Asymptotic spreading of a diffusive competition model with different free boundaries

Recently, the authors of [22] studied a diffusive prey–predator model with two different free boundaries. They first obtained the existence, uniqueness, regularity, uniform estimates and long time behaviors of global solution, and then established the conditions for spreading and vanishing. Especially, when spreading occurs, they provided accurate limits of two species as $t\to +\infty$, and gave some estimates of asymptotic spreading speeds of two species and asymptotic speeds of two free boundaries. Motivated by the paper [22], in this paper we discuss the diffusive competition model with two different free boundaries, which had been investigated by [7], [11], [15], [21]. The main purpose of this paper is to establish much sharper estimates of asymptotic spreading speeds of two species and asymptotic speeds of two free boundaries when spreading occurs. Furthermore, how the solution approaches the semi-wave when spreading happens is also described.     

Stabilization of the wave equations with localized Kelvin–Voigt type damping under optimal geometric conditions

The purpose of this note is to investigate the stabilization of the wave equation with Kelvin–Voigt damping in a bounded domain. Damping is localized via a non-smooth coefficient in a suitable subdomain. We prove a polynomial stability result in any space dimension, provided that the damping region satisfies some geometric conditions. The main novelty of this note is that the geometric situations covered here are richer than that considered in [25], [22], [16] and include in particular an example where the damping region is not localized in a neighborhood of the whole or a part of the boundary.     

Necessary conditions in stochastic linear quadratic problems and their applications

Stochastic linear quadratic optimal control problems are considered. A unified approach is proposed to treat the necessary optimality conditions of closed-loop optimal strategies and open-loop optimal controls. Notice that the former notion does not rely on initial wealth, while the later one does. Our conclusions of closed-loop optimal strategies are directly derived by suitable variational methods, the approach to which is different from [12], [11]. Moreover, the necessary conditions for closed-loop optimal strategies happen to be sufficient which takes us by surprise. Finally, two applications are given as illustration.     

Exact rates of convergence in some martingale central limit theorems

Renz [14], Ouchti [13], El Machkouri and Ouchti [3] and Mourrat [12] have established some tight bounds on the rate of convergence in the central limit theorem for martingales. In the present paper a modification of the methods, developed by Bolthausen [1] and Grama and Haeusler [7], is applied for obtaining exact rates of convergence in the central limit theorem for martingales with differences having conditional moments of order $2+\rho ,\rho >0$. Our results generalise and strengthen the bounds mentioned above.     

Real-analytic solvability for differential complexes associated to locally integrable structures

Inspired by the work of Suzuki [12] on the concept of real-analytic solvability for first-order analytic linear partial differential operators we extend his results for the differential complexes associated to analytic locally integrable structures of corank one. We prove that such notion of solvability is related to the smooth solvability condition introduced by F. Treves [13] in 1983. In our arguments the natural extension to closed forms of the well-known Baouendi–Treves approximation formula, the so-called “Approximate Poincaré Lemma” (cf. [1], [14]), plays a key role.     

An existence result for dissipative nonhomogeneous hyperbolic equations via a minimization approach

We discuss a purely variational approach to the study of a wide class of second order nonhomogeneous dissipative hyperbolic PDEs. Precisely, we focus on the wave-like equations that present also a nonzero source term and a first-order-in-time linear term. The paper carries on the research program initiated in [14], and developed in [15], [21], on the De Giorgi approach to hyperbolic equations.     

Interior regularity for fractional systems

We study the regularity of solutions of elliptic fractional systems of order 2s, $s\in (0,1)$, where the right hand side f depends on a nonlocal gradient and has the same scaling properties as the nonlocal operator. Under some structural conditions on the system we prove interior Hölder estimates in the spirit of [1]. Our results are stable in s allowing us to recover the classic results for elliptic systems due to S. Hildebrandt and K. Widman [11] and M. Wiegner [19].     

Reasoning about proof and knowledge

In previous work [15], we presented a hierarchy of classical modal systems, along with algebraic semantics, for the reasoning about intuitionistic truth, belief and knowledge. Deviating from Gödel's interpretation of IPC in S4, our modal systems contain IPC in the way established in [13]. The modal operator can be viewed as a predicate for intuitionistic truth, i.e. proof. Epistemic principles are partially adopted from Intuitionistic Epistemic Logic IEL [4]. In the present paper, we show that the S5-style systems of our hierarchy correspond to an extended Brouwer–Heyting–Kolmogorov interpretation and are complete w.r.t. a relational semantics based on intuitionistic general frames. In this sense, our S5-style logics are adequate and complete systems for the reasoning about proof combined with belief or knowledge. The proposed relational semantics is a uniform framework in which also IEL can be modeled. Verification-based intuitionistic knowledge formalized in IEL turns out to be a special case of the kind of knowledge described by our S5-style systems.     

