Powerful numbers in (1ℓ + qℓ)(2ℓ + qℓ)⋯(nℓ + qℓ)

Let q be a positive integer. Recently, Niu and Liu proved that, if $n\ge \max ?\{q,1198?q\}$, then the product $(1^3+q^3)(2^3+q^3)?(n^3+q^3)$ is not a powerful number. In this note, we prove (1) that, for any odd prime power ? and $n\ge \max ?\{q,11?q\}$, the product $(1^{?}+q^{?})(2^{?}+q^{?})?(n^{?}+q^{?})$ is not a powerful number, and (2) that, for any positive odd integer ?, there exists an integer $N_{q,?}$ such that, for any positive integer $n\ge N_{q,?}$, the product $(1^{?}+q^{?})(2^{?}+q^{?})?(n^{?}+q^{?})$ is not a powerful number.     

On the sub poly-harmonic property for solutions to (−Δ)pu < 0 in Rn

In this note, we mainly study the relation between the sign of ${(?\mathrm{\Delta })}^{p}u$ and ${(?\mathrm{\Delta })}^{p?i}u$ in ${\mathbb{R}}^n$ with $p?2$ and $n?2$ for $1?i?p?1$. Given the differential inequality ${(?\mathrm{\Delta })}^{p}u<0$, first we provide several sufficient conditions so that ${(?\mathrm{\Delta })}^{p?1}u<0$ holds. Then we provide conditions such that ${(?\mathrm{\Delta })}^{i}u<0$ for all $i=1,2,\dots ,p?1$, which is known as the sub poly-harmonic property for u. In the last part of the note, we revisit the super poly-harmonic property for solutions to ${(?\mathrm{\Delta })}^{p}u={\mathrm{e}}^{2pu}$ and ${(?\mathrm{\Delta })}^{p}u=u^q$ with $q>0$ in ${\mathbb{R}}^n$.     

On the Lp boundedness of wave operators for two-dimensional Schrödinger operators with threshold obstructions

Let $H=?\mathrm{\Delta }+V$ be a Schrödinger operator on $L^2({\mathbb{R}}^2)$ with real-valued potential V, and let $H_0=?\mathrm{\Delta }$. If V has sufficient pointwise decay, the wave operators $W_{\pm }=s?{\lim }_{t\to \pm \infty }?e^{itH}{e}^{?it{H}_0}$ are known to be bounded on $L^p({\mathbb{R}}^2)$ for all $1<p<\infty$ if zero is not an eigenvalue or resonance. We show that if there is an s-wave resonance or an eigenvalue only at zero, then the wave operators are bounded on $L^p({\mathbb{R}}^2)$ for $1<p<\infty$. This result stands in contrast to results in higher dimensions, where the presence of zero energy obstructions is known to shrink the range of valid exponents p.     

Space-time asymptotics of the two dimensional Navier–Stokes flow in the whole plane

We consider the space-time behavior of the two dimensional Navier–Stokes flow. Introducing some qualitative structure of initial data, we succeed to derive the first order asymptotic expansion of the Navier–Stokes flow without moment condition on initial data in $L^1({\mathbb{R}}^2)\cap L_{\sigma }^2({\mathbb{R}}^2)$. Moreover, we characterize the necessary and sufficient condition for the rapid energy decay ${6u(t)6}_2=o(t^{?1})$ as $t\to \infty$ motivated by Miyakawa–Schonbek [21]. By weighted estimated in Hardy spaces, we discuss the possibility of the second order asymptotic expansion of the Navier–Stokes flow assuming the first order moment condition on initial data. Moreover, observing that the Navier–Stokes flow $u(t)$ lies in the Hardy space $H^1({\mathbb{R}}^2)$ for $t>0$, we consider the asymptotic expansions in terms of Hardy-norm. Finally we consider the rapid time decay ${6u(t)6}_2=o(t^{?\frac{3}{2}})$ as $t\to \infty$ with cyclic symmetry introduced by Brandolese [2].     

