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.     

Bounds on multiplicities of spherical spaces over finite fields

Let G be a reductive group scheme of type A acting on a spherical scheme X. We prove that there exists a number C such that the multiplicity $\dim \mathrm{Hom}(\rho ,\mathbb{C}[X(F)])$ is bounded by C, for any finite field F and any irreducible representation ρ of $G(F)$. We give an explicit bound for C. We conjecture that this result is true for any reductive group scheme and when F ranges (in addition) over all local fields of characteristic 0.Different aspects of this conjecture were studied in [3], [11], [6], [7].     

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.     

Non-existence of extremals for the Adimurthi–Druet inequality

The Adimurthi–Druet [1] inequality is an improvement of the standard Moser–Trudinger inequality by adding a $L^2$-type perturbation, quantified by $\alpha \in [0,{\lambda }_1)$, where ${\lambda }_1$ is the first Dirichlet eigenvalue of Δ on a smooth bounded domain. It is known [3], [10], [14], [19] that this inequality admits extremal functions, when the perturbation parameter α is small. By contrast, we prove here that the Adimurthi–Druet inequality does not admit any extremal, when the perturbation parameter α approaches ${\lambda }_1$. Our result is based on sharp expansions of the Dirichlet energy for blowing sequences of solutions of the corresponding Euler–Lagrange equation, which take into account the fact that the problem becomes singular as $\alpha \to {\lambda }_1$.     

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].     

Blow-up of solutions to critical semilinear wave equations with variable coefficients

We verify the critical case $p=p_0(n)$ of Strauss' conjecture [30] concerning the blow-up of solutions to semilinear wave equations with variable coefficients in ${\mathbf{R}}^n$, where $n\ge 2$. The perturbations of Laplace operator are assumed to be smooth and decay exponentially fast at infinity. We also obtain a sharp lifespan upper bound for solutions with compactly supported data when $p=p_0(n)$. The unified approach to blow-up problems in all dimensions combines several classical ideas in order to generalize and simplify the method of Zhou [43] and Zhou & Han [45]: exponential “eigenfunctions” of the Laplacian [37] are used to construct the test function ${?}_q$ for linear wave equation with variable coefficients and John's method of iterations [13] is augmented with the “slicing method” of Agemi, Kurokawa and Takamura [1] for lower bounds in the critical case.     

Divisibility classes are seldom closed under flat covers

We show that the class of all divisible modules over an integral domain R is closed under flat covers if and only if R is almost perfect. Also, we show that if the class of all s-divisible modules, where s is a regular element of a commutative ring R, is closed under flat covers then the quotient ring $R/sR$ satisfies some rather restrictive properties. The question is motivated by the recent classification [11] of tilting classes over commutative rings.     

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].     

On extended graphical calculus for categorified quantum sl(n)

We study the properties of the extended graphical calculus for categorified quantum $sl(n)$. The main results include proofs of Reidemeister 2 and Reidemeister 3-like moves involving strands corresponding to arbitrary thicknesses and arbitrary colors – the results that were anounced in [12].     

Bases in the mod p Steenrod algebra, β included

In [3] well known results of Wall and Arnon on the monomial bases in the mod 2 Steenrod algebra (see [9], [1]) were generalized to the subalgebra ${\stackrel{￣}{A}}_p$ of the mod p Steenrod algebra, $p>2$, generated by the reduced powers. In the present paper we considered the case of the full Steenrod algebra $A_p$. We constructed βX-, βZ-, βC-, ZA-, and XC-bases. We proved extremal properties of the βX-, βZ-, ZA-, and XC-bases. Also we constructed a new polynomial generators of the ring $H^{?}(K(\mathbb{Z}/p,n),\mathbb{Z}/p)$ in terms of the βC-basis.     

New lower bounds on the radius of spatial analyticity for the KdV equation

The radius of spatial analyticity for solutions of the KdV equation is studied. It is shown that the analyticity radius does not decay faster than $t^{?1/4}$ as time t goes to infinity. This improves the works of Selberg and da Silva (2017) [30] and Tesfahun (2017) [34]. Our strategy mainly relies on a higher order almost conservation law in Gevrey spaces, which is inspired by the I-method.     

Ill-posedness of the Camassa–Holm and related equations in the critical space

We prove norm inflation and hence ill-posedness for a class of shallow water wave equations, such as the Camassa–Holm equation, Degasperis–Procesi equation and Novikov equation etc., in the critical Sobolev space $H^{3/2}$ and even in the Besov space $B_{p,r}^{1+1/p}$ for $p\in [1,\infty ],r\in (1,\infty ]$. Our results cover both real-line and torus cases (only real-line case for Novikov), solving an open problem left in the previous works ([5], [14], [16]).     

