Global well-posedness and decay estimates of strong solutions to a two-phase model with magnetic field

In this paper, we consider the Cauchy problem for a two-phase model with magnetic field in three dimensions. The global existence and uniqueness of strong solution as well as the time decay estimates in $H^2({\mathbb{R}}^3)$ are obtained by introducing a new linearized system with respect to $(n^{\gamma }?{\stackrel{?}{n}}^{\gamma },n?\stackrel{?}{n},P?\stackrel{?}{P},u,H)$ for constants $\stackrel{?}{n}\ge 0$ and $\stackrel{?}{P}>0$, and doing some new a priori estimates in Sobolev Spaces to get the uniform upper bound of $(n?\stackrel{?}{n},n^{\gamma }?{\stackrel{?}{n}}^{\gamma })$ in $H^2({\mathbb{R}}^3)$ norm.     

On Fano and weak Fano Bott–Samelson–Demazure–Hansen varieties

Let G be a simple algebraic group over the field of complex numbers. Fix a maximal torus T and a Borel subgroup B of G containing T. Let w be an element of the Weyl group W of G, and let $Z(\stackrel{?}{w})$ be the Bott–Samelson–Demazure–Hansen (BSDH) variety corresponding to a reduced expression $\stackrel{?}{w}$ of w with respect to the data $(G,B,T)$.In this article we give complete characterization of the expressions $\stackrel{?}{w}$ such that the corresponding BSDH variety $Z(\stackrel{?}{w})$ is Fano or weak Fano. As a consequence we prove vanishing theorems of the cohomology of tangent bundle of certain BSDH varieties and hence we get some local rigidity results.     

Corrigendum to “Scattering above energy norm of solutions of a loglog energy-supercritical Schrödinger equation with radial data”

The purpose of this corrigendum is to point out some errors that appear in [1]. Our main result remains valid, i.e scattering of ${\stackrel{?}{H}}^k:={\dot{H}}^k({\mathbb{R}}^n)\cap {\dot{H}}^1({\mathbb{R}}^n)$ solutions of the loglog energy-supercritical Schrödinger equation $i{?}_{t}u+△u=|u{|}^{\frac{4}{n?2}}u{\log }^c?(\log ?(10+|u{|}^2)$, $0<c<c_n$, $n\in \{3,4\}$, with $k>\frac{n}{2}$, radial data $u(0):=u_0\in {\stackrel{?}{H}}^k$ but with slightly different values of $c_n$, i.e $c_n=\frac{1}{5772}$ if $n=3$ and $c_n=\frac{3}{8024}$ if $n=4$. We propose some corrections.     

Approximation polynomiale des jets de Whitney ∂‾-plats de classe M

We study the problem of global approximation by polynomials in Whitney $\stackrel{￣}{?}$-flat jet classes on irregular compacts of ${\mathbb{C}}^N$.     

Une caractérisation de l'algèbre de Fourier pour certains groupes localement compacts

Let G be a separable locally compact group with type-I left regular representation, $\stackrel{?}{G}$ its dual and $A(G)$ its Fourier algebra. We prove an analogue of Parseval's theorem and that the mapping

$T?u(x):=\underset{\stackrel{?}{G}}{\int }Tr[T(\pi )\pi {(x)}^{?1}]{\mathrm{d}}_{\mu }(\pi )$

is an isometric isomorphism of Banach spaces from $L^1(\stackrel{?}{G})$ onto $A(G)$.     

On the remainder of the semialgebraic Stone–Cěch compactification of a semialgebraic set

In this work we analyze some topological properties of the remainder $?M:={\beta }_{\mathrm{s}}^{?}M?M$ of the semialgebraic Stone–Cěch compactification ${\beta }_{\mathrm{s}}^{?}M$ of a semialgebraic set $M?{\mathbb{R}}^m$ in order to ‘distinguish’ its points from those of M. To that end we prove that the set of points of ${\beta }_{\mathrm{s}}^{?}M$ that admit a metrizable neighborhood in ${\beta }_{\mathrm{s}}^{?}M$ equals $M_{\mathrm{lc}}\cup ({\mathrm{Cl}}_{{\beta }_{\mathrm{s}}^{?}M}({\stackrel{￣}{M}}_{\le 1})?{\stackrel{￣}{M}}_{\le 1})$ where $M_{\mathrm{lc}}$ is the largest locally compact dense subset of M and ${\stackrel{￣}{M}}_{\le 1}$ is the closure in M of the set of 1-dimensional points of M. In addition, we analyze the properties of the sets $\stackrel{?}{?}M$ and $\stackrel{?}{?}M$ of free maximal ideals associated with formal and semialgebraic paths. We prove that both are dense subsets of the remainder ?M and that the differences $?M?\stackrel{?}{?}M$ and $\stackrel{?}{?}M?\stackrel{?}{?}M$ are also dense subsets of ?M. It holds moreover that all the points of $\stackrel{?}{?}M$ have countable systems of neighborhoods in ${\beta }_{\mathrm{s}}^{?}M$.     

Bounding the number of limit cycles of discontinuous differential systems by using Picard–Fuchs equations

In this paper, by using Picard–Fuchs equations and Chebyshev criterion, we study the upper bounds of the number of limit cycles given by the first order Melnikov function for discontinuous differential systems, which can bifurcate from the periodic orbits of quadratic reversible centers of genus one (r19): $\dot{x}=y?12x^2+16y^2$, $\dot{y}=?x?16xy$, and (r20): $\dot{x}=y+4x^2$, $\dot{y}=?x+16xy$, and the periodic orbits of the quadratic isochronous centers $(S_1):\dot{x}=?y+x^2?y^2$, $\dot{y}=x+2xy$, and $(S_2):\dot{x}=?y+x^2$, $\dot{y}=x+xy$. The systems (r19) and (r20) are perturbed inside the class of polynomial differential systems of degree n and the system $(S_1)$ and $(S_2)$ are perturbed inside the class of quadratic polynomial differential systems. The discontinuity is the line $y=0$. It is proved that the upper bounds of the number of limit cycles for systems (r19) and (r20) are respectively $4n?3(n\ge 4)$ and $4n+3(n\ge 3)$ counting the multiplicity, and the maximum numbers of limit cycles bifurcating from the period annuluses of the isochronous centers $(S_1)$ and $(S_2)$ are exactly 5 and 6 (counting the multiplicity) on each period annulus respectively.