A Schauder and Riesz basis criterion for non-self-adjoint Schrödinger operators with periodic and antiperiodic boundary conditions

Under the assumption that $V\in L^2([0,\pi ];dx)$, we derive necessary and sufficient conditions in terms of spectral data for (non-self-adjoint) Schrödinger operators $?d^2/d{x}^2+V$ in $L^2([0,\pi ];dx)$ with periodic and antiperiodic boundary conditions to possess a Riesz basis of root vectors (i.e., eigenvectors and generalized eigenvectors spanning the range of the Riesz projection associated with the corresponding periodic and antiperiodic eigenvalues).We also discuss the case of a Schauder basis for periodic and antiperiodic Schrödinger operators $?d^2/d{x}^2+V$ in $L^p([0,\pi ];dx)$, $p\in (1,\infty )$.     

A sharp Balian–Low uncertainty principle for shift-invariant spaces

A sharp version of the Balian–Low theorem is proven for the generators of finitely generated shift-invariant spaces. If generators ${\{f_k\}}_{k=1}^K?L^2({\mathbb{R}}^d)$ are translated along a lattice to form a frame or Riesz basis for a shift-invariant space V, and if V has extra invariance by a suitable finer lattice, then one of the generators $f_k$ must satisfy ${\int }_{{\mathbb{R}}^d}|x||f_k(x){|}^{2}dx=\infty$, namely, $\stackrel{?}{f_k}?H^{1/2}({\mathbb{R}}^d)$. Similar results are proven for frames of translates that are not Riesz bases without the assumption of extra lattice invariance. The best previously existing results in the literature give a notably weaker conclusion using the Sobolev space $H^{d/2+?}({\mathbb{R}}^d)$; our results provide an absolutely sharp improvement with $H^{1/2}({\mathbb{R}}^d)$. Our results are sharp in the sense that $H^{1/2}({\mathbb{R}}^d)$ cannot be replaced by $H^s({\mathbb{R}}^d)$ for any $s<1/2$.     

An explicit counterexample for the Lp-maximal regularity problem

In this short Note we give a self-contained example of a consistent family of holomorphic semigroups $(T_p{(t))}_{t?0}$ such that $(T_p{(t))}_{t?0}$ does not have maximal regularity for $p>2$. This answers negatively the open question whether maximal regularity extrapolates from $L^2$ to the $L^p$-scale.     

Unique continuation for first-order systems with integrable coefficients and applications to elasticity and plasticity

Let $\Omega ?{\mathbb{R}}^N$ be a Lipschitz domain and Γ be a relatively open and non-empty subset of its boundary ?Ω. We show that the solution to the linear first-order system:(1)$\mathrm{?}\zeta =G\zeta ,\zeta {|}_{\Gamma }=0,$ vanishes if $G\in {\mathsf{L}}^1(\Omega ;{\mathbb{R}}^{(N\times N)\times N})$ and $\zeta \in {\mathsf{W}}^{1,1}(\Omega ;{\mathbb{R}}^N)$. In particular, square-integrable solutions ζ of (1) with $G\in {\mathsf{L}}^1\cap {\mathsf{L}}^2(\Omega ;{\mathbb{R}}^{(N\times N)\times N})$ vanish. As a consequence, we prove that:$???:{\mathsf{C}}_{°}^{\infty }(\Omega ,\Gamma ;{\mathbb{R}}^3)\to [0,\infty ),u?6\mathrm{sym}\left(\mathrm{?}u{P}^{?1}\right)6_{{\mathsf{L}}^2(\Omega )}$ is a norm if $P\in {\mathsf{L}}^{\infty }(\Omega ;{\mathbb{R}}^{3\times 3})$ with $\mathrm{Curl}P\in {\mathsf{L}}^p(\Omega ;{\mathbb{R}}^{3\times 3})$, $\mathrm{Curl}{P}^{?1}\in {\mathsf{L}}^q(\Omega ;{\mathbb{R}}^{3\times 3})$ for some $p,q>1$ with $1/p+1/q=1$ as well as $\det P?c^{+}>0$. We also give a new and different proof for the so-called ‘infinitesimal rigid displacement lemma’ in curvilinear coordinates: Let $\Phi \in {\mathsf{H}}^1(\Omega ;{\mathbb{R}}^3)$, $\Omega ?{\mathbb{R}}^3$, satisfy $\mathrm{sym}(\mathrm{?}{\Phi }^{?}\mathrm{?}\Psi )=0$ for some $\Psi \in {\mathsf{W}}^{1,\infty }(\Omega ;{\mathbb{R}}^3)\cap {\mathsf{H}}^2(\Omega ;{\mathbb{R}}^3)$ with $\det \mathrm{?}\Psi ?c^{+}>0$. Then there exists a constant translation vector $a\in {\mathbb{R}}^3$ and a constant skew-symmetric matrix $A\in \mathfrak{so}(3)$, such that $\Phi =A\Psi +a$.     