The Balmer spectrum of rational equivariant cohomology theories

The category of rational G-equivariant cohomology theories for a compact Lie group G is the homotopy category of rational G-spectra and therefore tensor-triangulated. We show that its Balmer spectrum is the set of conjugacy classes of closed subgroups of G, with the topology corresponding to the topological poset of [7]. This is used to classify the collections of subgroups arising as the geometric isotropy of finite G-spectra. The ingredients for this classification are (i) the algebraic model of free spectra of the author and B. Shipley [14], (ii) the Localization Theorem of Borel–Hsiang–Quillen [21] and (iii) tom Dieck's calculation of the rational Burnside ring [4].     

Singularities and semistable degenerations for symplectic topology

We overview our work [7], [8], [9], [10], [11], [6] defining and studying normal crossings varieties and subvarieties in symplectic topology. This work answers a question of Gromov on the feasibility of introducing singular (sub)varieties into symplectic topology in the case of normal crossings singularities. It also provides a necessary and sufficient condition for smoothing normal crossings symplectic varieties. In addition, we explain some connections with other areas of mathematics and discuss a few directions for further research.     

Non-degeneracy and uniqueness of solutions to singular mean field equations on bounded domains

The aim of this paper is to complete the program initiated in [51], [23] and then carried out by several authors concerning non-degeneracy and uniqueness of solutions to mean field equations. In particular, we consider mean field equations with general singular data on non-smooth domains. The argument is based on the Alexandrov–Bol inequality and on the eigenvalues analysis of linearized singular Liouville-type problems.     

An Ikehara-type theorem for functions convergent to zero

We establish an analogue of the Ikehara theorem for positive non-increasing functions convergent to zero. In particular, this provides a complete proof of the results formulated in Diekmann & Kaper (1978) [5] and Carr & Chmaj (2004) [1], which are widely used nowadays to prove the uniqueness of traveling waves for various reaction–diffusion equations.     

Existence,uniqueness and positivity of solutions for BGK models for mixtures

We consider kinetic models for a multi component gas mixture without chemical reactions. In the literature, one can find two types of BGK models in order to describe gas mixtures. One type has a sum of BGK type interaction terms in the relaxation operator, for example the model described by Klingenberg, Pirner and Puppo [20] which contains well-known models of physicists and engineers for example Hamel [16] and Gross and Krook [15] as special cases. The other type contains only one collision term on the right-hand side, for example the well-known model of Andries, Aoki and Perthame [1]. For each of these two models [20] and [1], we prove existence, uniqueness and positivity of solutions in the first part of the paper. In the second part, we use the first model [20] in order to determine an unknown function in the energy exchange of the macroscopic equations for gas mixtures described by Dellacherie [11].     

On the instability of elliptic traveling wave solutions of the modified Camassa–Holm equation

This paper is concerned with the orbital instability for a specific class of periodic traveling wave solutions with the mean zero property and large spatial period related to the modified Camassa–Holm equation. These solutions, called snoidal waves, are written in terms of the Jacobi elliptic functions. To prove our result we use the abstract method of Grillakis, Shatah and Strauss [23], the Floquet theory for periodic eigenvalue problems and the n-gaps potentials theory of Dubrovin, Matveev and Novikov [19].     

A new path to the non blow-up of incompressible flows

One of the most challenging questions in fluid dynamics is whether the three-dimensional (3D) incompressible Navier-Stokes, 3D Euler and two-dimensional Quasi-Geostrophic (2D QG) equations can develop a finite-time singularity from smooth initial data. Recently, from a numerical point of view, Luo & Hou presented a class of potentially singular solutions to the Euler equations in a fluid with solid boundary [1], [2]. Furthermore, in two recent papers [3], [4], Tao indicates a significant barrier to establishing global regularity for the 3D Euler and Navier-Stokes equations, in that any method for achieving this, must use the finer geometric structure of these equations. In this paper, we show that the singularity discovered by Luo & Hou which lies right on the boundary is not relevant in the case of the whole domain ${\mathbb{R}}^3$. We reveal also that the translation and rotation invariance present in the Euler, Navier-Stokes and 2D QG equations are the key for the non blow-up in finite time of the solutions. The translation and rotation invariance of these equations combined with the anisotropic structure of regions of high vorticity allowed to establish a new geometric non blow-up criterion which yield us to the non blow-up of the solutions in all the Kerr's numerical experiments and to show that the potential mechanism of blow-up introduced in [5] cannot lead to the blow-up in finite time of solutions of Euler equations.     

