation, but also a way to proceed analytically.
Homogenization of an elastic medium reinforced by anisotropic fibers
Authors: Sili Ali
Published: December 2024

We study the homogenization of an elastic medium periodically reinforced by thin anisotropic elastic fibers, the radius of the fibers being of the order of the size ε of the period, while 1/ε2 is the ratio between the coefficients of the tensor inside the fibers with those of the tensor outside them. In this case, the sequence uε of the displacements is not apriori bounded in (L2(Ω))3. We prove that the homogenized problem is a strongly coupled nonlocal problem which involves variables due to the anisotropy of the fibers.
On the stability of solutions of some non‐autonomous delay differential equations of the third order
Authors: Sadek A.I.
Published: December 2024

The paper gives sufficient conditions for the asymptotic stability of the zero solution of system (1.1). By constructing Lyapunov functionals, we obtained the results in a simple form.
A higher‐order energy expansion to two‐dimensional singularly perturbed Neumann problems
Authors: Wei Juncheng , Winter Matthias , Yeung Wai‐Kong
Published: December 2024

Of concern is the following singularly perturbed semilinear elliptic problem [Formula: see text] where Ω is a bounded domain in RN with smooth boundary $\[$\curpartial \varOmega $$ , ε>0 is a small constant and $\[$1
0 is a generic constant, Pε is the unique local maximum point of uε and H(Pε) is the boundary mean curvature function at $\[$P_{\varepsilon }\in \curpartial \varOmega $$ . Later, Wei and Winter [C. R. Acad. Sci. Paris, Ser. I 337 (2003), 37–42; Calc. Var. Partial Differential Equations 20 (2004), 403–430] improved the result and obtained a higher‐order expansion of Jε[uε]: [Formula: see text] where c2 and c3>0 are generic constants and R(Pε) is the scalar curvature at Pε. However, if N=2, the scalar curvature is always zero. The expansion (2) is no longer sufficient to distinguish spike locations with same mean curvature. In this paper, we consider this case and assume that 2≤p<+∞. Without loss of generality, we may assume that the boundary near $\[$P\in \curpartial \varOmega $$ is represented by the graph {x2=ρP(x1)}. Then we have the following higher order expansion of Jε[uε]: [Formula: see text] where H(Pε)=ρ″Pε(0) is the curvature, P(t)=A1t+A2t2+A3t3 is a polynomial, c1, c2, c3 and A1, A2, A3 are generic real constants and S(Pε)=ρPε(4)(0). In particular c3<0. Some applications of this expansion are given.
WKB asymptotics for the Euler–Maxwell equations
Authors: Texier Benjamin
Published: December 2024

The Euler–Maxwell system of equations is a complex, hydrodynamical model for the description of laser–plasma interactions. We introduce non‐dimensional variables, exhibit a small parameter ε and study various WKB approximations with respect to ε for this system, under a polarization condition for the initial data. We justify an approximation by a weak Zakharov equation for times O(1) and an approximation by a Davey–Stewartson equation for times O(|log ε|). Our key observation is that the Euler–Maxwell exhibit transparency properties, similar to the properties exhibited by Joly, Métivier and Rauch for Maxwell–Bloch systems. These properties imply in particular that in a weakly nonlinear regime, the geometric optics approximation is given by a linear equation.
On quasimodes of small diffusion operators corresponding to stable invariant tori with nonregular neighborhoods
Authors: Albeverio Sergio , Dobrokhotov Sergey , Poteryakhin Michael
Published: December 2024

Smooth vector fields V in $\[$\mathbb{R}$$ n having a k-dimensional invariant torus are considered and the possibility of constructing the asymptotic eigenfunctions localized in the neighborhood of this torus is analyzed for the case of a corresponding small diffusion operator V·∇−εΔ, ε>0. The asymptotic stability property of the invariant torus is a sufficient condition for the existence of such eigenfunctions. Both the regular case (where the variational system is reducible) and the nonregular case (where the variational system is nonreducible) are considered.
Homogenisation of exponential order for elliptic systems in infinite cylinders
Authors: Matthies Karsten
Published: December 2024

We consider systems of semilinear elliptic equations on infinite cylinders with a nonlinear rapid periodic inhomogeneity in the unbounded direction. We transform the equation, such that the inhomogeneous term is exponentially small in the period of the inhomogeneity for bounded solutions. The results can be used to show that equilibrium solutions persist as periodic solutions with exponentially small modulation. The analytic tools of the paper include the dynamical systems approach to elliptic equations, averaging of exponential order for ordinary differential equations and extreme regularity (Gevrey classes).
Stark–Wannier type operators with purely singular spectrum
Authors: Perelman Galina
Published: December 2024

