The Cauchy problem for a generalized Camassa–Holm equation

In this paper, we are concerned with the Cauchy problem of the generalized Camassa–Holm equation. Using a Galerkin-type approximation scheme, it is shown that this equation is well-posed in Sobolev spaces $H^s$, $s>3/2$ for both the periodic and the nonperiodic case in the sense of Hadamard. That is, the data-to-solution map is continuous. Furthermore, it is proved that this dependence is sharp by showing that the solution map is not uniformly continuous. The nonuniform dependence is proved using the method of approximate solutions and well-posedness estimates. Moreover, it is shown that the solution map for the generalized Camassa–Holm equation is Hölder continuous in $H^r$-topology. Finally, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time.     

Global existence and time decay of the non-cutoff Boltzmann equation with hard potential

This paper is concerned with the Cauchy problem on the Boltzmann equation without angular cutoff assumption for hard potential in the whole space. When the initial data is a small perturbation of a global Maxwellian, the global existence of solution to this problem is proved in unweighted Sobolev spaces $H^N\left({\mathbb{R}}_{x,v}^6\right)$ with $N\ge 2$. But if we want to obtain the optimal temporal decay estimates, we need to add the velocity weight function, in this case the global existence and the optimal temporal decay estimate of the Boltzmann equation are all established. Meanwhile, we further gain a more accurate energy estimate, which can guarantee the validity of the assumption in Chen et al. (0000).     

Dilations of semigroups on von Neumann algebras and noncommutative Lp-spaces

We prove that any weak* continuous semigroup ${(T_t)}_{t?0}$ of factorizable Markov maps acting on a von Neumann algebra M equipped with a normal faithful state can be dilated by a group of Markov ?-automorphisms analogous to the case of a single factorizable Markov operator, which is an optimal result. We also give a version of this result for strongly continuous semigroups of operators acting on noncommutative ${\mathrm{L}}^p$-spaces and examples of semigroups to which the results of this paper can be applied. Our results imply the boundedness of the McIntosh's ${\mathrm{H}}^{\infty }$ functional calculus of the generators of these semigroups on the associated noncommutative ${\mathrm{L}}^p$-spaces generalising some previous work from Junge, Le Merdy and Xu. Finally, we also give concrete dilations for Poisson semigroups which are even new in the case of ${\mathbb{R}}^n$.     

Signed graphs and the freeness of the Weyl subarrangements of type Bℓ

A Weyl arrangement is the hyperplane arrangement defined by a root system. Saito proved that every Weyl arrangement is free. The Weyl subarrangements of type $A_{\ell }$ are represented by simple graphs. Stanley gave a characterization of freeness for this type of arrangements in terms of their graph. In addition, the Weyl subarrangements of type $B_{\ell }$ can be represented by signed graphs. A characterization of freeness for them is not known. However, characterizations of freeness for a few restricted classes are known. For instance, Edelman and Reiner characterized the freeness of the arrangements between type $A_{\ell -1}$ and type $B_{\ell }$. In this paper, we give a characterization of the freeness and supersolvability of the Weyl subarrangements of type $B_{\ell }$ under certain assumption.     

Exponential sums formed with the Möbius function

Let $\mu \left(n\right)$ be the Möbius function and $x$ real. In this paper, we investigated the best possible estimates for the sum ${\sum }_{n\le x}\mu \left(n\right)e\left(n^k\theta \right)$ under the weak Generalized Riemann Hypothesis. A similar result also holds for the Liouville function $\lambda \left(n\right)$.     

Lp Sobolev regularity of averaging operators over hypersurfaces and the Newton polyhedron

$L^p$ to $L_{\beta }^p$ boundedness results are proven for translation invariant averaging operators over hypersurfaces in Euclidean space. The operators can either be Radon transforms or averaging operators with multiparameter fractional integral kernel. In many cases, the amount $\beta >0$ of smoothing proven is optimal up to endpoints, and in such situations this amount of smoothing can be computed explicitly through the use of appropriate Newton polyhedra.     

On the local geometry of graphs in terms of their spectra

In this paper we consider the relation between the spectrum and the number of short cycles in large graphs. Suppose $G_1,G_2,G_3,\dots$ is a sequence of finite and connected graphs that share a common universal cover $T$ and such that the proportion of eigenvalues of $G_n$ that lie within the support of the spectrum of $T$ tends to 1 in the large $n$ limit. This is a weak notion of being Ramanujan. We prove such a sequence of graphs is asymptotically locally tree-like. This is deduced by way of an analogous theorem proved for certain infinite sofic graphs and unimodular networks, which extends results for regular graphs and certain infinite Cayley graphs.     

Quasi-pure E0-semigroups

We introduce a class of $E_0$-semigroups that is broader and more flexible than the class of pure $E_0$-semigroups, and characterize the states of the spectral $C^{?}$-algebra $C^{?}(E)$ of a product system $E={\{E_t\}}_{t>0}$ that give rise to them.     

On the inhomogeneous biharmonic nonlinear Schrödinger equation: Local,global and stability results

We consider the inhomogeneous biharmonic nonlinear Schrödinger equation (IBNLS) $i{u}_t+{\Delta }^{2}u+\lambda {\left|x\right|}^{-b}{\left|u\right|}^{\alpha }u=0,$ where $\lambda =\pm 1$ and $\alpha$, $b>0$. We show local and global well-posedness in $H^s\left({\mathbb{R}}^N\right)$ in the $H^s$-subcritical case, with $s=0,2$. Moreover, we prove a stability result in $H^2\left({\mathbb{R}}^N\right)$, in the mass-supercritical and energy-subcritical case. The fundamental tools to prove these results are the standard Strichartz estimates related to the linear problem.