Estimating the Gerber–Shiu function in the perturbed compound Poisson model by Laguerre series expansion

In this paper, we study the statistical estimation of the Gerber–Shiu function in the compound Poisson risk model perturbed by diffusion. This problem has been solved in [32] by the Fourier–Sinc series expansion method. Different from [32], we use the Laguerre series to expand the Gerber–Shiu function and propose a relevant estimator. The estimator is easily computed and has fast convergence rate. Various simulation studies are presented to confirm that the estimator performs well when the sample size is finite.     

Approximation locale précisée dans des problèmes multi-échelles avec défauts localisés

We proceed here with our systematic study, initiated in [3], of multiscale problems with defects, within the context of homogenization theory. The case under consideration here is that of a diffusion equation with a diffusion coefficient of the form of a periodic function perturbed by an $L^r({\mathbb{R}}^d)$, $1<r<+\infty$, function modelling a localized defect. We outline the proof of the following approximation result: the corrector function, the existence of which has been established in [3], [4], allows us to approximate the solution to the original multiscale equation with essentially the same accuracy as in the purely periodic case. The rates of convergence may however vary, and are made precise, depending upon the $L^r$ integrability of the defect. The generalization to an abstract setting is mentioned. Our proof exactly follows, step by step, the pattern of the original proof of Avellaneda and Lin in [1] in the periodic case, extended in the works of Kenig and collaborators [12], and borrows a lot from it. The details of the results announced in this Note are given in our publications [2], [11].     

A symmetric monoidal and equivariant Segal infinite loop space machine

In [12], we reworked and generalized equivariant infinite loop space theory, which shows how to construct G-spectra from G-spaces with suitable structure. In this paper, we construct a new variant of the equivariant Segal machine that starts from the category

of finite sets rather than from the category

of finite G-sets and which is equivalent to the machine studied in [19], [12]. In contrast to the machine in [19], [12], the new machine gives a lax symmetric monoidal functor from the symmetric monoidal category of

–G-spaces to the symmetric monoidal category of orthogonal G-spectra. We relate it multiplicatively to suspension G-spectra and to Eilenberg–Mac?Lane G-spectra via lax symmetric monoidal functors from based G-spaces and from abelian groups to

–G-spaces. Even non-equivariantly, this gives an appealing new variant of the Segal machine. This new variant makes the equivariant generalization of the theory essentially formal, hence likely to be applicable in other contexts.     

Stable ground states for the HMF Poisson model

In this paper we prove the nonlinear orbital stability of a large class of steady state solutions to the Hamiltonian Mean Field (HMF) system with a Poisson interaction potential. These steady states are obtained as minimizers of an energy functional under one, two or infinitely many constraints. The singularity of the Poisson potential prevents from a direct run of the general strategy in [16], [19] which was based on generalized rearrangement techniques, and which has been recently extended to the case of the usual (smooth) cosine potential [17]. Our strategy is rather based on variational techniques. However, due to the boundedness of the space domain, our variational problems do not enjoy the usual scaling invariances which are, in general, very important in the analysis of variational problems. To replace these scaling arguments, we introduce new transformations which, although specific to our context, remain somehow in the same spirit of rearrangements tools introduced in the references above. In particular, these transformations allow for the incorporation of an arbitrary number of constraints, and yield a stability result for a large class of steady states.     

Local well-posedness of the incompressible Euler equations in B∞,11 and the inviscid limit of the Navier–Stokes equations

We prove the inviscid limit of the incompressible Navier–Stokes equations in the same topology of Besov spaces as the initial data. The proof is based on proving the continuous dependence of the Navier–Stokes equations uniformly with respect to the viscosity. To show the latter, we rely on some Bona–Smith type argument in the $L^p$ setting. Our obtained result implies a new result that the Cauchy problem of the Euler equations is locally well-posed in the borderline Besov space $B_{\infty ,1}^1({\mathbb{R}}^d)$, $d\ge 2$, in the sense of Hadmard, which is an open problem left in recent works by Bourgain and Li in [3], [4] and by Misio?ek and Yoneda in [12], [13], [14].