We consider the one-dimensional Stark–Wannier type operators [Formula: see text] where q is a smooth function slowly growing at infinity and v is periodic, $\[$v\in L_{1}(\mathbb{T})$$ , with the Fourier coefficients of the form (ln |n|)−β, 0<β<1/2, as n→∞. We show that for suitable q and F the spectrum of the corresponding operator is purely singular continuous. This proves the sharpness of the a.c. spectrum stability result obtained in Comm. Math. Phys. 234 (2003), 359–381.
Reduced equations for models of laminated materials in thin domains. I
Authors: DeVille R.E. Lee , Wayne C. Eugene
Published: December 2024

In this paper, we consider a scalar wave equation on a thin, laminated, three‐dimensional plate. We show that if the plate is sufficiently thin, then there is a hierarchy of two‐dimensional equations whose dynamics model the dynamics of the full plate, each of which successively lengthens the time interval over which the approximation holds.
Regularity results via duality for minimizers of degenerate functionals
Authors: Carozza Menita , Moscariello Gioconda , Passarelli di Napoli Antonia
Published: December 2024

We prove a regularity result for local minimizers of degenerate variational integrals, whose model arises in the study of mappings with finite distortion. The degeneracy function 𝒦(x) lies in the exponential class, i.e., exp (λ𝒦(x)) is integrable for some λ>0. The right space of the gradient of a local minimizer u turns out to be the Zygmund class Lp log −1L. Our result states that if λ is sufficiently large, then Du belongs to the Zygmund space Lp log αL, α≥1 and α increases with λ.
On a wide class of nonlinear models for non-Newtonian fluids with mixed boundary conditions in thin domains
Authors: Sac-Épée J.-M. , Taous K.
Published: December 2024

The behaviour of Newtonian and non-Newtonian flows through a thin three-dimensional domain are widely studied in the literature. Usually, authors deal with special models related to particular concrete fluids. In this work, our aim is to present a general model, governing the behaviour of a large class of Newtonian and non-Newtonian fluids. Moreover, we deal with mixed boundary conditions, which are not often studied in the literature related to flows in thin domains. We consider a nonlinear model of a flow in a thin three-dimensional domain, and we study its behaviour when the thickness in one direction tends to zero. At the limit, we obtain a quasilinear two-dimensional problem for the pressure, a nonlinear Reynolds's law for the velocity and a nonlinear Darcy's law for the averaged velocity. Finally, we check that our results hold for a large class of non-Newtonian fluids by producing concrete examples.
An interface evolution problem for axisymmetric stressed pore channels
Authors: Boutat M. , D'Angelo Y. , Hilout S. , Lods V.
Published: December 2024

The aim of this paper is the mathematical study of the time evolution of a stressed pore channel in an axisymmetric configuration. Under some conditions, morphological instabilities may appear at the material–vacuum interface. Assuming some formal asymptotic assumptions, we derive a nonlinear parabolic PDE (19) governing the cylindrical surface evolution. Local existence and unity of the solution of this PDE are shown and we also perform some numerical computations (with different parameters and initial condition), using a pseudo-spectral Galerkin method, yielding different behaviours for the solution to (19). In particular, we numerically observe what appears to be a finite time pinch-off.
Diffusion approximation and entropy-based moment closure for kinetic equations
Authors: Coulombel Jean-François , Golse François , Goudon Thierry
Published: December 2024

It is a well-known fact that, in small mean free path regimes, kinetic equations can lead to diffusion equations. Besides, kinetic equations can be approached by a closed system of moments equations. In this paper, we are interested in a special closure based on an entropy minimization principle, as introduced earlier by Levermore. We investigate the behavior of the resulting nonlinear hyperbolic system in the diffusive scaling. We first establish various fundamental facts on this system, then we show that the hyperbolic system admits global smooth solutions, and is consistent with the diffusion limit. Similar features are also discussed for a simpler limited flux equation.
A piezoelectric anisotropic plate model
Authors: Figueiredo Isabel M. Narra , Leal Carlos M. Franco
Published: December 2024