Construction of solutions via local Pohozaev identities

This paper deals with the following nonlinear elliptic equation

$?\mathrm{\Delta }u+V(|y^{\prime }|,y^{″})u=u^{\frac{N+2}{N?2}},u>0,u\in H^1({\mathbb{R}}^N),$

where $(y^{\prime },y^{″})\in {\mathbb{R}}^2\times {\mathbb{R}}^{N?2}$, $V(|y^{\prime }|,y^{″})$ is a bounded non-negative function in ${\mathbb{R}}^{+}\times {\mathbb{R}}^{N?2}$. By combining a finite reduction argument and local Pohozaev type of identities, we prove that if $N\ge 5$ and $r^{2}V(r,y^{″})$ has a stable critical point $(r_0,y_0^{″})$ with $r_0>0$ and $V(r_0,y_0^{″})>0$, then the above problem has infinitely many solutions. This paper overcomes the difficulty appearing in using the standard reduction method to locate the concentrating points of the solutions.     

Weak version of restriction estimates for spheres and paraboloids in finite fields

We study $L^p-L^r$ restriction estimates for algebraic varieties in d-dimensional vector spaces over finite fields. Unlike the Euclidean case, if the dimension d is even, then it is conjectured that the $L^{(2d+2)/(d+3)}-L^2$ Stein–Tomas restriction result can be improved to the $L^{(2d+4)/(d+4)}-L^2$ estimate for both spheres and paraboloids in finite fields. In this paper we show that the conjectured $L^p-L^2$ restriction estimate holds in the specific case when test functions under consideration are restricted to d-coordinate functions or homogeneous functions of degree zero. To deduce our result, we use the connection between the restriction phenomena for our varieties in d dimensions and those for homogeneous varieties in $(d+1)$ dimensions.     

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 a p-Laplace equation with multiple critical nonlinearities

Using the Mountain-Pass Theorem of Ambrosetti and Rabinowitz we prove that $?{\mathrm{\Delta }}_{p}u?\mu {|x|}^{?p}{u}^{p?1}={|x|}^{?s}{u}^{p^{?}(s)?1}+u^{p^{?}?1}$ admits a positive weak solution in ${\mathbb{R}}^n$ of class $D_1^p({\mathbb{R}}^n)\cap C^1({\mathbb{R}}^n?\{0\})$, whenever $\mu <{\mu }_1$, and ${\mu }_1={[(n?p)/p]}^p$. The technique is based on the existence of extremals of some Hardy–Sobolev type embeddings of independent interest. We also show that if $u\in D_1^p({\mathbb{R}}^n)$ is a weak solution in ${\mathbb{R}}^n$ of $?{\mathrm{\Delta }}_{p}u?\mu {|x|}^{?p}{|u|}^{p?2}u={|x|}^{?s}{|u|}^{p^{?}(s)?2}u+{|u|}^{q?2}u$, then $u\equiv 0$ when either $1<q<p^{?}$, or $q>p^{?}$ and u is also of class $L_{\mathrm{loc}}^{\infty }({\mathbb{R}}^n?\{0\})$.