Random version of Dvoretzky’s theorem in ℓpn

We study the dependence on $\epsilon$ in the critical dimension $k(n,p,\epsilon )$ for which one can find random sections of the ${?}_p^n$-ball which are $(1+\epsilon )$-spherical. We give lower (and upper) estimates for $k(n,p,\epsilon )$ for all eligible values $p$ and $\epsilon$ as $n\to \infty$, which agree with the sharp estimates for the extreme values $p=1$ and $p=\infty$. Toward this end, we provide tight bounds for the Gaussian concentration of the ${?}_p$-norm.     

On uniqueness of stationary solutions to the Navier–Stokes equations in exterior domains

The aim of this paper is to prove a uniqueness criterion for solutions to the stationary Navier–Stokes equation in 3-dimensional exterior domains within the class $u\in L_{3,\infty }$ with $?u\in L_{3/2,\infty }$, where $L_{3,\infty }$ and $L_{3/2,\infty }$ are the Lorentz spaces. Our criterion asserts that if $u$ and $v$ are the solutions, $u$ is small in $L_{3,\infty }$ and $u,v\in L_p$ for some $p>3$, then $u=v$. The proof is based on analysis of the dual equation with the aid of the bootstrap argument.     

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.     

Subspace partitions of Fqn containing direct sums

Let $q$ be a prime power and $n$ be a positive integer. A subspace partition of $V={\mathbb{F}}_q^n$, the vector space of dimension $n$ over ${\mathbb{F}}_q$, is a collection $\Pi$ of subspaces of $V$ such that each nonzero vector of $V$ is contained in exactly one subspace in $\Pi$; the multiset of dimensions of subspaces in $\Pi$ is then called a Gaussian partition of $V$. We say that $\Pi$contains a direct sum if there exist subspaces $W_1,\dots ,W_k\in \Pi$ such that $W_1\oplus ?\oplus W_k=V$. In this paper, we study the problem of classifying the subspace partitions that contain a direct sum. In particular, given integers $a_1$ and $a_2$ with $n>a_1>a_2\ge 1$, our main theorem shows that if $\Pi$ is a subspace partition of ${\mathbb{F}}_q^n$ with $m_i$ subspaces of dimension $a_i$ for $i=1,2$, then $\Pi$ contains a direct sum when $a_1{x}_1+a_2{x}_2=n$ has a solution $(x_1,x_2)$ for some integers $x_1,x_2\ge 0$ and $m_2$ belongs to the union $I$ of two natural intervals. The lower bound of $I$ captures all subspace partitions with dimensions in $\{a_1,a_2\}$ that are currently known to exist. Moreover, we show the existence of infinite classes of subspace partitions without a direct sum when $m_2?I$ or when the condition on the existence of a nonnegative integral solution $(x_1,x_2)$ is not satisfied. We further conjecture that this theorem can be extended to any number of distinct dimensions, where the number of subspaces in each dimension has appropriate bounds. These results offer further evidence of the natural combinatorial relationship between Gaussian and integer partitions (when $q\to 1$) as well as subspace and set partitions.     

Infinite Fujita exponents for nonlinear diffusion equations with localized sources

This paper deals with Cauchy problem to nonlinear diffusion $u_t=\Delta u^m+{\lambda }_1{u}^{p_1}(x,t)+{\lambda }_2{u}^{p_2}(x^1\left(t\right),t)$ with $m\ge 1$, $p_i,{\lambda }_i\ge 0$ ($i=1,2$) and $x^1\left(t\right)$ Hölder continuous. A new phenomenon is observed that the critical Fujita exponent $p_c=+\infty$ whenever ${\lambda }_2>0$. More precisely, the solution blows up under any nontrivial and nonnegative initial data for all $p=max\{p_1,p_2\}\in (1,+\infty )$. This result is then extended to a coupled system with localized sources as well as the cases with other nonlinearities.     

Lp solutions of reflected BSDEs under monotonicity condition

We prove existence and uniqueness of ${\mathbb{L}}^p$ solutions, $p\in [1,2]$, of reflected backward stochastic differential equations with $p$-integrable data and generators satisfying the monotonicity condition. We also show that the solution may be approximated by the penalization method. Our results are new even in the classical case $p=2$.     

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.     

The zero-one law for a complete orthonormal system

A complete orthonormal system of functions $\Theta ={\left\{{\theta }_n\right\}}_{n=1}^{\infty }\text{,}{\theta }_n\in L_{[0\text{,}1]}^{\infty }$ is constructed such that ${\sum }_{n=1}^{\infty }{a}_n{\theta }_n$ converges almost everywhere on $[0\text{,}1]$ if ${\left\{a_n\right\}}_{n=1}^{\infty }\in l^2$ and ${\sum }_{n=1}^{\infty }{a}_n{\theta }_n$ diverges a.e. for any ${\left\{a_n\right\}}_{n=1}^{\infty }?l^2$. We also show that for any complete ONS ${\left\{f_n\right\}}_{n=1}^{\infty }$ of functions defined on $[0\text{,}1]$ there exists a fixed non decreasing subsequence ${\left\{n_k\right\}}_{k=1}^{\infty }$ of natural numbers such that for any $f\in L_{[0\text{,}1]}^0$ and some sequence of coefficients ${\left\{b_n\right\}}_{n=1}^{\infty }$,

$\sum _{n=1}^{n_k}{b}_n{f}_n\to f\text{a.e. when}k\to \infty \text{.}$

To cite this article: K. Kazarian, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Singular discrete and continuous mixed boundary value problems

For each $n\in \mathbb{N}$, $n\ge 2$, we prove the existence of a solution $(u_0,\dots ,u_n)\in {\mathbb{R}}^{n+1}$ of the singular discrete problem $\frac{1}{h^2}{\Delta }^2{u}_{k?1}+f(t_k,u_k)=0\text{,}k=1,\dots ,n?1\text{,}$$\Delta u_0=0\text{,}{u}_n=0\text{,}$ where $u_k>0$ for $k=0,\dots ,n?1$. Here $T\in (0,\infty )$, $h=\frac{T}{n}$, $t_k=hk$, $f(t,x):[0,T]\times (0,\infty )\to \mathbb{R}$ is continuous and has a singularity at $x=0$. We prove that for $n\to \infty$ the sequence of solutions of the above discrete problems converges to a solution $y$ of the corresponding continuous boundary value problem $y^{″}\left(t\right)+f(t,y\left(t\right))=0\text{,}$$y^{\prime }\left(0\right)=0\text{,}y\left(T\right)=0\text{.}$     

On some cycles in linearized Wenger graphs

Let ${\mathbb{F}}_q$ be the finite field of $q$ elements for prime power $q$ and let $p$ be the character of ${\mathbb{F}}_q$. For any positive integer $m$, the linearized Wenger graph $L_m\left(q\right)$ is defined as follows: it is a bipartite graph with the vertex partitions being two copies of the $(m+1)$-dimensional vector space ${\mathbb{F}}_q^{m+1}$, and two vertices $p=\left(p\left(1\right),\dots ,p(m+1)\right)$ and $l=\left[l\left(1\right),\dots ,l(m+1)\right]$ being adjacent if $p\left(i\right)+l\left(i\right)=p\left(1\right){\left(l\left(1\right)\right)}^{p^{i?2}}$, for all $i=2,3,\dots ,m+1$. In this paper, we show that for any positive integers $m$ and $k$ with $3\le k\le p^2$, $L_m\left(q\right)$ contains even cycles of length $2k$ which is an open problem put forward by Cao et al. (2015).