We mathematically justify a reduced piezoelectric plate model. This is achieved considering the three-dimensional static equations of piezoelectricity, for a nonhomogeneous anisotropic thin plate, and using the asymptotic analysis to compute the limit of the displacement vector and electric potential, as the thickness of the plate approaches zero. We prove that the three-dimensional displacement vector converges to a Kirchhoff–Love displacement, that solves a two-dimensional piezoelectric plate model, defined on the middle surface of the plate. Moreover, the three-dimensional electric potential converges to a scalar function that is a second-order polynomial with respect to the thickness variable, with coefficients that depend on the transverse component of the Kirchhoff–Love displacement. We remark that the results of this paper generalize a previous work of A. Sene (Asymptotic Anal. 25(1) (2001), 1–20) for homogeneous and isotropic materials.
On the coexistence of the strings and antistrings of the nontopological type
Authors: Chae Dongho
Published: December 2024

We study a semilinear elliptic system modeling the physical system strings and antistrings in cosmology under the boundary condition of the symmetric vacuum (the nontopological type). We construct solutions with the representation having precise informations on the asymptotic behaviors near infinity for arbitrary location of strings and antistrings satisfying $\[$1\leqslant M-N<\tfrac{1}{4\pi G}$$ , where M and N are the total string number and the total antistring number respectively, and G is the gravitational constant. The asymptotic properties, in particular, are completely different to the solutions under the boundary condition of the asymmetric vacuum (the topological type) constructed previously by Y. Yang [Phys. Rev. Lett. 80 (1999), 26–29]. We also compute the total magnetic flux, total energy and the total Gaussian curvature of our solutions.
Blowing up solutions for a biharmonic equation with critical nonlinearity
Authors: El Mehdi Khalil , Hammami Mokhless
Published: December 2024

In this paper we consider the following biharmonic equation with critical exponent $\[$(P_{\varepsilon})\dvt\Delta^{2}u=Ku^{\frac{n+4}{n-4}-\varepsilon}$$ , u>0 in Ω and u=Δu=0 on $\[$\curpartial \varOmega $$ , where Ω is a smooth bounded domain in $\[$\mathbb{R}^{n}$$ , n≥5, ε is a small positive parameter, and K is a smooth positive function in $\[$\overline{ \varOmega }$$ . We construct solutions of (Pε) which blow up and concentrate at strict local maximum of K either at the boundary or in the interior of Ω. We also construct solutions of (Pε) concentrating at an interior strict local minimum point of K. Finally, we prove a nonexistence result for the corresponding supercritical problem which is in sharp contrast to what happened for (Pε).
Application of the averaging method to the gyrokinetic plasma
Authors: Frénod E.
Published: December 2024

We show that the solution to an oscillatory–singularly perturbed ordinary differential equation may be asymptotically expanded into a sum of oscillating terms. Each of those terms writes as an oscillating operator acting on the solution to a nonoscillating ordinary differential equation with an oscillating correction added to it. The expression of the nonoscillating ordinary differential equations are defined by a recurrence relation. We then apply this result to problems where charged particles are submitted to large magnetic field.
Lp′–Lp decay estimates of solutions to the wave equation with a short-range potential
Authors: Cuevas Claudio , Vodev Georgi
Published: December 2024

We obtain large time decay of the Lp norms, 2<p<+∞, of solutions to the wave equation with a real-valued potential which decays slowly at infinity. This extends previous results by Beals and Strauss [Comm. Partial Differential Equations 18 (1993), 1365–1397] and Georgiev and Visciglia [Comm. Partial Differential Equations 28 (2003), 1325–1369].
Spectral asymptotics stemming from parabolic equations under dynamical boundary conditions
Authors: François Gilles
Published: December 2024

This paper deals with a spectral problem for a second-order elliptic operator A stemming from a parabolic problem under a dynamical boundary condition. The discrete character and the convergence to infinity of the eigenvalue sequence of the problem [Formula: see text] are shown. By means of min–max formulae, a comparison of the eigenvalue sequence with the spectra of the Dirichlet and Neumann problem is obtained and yields an upper bound for λk. On the other hand, comparing with the Steklov problem leads to a lower bound. In the two-dimensional case, this yields the exact growth order of the eigenvalue sequence and leads to inequalities about the constant of the leading term in its asymptotic behavior. For higher dimensions, the same arguments hold for a sufficiently smooth domain, while for a Lipschitz boundary, lower and upper bounds for the growth exponent are obtained. Finally, the variational characterization of the eigenvalues yields an upper bound for the number of nodal domains of the k-th eigenfunction. The one-dimensional case is also discussed.
On internal and boundary layers with unbounded energy in thin shell theory. Parabolic characteristic and non-characteristic cases
Authors: Caillerie D. , Raoult A. , Sanchez-Palencia E.
Published: December 2024

