Modified scattering for the critical nonlinear Schrödinger equation

We consider the nonlinear Schrödinger equation

$i{u}_t+\mathrm{\Delta }u=\lambda |u{|}^{\frac{2}{N}}u$

in all dimensions $N\ge 1$, where $\lambda \in \mathbb{C}$ and $?\lambda \le 0$. We construct a class of initial values for which the corresponding solution is global and decays as $t\to \infty$, like $t^{?\frac{N}{2}}$ if $?\lambda =0$ and like ${(t\log ?t)}^{?\frac{N}{2}}$ if $?\lambda <0$. Moreover, we give an asymptotic expansion of those solutions as $t\to \infty$. We construct solutions that do not vanish, so as to avoid any issue related to the lack of regularity of the nonlinearity at $u=0$. To study the asymptotic behavior, we apply the pseudo-conformal transformation and estimate the solutions by allowing a certain growth of the Sobolev norms which depends on the order of regularity through a cascade of exponents.     

Rough path properties for local time of symmetric α stable process

In this paper, we first prove that the local time associated with symmetric $\alpha$-stable processes is of bounded $p$-variation for any $p>\frac{2}{\alpha ?1}$ partly based on Barlow’s estimation of the modulus of the local time of such processes.  The fact that the local time is of bounded $p$-variation for any $p>\frac{2}{\alpha ?1}$ enables us to define the integral of the local time ${\int }_{?\infty }^{\infty }{?}_{?}^{\alpha ?1}f\left(x\right)d_x{L}_t^x$ as a Young integral for less smooth functions being of bounded $q$-variation with $1\le q<\frac{2}{3?\alpha }$. When $q\ge \frac{2}{3?\alpha }$, Young’s integration theory is no longer applicable. However, rough path theory is useful in this case. The main purpose of this paper is to establish a rough path theory for the integration with respect to the local times of symmetric $\alpha$-stable processes for $\frac{2}{3?\alpha }\le q<4$.     

Sharp well-posedness of the cauchy problem for the higher-order dispersive equation

This current paper is devoted to the Cauchy problem for higher order dispersive equation u_t+ ?_x~(2n+1)u = ?_x(u?_x~nu) + ?_x~(n-1)(u_x~2), n ≥ 2, n ∈ N~+.By using Besov-type spaces, we prove that the associated problem is locally well-posed in H~(-n/2+3/4,-1/(2n))(R). The new ingredient is that we establish some new dyadic bilinear estimates. When n is even, we also prove that the associated equation is ill-posed in H~(s,a)(R) with s -n/2+3/4 and all a∈R.     

Bound states of schrödinger equations with electromagnetic fields and vanishing potentials

We study the bound states to nonlinear Schrödinger equations with electro-magnetic fields $\text{i}h\frac{?\psi }{?t}={(\frac{h}{i}?-A(x))}^2\psi +V(x)\psi -K(x){|}^{p-1}\psi =0,\text{on}{?}_{+}\times {?}^N$. Let $G(x)={[V(x)]}^{\frac{p+1}{p-1}-\frac{N}{2}}{[K(x)]}^{-\frac{2}{p-1}}$ and suppose that G(x) has k local minimum points. For h > 0 small, we find multi-bump bound states ψh(x,t) = _e^?lEt/^h_Uh(χ) with Uh concentrating at the local minimum points of G(x) simultaneously as h → 0. The potentials V(x) and K(x) are allowed to be either compactly supported or unbounded at infinity.     

Existence et unicité pour un fluide inhomogène

Recently R. Danchin showed the existence and uniqueness for an inhomogenous fluid in the homogeneous Besov space ${\dot{B}}_{21}^{\frac{N}{2}}({\mathbb{R}}^N)\times {\dot{B}}_{21}^{?1+\frac{N}{2}}({\mathbb{R}}^N)$, under the condition that ${\rho }_0?1$ is small in ${\dot{B}}_{2\infty }^{\frac{N}{2}}\cap L^{\infty }$ if $2<N,$ in ${\dot{B}}_{21}^{\frac{N}{2}}$ if $N=2.$ In this Note, one shows that the condition $6{\rho }_0?16_{L^{\infty }}?1$ is sufficient to have the existence and uniqueness. To cite this article: H. Abidi, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

