Hölder estimates for fractional parabolic equations with critical divergence free drifts

This work focuses on drift-diffusion equations with fractional dissipation ${(?\mathrm{\Delta })}^{\alpha }$ in the regime $\alpha \in (1/2,1)$. Our main result is an a priori Hölder estimate on smooth solutions to the Cauchy problem, starting from initial data with finite energy. We prove that for some $\beta \in (0,1)$, the $C^{\beta }$ norm of the solution depends only on the size of the drift in critical spaces of the form $L_t^q({\mathrm{BMO}}_x^{?\gamma })$ with $q>2$ and $\gamma \in (0,2\alpha ?1]$, along with the $L_x^2$ norm of the initial datum. The proof uses the Caffarelli/Vasseur variant of De Giorgi's method for non-local equations.     

Local kinetic energy and singularities of the incompressible Navier–Stokes equations

We study the partial regularity problem of the incompressible Navier–Stokes equations. A reverse Hölder inequality of velocity gradient with increasing support is obtained under the condition that a scaled functional corresponding the local kinetic energy is uniformly bounded. As an application, we give a new bound for the Hausdorff dimension and the Minkowski dimension of singular set when weak solutions v belong to $L^{\infty }(0,T;L^{3,w}({\mathbb{R}}^3))$ where $L^{3,w}({\mathbb{R}}^3)$ denotes the standard weak Lebesgue space.     

On Birman's sequence of Hardy–Rellich-type inequalities

In 1961, Birman proved a sequence of inequalities $\{I_n\}$, for $n\in \mathbb{N}$, valid for functions in $C_0^n((0,\infty ))?L^2((0,\infty ))$. In particular, $I_1$ is the classical (integral) Hardy inequality and $I_2$ is the well-known Rellich inequality. In this paper, we give a proof of this sequence of inequalities valid on a certain Hilbert space $H_n([0,\infty ))$ of functions defined on $[0,\infty )$. Moreover, $f\in H_n([0,\infty ))$ implies $f^{\prime }\in H_{n?1}([0,\infty ))$; as a consequence of this inclusion, we see that the classical Hardy inequality implies each of the inequalities in Birman's sequence. We also show that for any finite $b>0$, these inequalities hold on the standard Sobolev space $H_0^n((0,b))$. Furthermore, in all cases, the Birman constants ${[(2n?1)!!]}^2/2^{2n}$ in these inequalities are sharp and the only function that gives equality in any of these inequalities is the trivial function in $L^2((0,\infty ))$ (resp., $L^2((0,b))$). We also show that these Birman constants are related to the norm of a generalized continuous Cesàro averaging operator whose spectral properties we determine in detail.     

On the positive radial solutions of a class of singular semilinear elliptic equations

In this paper, we consider the following elliptic equation(0.1)$\mathit{div}\left(A\right(|x|\left)\mathrm{?}u\right)+B(|x|)u^p=0\text{in}{\mathbb{R}}^n,$ where $p>1$, $n?3$, $A(|x|)>0$ is differentiable in ${\mathbb{R}}^n?\{0\}$ and $B(|x|)$ is a given nonnegative Hölder continuous function in ${\mathbb{R}}^n?\{0\}$. The asymptotic behavior at infinity and structure of separation property of positive radial solutions with different initial data for (0.1) are discussed. Moreover, the existence and separation property of infinitely many positive solutions for Hardy equation and an equation related to Caffarelli–Kohn–Nirenberg inequality are obtained respectively, as special cases.     

On a discrete bilinear singular operator

We prove that for a large class of functions P and Q, the discrete bilinear operator $T_{P,Q}(f,g)(n)={\sum }_{m\in \mathbb{Z}?\{0\}}f(n?P(m))g(n?Q(m))\frac{1}{m}$ is bounded from $l^2\times l^2$ into $l^{1+?,\infty }$ for any $?\in (0,1]$.     

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.     

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 $L^p([0,\pi ];dx)$, $p\in (1,\infty )$.     