We consider the system of equations of Koiter shell theory in a slightly simplified form, in the case when the limit (for small thickness) problem is parabolic, i.e., the directions of the principal curvatures of the middle surface coincide everywhere. Under loadings not belonging to the dual of the energy space of the limit problem, the energy of the solutions grows without limit as the thickness tends to zero and concentrates on internal or boundary layers. Two cases are considered, when the singular loadings are applied either along a non-characteristic or a characteristic curve. In both cases we define and prove the convergence to the leading order in the layers.
Some remarks on stability of semigroups arising from linear viscoelasticity
Authors: Chepyzhov V.V. , Pata V.
Published: December 2024

An abstract integrodifferential equation arising from linear viscoelasticity is considered. The stability properties of the related C0-semigroup are discussed, in dependence on the form of the convolution (memory) kernel.
Strichartz estimates for the Schrödinger and heat equations perturbed with singular and time dependent potentials
Authors: Pierfelice Vittoria
Published: December 2024

We prove Strichartz estimates for the Schrödinger equation perturbed with a small time dependent potential V(t,x) belonging to the weak Lebesgue space L∞tL(n/2,∞)x. We also consider the heat equation perturbed with a time independent singular potential V(x)∈L(n/2,∞) and we prove both the maximum principle and the Strichartz estimates under a suitable smallness condition on the negative part of the potential.
Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation
Authors: Hérau Frédéric
Published: December 2024

We consider an inhomogeneous linear Boltzmann equation, with an external confining potential. The collision operator is a simple relaxation toward a local Maxwellian, therefore without diffusion. We prove the exponential time decay toward the global Maxwellian, with an explicit rate of decay. The methods are based on hypoelliptic methods transposed here to get spectral information. They were inspired by former works on the Fokker–Planck equation and the main feature of this work is that they are relevant although the equation itself has no regularizing properties.
Multisummability of formal power series solutions of nonlinear partial differential equations in complex domains
Authors: Ōuchi Sunao
Published: December 2024

Let $\hat{u}(t,x)=\sum_{n=1}^{\infty}u_{n}(x)t^{n},\ (t,x)\in \mathbb{C}\times \mathbb{C}^{d}$ , be a formal power series solution of a nonlinear partial differential equation. We study multisummability of $\hat{u}(t,x)$ . This paper is a continuation of [S. Ōuchi, Multisummability of formal solutions of some linear partial differential equations, J. Differential Equations 185 (2002), 513–549], where linear partial differential equations were treated.
Weighted dispersion and Strichartz estimates for the Liouville equation in 1D
Authors: Salort Delphine
Published: December 2024

We consider in dimension 1 the Liouville equation associated to a nonnegative metric g∈𝒞2b( $\mathbb {R}$ ). We prove global in time dispersion and Strichartz estimates with a weight dependent on g. This weighted norm appears because of a phenomenon of concentration of trajectories when g is close to 0.
Modelization of a split in a ferromagnetic body by an equivalent boundary condition: Part I
Authors: Santugini-Repiquet Kévin
Published: December 2024

We study the influence of a thin split over the dynamic evolution of a ferromagnetic body. A naive numerical simulation would require a huge number of cells to model the split. To avoid this problem, we introduce an equivalent boundary condition which is obtained by a Taylor expansion in the thickness of the split. We prove the existence of solutions to this new problem and then rigorously establish the convergence of the expansion.
The asymptotic behavior of eigenvalues of a modified Jaynes–Cummings model
Authors: Boutet de Monvel Anne , Naboko Serguei , Silva Luis O.
Published: December 2024

We obtain the asymptotic behavior of eigenvalues of Jacobi matrices corresponding to a modified Jaynes–Cummings model with additive and multiplicative modulations using the so-called successive diagonalization method. By comparing the cases with and without modulated entries, we show the influence of periodic modulations on the eigenvalue asymptotics.
On regions of existence and nonexistence of solutions for a system of p–q-Laplacians
Authors: Clément Philippe , García-Huidobro Marta , Guerra Ignacio , Manásevich Raúl
Published: December 2024

We give a new region of existence of solutions to the superhomogeneous Dirichlet problem [Formula: see text] where B is the ball of radius R>0 centered at the origin in $\mathbb {R}^{N}$ . Here δ,μ>0 and Δmu=div(|∇u|m−2∇u) is the m-Laplacian operator for m>1.
Scale invariant energy smoothing estimates for the Schrödinger equation with small magnetic potential
Authors: Georgiev Vladimir , Tarulli Mirko
Published: December 2024