On the sum of reciprocals of least common multiples

Let ${\{a_i\}}_{i=1}^{\infty }$ be a strictly increasing sequence of positive integers ($a_i<a_j$ if $i<j$). In 1978, Borwein showed that for any positive integer n, we have ${\sum }_{i=1}^n\frac{1}{\mathrm{lcm}(a_i,a_{i+1})}\le 1?\frac{1}{2^n}$, with equality occurring if and only if $a_i=2^{i?1}$ for $1\le i\le n+1$. Let $3\le r\le 7$ be an integer. In this paper, we investigate the sum ${\sum }_{i=1}^n\frac{1}{\mathrm{lcm}(a_i,...,a_{i+r?1})}$ and show that ${\sum }_{i=1}^n\frac{1}{\mathrm{lcm}(a_i,...,a_{i+r?1})}\le U_r(n)$ for any positive integer n, where $U_r(n)$ is a constant depending on r and n. Further, for any integer $n\ge 2$, we also give a characterization of the sequence ${\{a_i\}}_{i=1}^{\infty }$ such that the equality ${\sum }_{i=1}^n\frac{1}{\mathrm{lcm}(a_i,...,a_{i+r?1})}=U_r(n)$ holds.     

Three new results on continuation criteria for the 3D relativistic Vlasov–Maxwell system

We consider continuation criteria for the three-dimensional relativistic Vlasov–Maxwell system. When the particle density, $f(t,x,p)$, is compactly supported at $t=0$, we prove ${6p_0^{\frac{18}{5r}?1+\beta }f6}_{L_t^{\infty }{L}_x^r{L}_p^1}?1$, where $1\le r\le 2$ and $\beta >0$ is arbitrarily small, is a continuation criteria. Our continuation criteria is an improvement in the $1\le r\le 2$ range to the previously best known criteria ${6p_0^{\frac{4}{r}?1+\beta }f6}_{L_t^{\infty }{L}_x^r{L^1}_p}?1$ due to Kunze [7]. We also consider continuation criteria when $f(0,x,p)$ has noncompact support. In this regime, Luk–Strain [9] proved that ${6p_0^{\theta }f6}_{L_x^1{L}_p^1}?1$ is a continuation criteria for $\theta >5$. We improve this result to $\theta >3$. Finally, we build on another result by Luk–Strain [8]. The authors proved boundedness of momentum support on a fixed two-dimensional plane is a sufficient continuation criteria. We prove the same result even if the plane varies continuously in time.     

EXISTENCE AND BLOW-UP BEHAVIOR OF CONSTRAINED MINIMIZERS FOR SCHRDINGER-POISSON-SLATER SYSTEM

In this article,we study constrained minimizers of the following variational problem e(p):=inf{u∈H1(R3),||u||22=p}E(u),p〉0,where E(u)is the Schrdinger-Poisson-Slater(SPS)energy functional E(u):=1/2∫R3︱▽u(x)︱2dx-1/4∫R3∫R3u2(y)u2(x)/︱x-y︱dydx-1/p∫R3︱u(x)︱pdx in R3 and p∈(2,6).We prove the existence of minimizers for the cases 2p10/3,ρ0,and p=10/3,0ρρ~*,and show that e(ρ)=-∞for the other cases,whereρ~*=||φ||_2~2 andφ(x)is the unique(up to translations)positive radially symmetric solution of-△u+u=u~(7/3)in R~3.Moreover,when e(ρ~*)=-∞,the blow-up behavior of minimizers asρ↗ρ~*is also analyzed rigorously.     

Effect of stochastic perturbations for front propagation in Kolmogorov Petrovskii Piscunov equations

This article considers equations of Kolmogorov Petrovskii Piscunov type in one space dimension, with stochastic perturbation:

where the stochastic differential is taken in the sense of Itô and $\zeta$ is a Gaussian random field satisfying $\mathbb{E}\left[\zeta \right]=0$ and $\mathbb{E}\left[\zeta (s,x)\zeta (t,y)\right]=(s\wedge t)\Gamma (x?y)$. Two situations are considered: firstly, $\zeta$ is simply a standard Wiener process (i.e. $\Gamma \equiv 1$): secondly, $\Gamma \in C^{\infty }\left(\mathbb{R}\right)$ with ${lim}_{\left|z\right|\to +\infty }\left|\Gamma \left(z\right)\right|=0$.The results are as follows: in the first situation (standard Wiener process: i.e. $\Gamma \left(x\right)\equiv 1$), there is a non-degenerate travelling wave front if and only if $\frac{{?}^2}{2}<1$, with asymptotic wave speed $max\left(\sqrt{2\kappa (1?\frac{{?}^2}{2})},\left(\frac{1}{N}(1?\frac{{?}^2}{2})+\frac{\kappa N}{2}\right){\mathbf{1}}_{\{N<\sqrt{\frac{2}{\kappa }(1?\frac{{?}^2}{2})}\}}\right)$; the noise slows the wave speed. If the stochastic integral is taken instead in the sense of Stratonovich, then the asymptotic wave speed is the classical McKean wave speed and does not depend on $?$.In the second situation (noise with spatial covariance which decays to 0 at $\pm \infty$, stochastic integral taken in the sense of Itô), a travelling front can be defined for all $?>0$. Its average asymptotic speed does not depend on $?$ and is the classical wave speed of the unperturbed KPP equation.     