Approximations in Sobolev spaces by prolate spheroidal wave functions

Recently, there is a growing interest in the spectral approximation by the Prolate Spheroidal Wave Functions (PSWFs) ${\psi }_{n,c},c>0$. This is due to the promising new contributions of these functions in various classical as well as emerging applications from Signal Processing, Geophysics, Numerical Analysis, etc. The PSWFs form a basis with remarkable properties not only for the space of band-limited functions with bandwidth c, but also for the Sobolev space $H^s([?1,1])$. The quality of the spectral approximation and the choice of the parameter c when approximating a function in $H^s([?1,1])$ by its truncated PSWFs series expansion, are the main issues. By considering a function $f\in H^s([?1,1])$ as the restriction to $[?1,1]$ of an almost time-limited and band-limited function, we try to give satisfactory answers to these two issues. Also, we illustrate the different results of this work by some numerical examples.     

The nonlinear complexity of level sequences over Z/(4)

For any sequence $\underline{a}$ over $\mathrm{Z}/(2^2)$, there is an unique 2-adic expansion $\underline{a}={\underline{a}}_0+{\underline{a}}_1·2$, where ${\underline{a}}_0$ and ${\underline{a}}_1$ are sequences over $\{0,1\}$ and can be regarded as sequences over the binary field $\mathit{GF}(2)$ naturally. We call ${\underline{a}}_0$ and ${\underline{a}}_1$ the level sequences of $\underline{a}$. Let $f(x)$ be a primitive polynomial of degree $n$ over $\mathrm{Z}/(2^2)$, and $\underline{a}$ be a primitive sequence generated by $f(x)$. In this paper, we discuss how many bits of ${\underline{a}}_1$ can determine uniquely the original primitive sequence $\underline{a}$. This issue is equivalent with one to estimate the whole nonlinear complexity, $\mathit{NL}(f(x),2^2)$, of all level sequences of $f(x)$. We prove that $4n$ is a tight upper bound of $\mathit{NL}(f(x),2^2)$ if $f(x)(\mathrm{mod}2)$ is a primitive trinomial over $\mathit{GF}(2)$. Moreover, the experimental result shows that $\mathit{NL}(f(x),2^2)$ varies around $4n$ if $f(x)(\mathrm{mod}2)$ is a primitive polynomial over $\mathit{GF}(2)$. From this result, we can deduce that $\mathit{NL}(f(x),2^2)$ is much smaller than $L(f(x),2^2)$, where $L(f(x),2^2)$ is the linear complexity of level sequences of $f(x)$.     

Robust local Hölder rigidity of circle maps with breaks

We prove that, for every $\epsilon \in (0,1)$, every two $C^{2+\alpha }$-smooth $(\alpha >0)$ circle diffeomorphisms with a break point, i.e. circle diffeomorphisms with a single singular point where the derivative has a jump discontinuity, with the same irrational rotation number $\rho \in (0,1)$ and the same size of the break $c\in {\mathbb{R}}_{+}\backslash \{1\}$, are conjugate to each other via a conjugacy which is $(1?\epsilon )$-Hölder continuous at the break points. An analogous result does not hold for circle diffeomorphisms even when they are analytic.     

Existence and convexity of local solutions to degenerate Hessian equations

In this work, we prove the existence of convex solutions to the following k-Hessian equation

$S_k[u]=K(y)g(y,u,Du)$

in the neighborhood of a point $(y_0,u_0,p_0)\in {\mathbb{R}}^n\times \mathbb{R}\times {\mathbb{R}}^n$, where $g\in C^{\infty },g(y_0,u_0,p_0)>0$, $K\in C^{\infty }$ is nonnegative near $y_0$, $K(y_0)=0$ and $\text{Rank}(D_y^{2}K)(y_0)\ge n?k+1$.     

The extended Bloch groups of biquadratic and dihedral number fields

In this paper, we study the Galois action on the extended Bloch groups of biquadratic and dihedral number fields. We prove that if F is a biquadratic number field, then the index $Q_2(F)$ in Browkin and Gangl's formulas on the Brauer–Kuroda relation can only be 1 or 2. This is exactly what Browkin and Gangl predicted in their paper. Moreover we give the explicit criteria for $Q_2(F)=1$ or 2 in terms of the Tate kernels. We also prove that $Q_2(F)=1$ or p for any dihedral extension $F/\mathbb{Q}$ whose Galois group is the dihedral group of order 2p, where p is an odd prime.