We consider some scale invariant generalizations of the smoothing estimates for the free Schrödinger equation obtained by Kenig, Ponce and Vega (Ann. Inst. H. Poincaré Anal. Non Lineaire 10(3) (1993), 255–288; Invent. Math. 134(3) (1998), 489–545). Applying these estimates and using appropriate commutator estimates, we obtain similar scale invariant smoothing estimates for perturbed Schrödinger equation with small magnetic potential.
Homogenization of degenerate quasilinear parabolic equations with periodic structure
Authors: Huang Yong , Su Ning , Zhang Xingyou
Published: December 2024

In this paper the homogenization of degenerate quasilinear parabolic equations [Formula: see text] is studied via a weighted compensated compactness result, where a(t,y,α,λ) is periodic in (t,y).
The nonlinear bending–torsion theory for curved rods as $\varGamma $-limit of three-dimensional elasticity
Authors: Scardia Lucia
Published: December 2024

The problem of the rigorous derivation of one-dimensional models for nonlinearly elastic curved beams is studied in a variational setting. Considering different scalings of the three-dimensional energy and passing to the limit as the diameter of the beam goes to zero, a nonlinear model for strings and a bending–torsion theory for rods are deduced.
Phase boundaries motion preserving the volume of each connected component
Authors: Karali Georgia
Published: December 2024

We consider a finite number of particles initially close to spherical but of varying size exhibiting the property of preserving the measure of each connected component. We show that under the assumptions (i) particle size/interparticle distance «1 and (ii) initial deviation from sphericity/particle size «1 the particles retain their almost spherical shape and the dynamics of the system is determined by the motion of the centers. This is in sharp contrast with the Mullins–Sekerka free boundary problem where the particle centers remain almost fixed and the dynamics of the system is determined by the evolution of the radii.
Risk-sensitive quasi-variational inequalities for optimal investment with general transaction costs
Authors: Nagai Hideo
Published: December 2024

Risk-sensitive variational inequalities (QVIs) for optimal investment with general transaction costs are studied. The QVIs are derived to solve impulse control problems formulated for power utility maximization on infinite time horizon with general transaction costs. The QVI for such kind of problem is of “ergodic type” in which the pair (u,l) of a function u and a constant l is considered to be a solution. The constant determines the value maximizing the growth rate of expected power utility to the investor's total wealth. An optimal strategy is constructed from the function u. The difficulty in solving the QVI lies in that its related stopping problem is of a multiplicative functional, which enforce us to comprehend how to treat the quadratic growth nonlinear term appearing in the QVI from analytical, or probabilistic view points. Indeed we cannot employ the methods based on monotonicity which were effective in classical cases and need to invent other scheme constructing a solution instead. Besides, we would note, in spite of the name of “ergodic type”, the underlying diffusion process of the relevant QVI is not ergodic and it may include different features from the one studied by M. Robin (SIAM J. Control Optim. 19 (1981), 333–358) in this regard as well.
Energy decay for the nonlinear wave equation with a half-linear dissipation in $\mathbb{R}^{N}$
Authors: Jung Il Hyo , Kang Yong Han
Published: December 2024

We study decay estimates of the energy for the nonlinear wave equation in the whole space $\mathbb{R}^{N}$ . The dissipative term consists of the following two parts: The one part is nonlinear in a suitable ball; the other part is linear in the outside of the ball and it may be effective only at infinity. So we may call such a dissipation as a half-linear dissipation. We note that the method of proof is based on the multiplier technique and the unique continuation.
On the decay of the energy for systems with memory and indefinite dissipation
Authors: Muñoz Rivera Jaime E. , Naso Maria Grazia
Published: December 2024

In this paper we study the asymptotic behavior of the viscoelastic system with nondissipative kernels. We show that the uniform decay of the energy depends on the decay of the kernel, the positivity of the kernel in t=0 and some smallness condition. That is, if the kernel g∈C2( ${\mathbb{R}}$ +) with g(0)>0, decays exponentially to zero then the solution decays exponentially to zero. On the other hand, if the kernel decays polynomially as t−p then the corresponding solutions also decays polynomially to zero with the same rate of decay.
Stationary solutions to a Keller–Segel chemotaxis system
Authors: Musso Monica , Wei Juncheng
Published: December 2024

We consider the following stationary Keller–Segel system from chemotaxis [Formula: see text] where Ω⊂ $\mathbb{R}$ 2 is a smooth and bounded domain. We show that given any two positive integers K,L, for p sufficiently large, there exists a solution concentrating in K interior points and L boundary points. The location of the blow-up points is related to the Green function. The solutions are obtained as critical points of some finite-dimensional reduced energy functional. No assumption on the symmetry, geometry nor topology of the domain is needed.