On the well-posedness of 3-D inhomogeneous incompressible Navier–Stokes equations with variable viscosity

In this paper, we mainly study the well-posedness for the 3-D inhomogeneous incompressible Navier–Stokes equations with variable viscosity. With some smallness assumption on the BMO-norm of the initial density, we first get the local well-posedness of (1.1) in the critical Besov spaces. Moreover, if the viscosity coefficient is a constant, we can extend this local solution to be a global one. Our theorem implies that we have successfully extended the integrability index p of the initial velocity which has been obtained by Abidi, Gui and Zhang in [3], Burtea in [8] and Zhai and Yin in [32] to approach the ideal one i.e. $1<p<6$. The main novelty of this work is to apply the CRW theorem obtained by Coifman, Rochberg, Weiss in [11] to get a new a priori estimate for an elliptic equation with variable coefficients. The uniqueness of the solution also relies on a Lagrangian approach as in [16], [17], [18].     

A refined convergence result in homogenization of second order parabolic systems

We derive the sharp $O(\epsilon )$ convergence rate in $L^2(0,T;L^{q_0}(\mathrm{\Omega })),q_0=2d/(d?1)$ in periodic homogenization of second order parabolic systems with bounded measurable coefficients in Lipschitz cylinders. This extends the corresponding result for elliptic systems established in [20] to parabolic systems and improves the corresponding result in $L^2$ settings derived in [7], [28] for second order parabolic systems with time-dependent coefficients.     

A uniform bound of reducibility index of good parameter ideals for certain class of modules

Let $(R,\mathfrak{m})$ be a Noetherian local ring and M a finitely generated R-module. The invariants $\mathrm{p}(M)$ and $\mathrm{sp}(M)$ of M were introduced in [3] and [17] in order to measure the non-Cohen–Macaulayness and the non-sequential-Cohen–Macaulayness of M, respectively. Let $M=D_0?D_1?\dots ?D_k$ be the filtration of M such that $D_i$ is the largest submodule of M of dimension less than $\dim ?D_{i?1}$ for all $i\le k$ and $\mathrm{p}(D_k)\le 1$. In this paper we prove that if $\mathrm{sp}(M)\le 1$, then there exists a constant c such that ${\mathrm{ir}}_M(\mathfrak{q}M)\le c$ for all good parameter ideals $\mathfrak{q}$ of M with respect to this filtration. Here ${\mathrm{ir}}_M(\mathfrak{q}M)$ is the reducibility index of $\mathfrak{q}$ on M. This is an extension of the main results of [19], [20], [24].     

Three solutions for a nonlocal problem with critical growth

The main goal of this work is to prove the existence of three different solutions (one positive, one negative and one with nonconstant sign) for the equation ${(?{\mathrm{\Delta }}_p)}^{s}u=|u{|}^{p_s^{?}?2}u+\lambda f(x,u)$ in a bounded domain with Dirichlet condition, where ${(?{\mathrm{\Delta }}_p)}^s$ is the well known p-fractional Laplacian and $p_s^{?}=\frac{np}{n?sp}$ is the critical Sobolev exponent for the non local case. The proof follows the ideas of [28] and is based in the extension of the Concentration Compactness Principle for the p-fractional Laplacian [20] and Ekeland's variational Principle [7].     

Symmetry and rigidity for the hinged composite plate problem

The composite plate problem is an eigenvalue optimization problem related to the fourth order operator ${(?\mathrm{\Delta })}^2$. In this paper we continue the study started in [10], focusing on symmetry and rigidity issues in the case of the hinged composite plate problem, a specific situation that allows us to exploit classical techniques like the moving plane method.     

On modal logics arising from scattered locally compact Hausdorff spaces

For a topological space X, let $\mathsf{L}(X)$ be the modal logic of X where □ is interpreted as interior (and hence ◇ as closure) in X. It was shown in [3] that the modal logics S4, S4.1, S4.2, S4.1.2, S4.Grz, ${\mathsf{S4}.\mathsf{Grz}}_n$ ($n\ge 1$), and their intersections arise as $\mathsf{L}(X)$ for some Stone space X. We give an example of a scattered Stone space whose logic is not such an intersection. This gives an affirmative answer to [3, Question 6.2]. On the other hand, we show that a scattered Stone space that is in addition hereditarily paracompact does not give rise to a new logic; namely we show that the logic of such a space is either S4.Grz or ${\mathsf{S4}.\mathsf{Grz}}_n$ for some $n\ge 1$. In fact, we prove this result for any scattered locally compact open hereditarily collectionwise normal and open hereditarily strongly zero-dimensional space.     

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].