An exact upper bound for sums of element orders in non-cyclic finite groups

Denote the sum of element orders in a finite group G by $\psi (G)$ and let $C_n$ denote the cyclic group of order n. Suppose that G is a non-cyclic finite group of order n and q is the least prime divisor of n. We proved that $\psi (G)\le \frac{7}{11}\psi (C_n)$ and $\psi (G)<\frac{1}{q?1}\psi (C_n)$. The first result is best possible, since for each $n=4k$, k odd, there exists a group G of order n satisfying $\psi (G)=\frac{7}{11}\psi (C_n)$ and the second result implies that if G is of odd order, then $\psi (G)<\frac{1}{2}\psi (C_n)$. Our results improve the inequality $\psi (G)<\psi (C_n)$ obtained by H. Amiri, S.M. Jafarian Amiri and I.M. Isaacs in 2009, as well as other results obtained by S.M. Jafarian Amiri and M. Amiri in 2014 and by R. Shen, G. Chen and C. Wu in 2015. Furthermore, we obtained some $\psi (G)$-based sufficient conditions for the solvability of G.     

Global existence of classical solutions to the hyperbolic geometry flow with time-dependent dissipation

In this article, we investigate the hyperbolic geometry flow with time-dependent dissipation(?~2 g_(ij))/? t~2+μ/((1 + t)~λ)(? g_(ij))/? t=-2 R_(ij),on Riemann surface. On the basis of the energy method, for 0 λ≤ 1, μ λ + 1, we show that there exists a global solution gij to the hyperbolic geometry flow with time-dependent dissipation with asymptotic flat initial Riemann surfaces. Moreover, we prove that the scalar curvature R(t, x) of the solution metric g_(ij) remains uniformly bounded.     

A sharp bound on the expected number of upcrossings of an L2-bounded Martingale

For a martingale $M$ starting at $x$ with final variance ${\sigma }^2$, and an interval $(a,b)$, let $\Delta =\frac{b?a}{\sigma }$ be the normalized length of the interval and let $\delta =\frac{|x?a|}{\sigma }$ be the normalized distance from the initial point to the lower endpoint of the interval. The expected number of upcrossings of $(a,b)$ by $M$ is at most $\frac{\sqrt{1+{\delta }^2}?\delta }{2\Delta }$ if ${\Delta }^2\le 1+{\delta }^2$ and at most $\frac{1}{1+{(\Delta +\delta )}^2}$ otherwise. Both bounds are sharp, attained by Standard Brownian Motion stopped at appropriate stopping times. Both bounds also attain the Doob upper bound on the expected number of upcrossings of $(a,b)$ for submartingales with the corresponding final distribution. Each of these two bounds is at most $\frac{\sigma }{2(b?a)}$, with equality in the first bound for $\delta =0$. The upper bound $\frac{\sigma }{2}$ on the length covered by $M$ during upcrossings of an interval restricts the possible variability of a martingale in terms of its final variance. This is in the same spirit as the Dubins & Schwarz sharp upper bound $\sigma$ on the expected maximum of $M$ above $x$, the Dubins & Schwarz sharp upper bound $\sigma \sqrt{2}$ on the expected maximal distance of $M$ from $x$, and the Dubins, Gilat & Meilijson sharp upper bound $\sigma \sqrt{3}$ on the expected diameter of $M$.     

Weak solutions of semilinear elliptic equation involving Dirac mass

In this paper, we study the elliptic problem with Dirac mass

(1)$\{\begin{array}{c}?\mathrm{\Delta }u=V{u}^p+k{\delta }_0\text{in}{\mathbb{R}}^N,\hfill \\ \underset{|x|\to +\infty }{\lim }?u(x)=0,\hfill \end{array}$

where $N>2$, $p>0$, $k>0$, ${\delta }_0$ is the Dirac mass at the origin and the potential V is locally Lipchitz continuous in ${\mathbb{R}}^N?\{0\}$, with non-empty support and satisfying

$0\le V(x)\le \frac{{\sigma }_1}{|x{|}^{a_0}(1+|x{|}^{a_{\infty }?a_0})},$

with $a_0<N$, $a_0<a_{\infty }$ and ${\sigma }_1>0$. We obtain two positive solutions of (1) with additional conditions for parameters on $a_{\infty },a_0$, p and k. The first solution is a minimal positive solution and the second solution is constructed via Mountain Pass Theorem.