Strichartz Estimates for Charge Transfer Models

In this note, we prove Strichartz estimates for scattering states of scalar charge transfer models in $\mathbb{R}^{3}$. Based on the idea of the proof of Strichartz estimates which follows \cite{CM,RSS}, we also show the energy of the whole evolution is bounded independently of time without using the phase space method, for example, in \cite{Graf}. One can easily generalize our argument to $\mathbb{R}^{n}$ for $n\geq3$. Finally, in the last section, we discuss the extension of these results to matrix charge transfer models in $\mathbb{R}^{3}$.Comment: 26 pages, this is a revised version based on the comments of an anonymous refere

Multisolitons for the defocusing energy critical wave equation with potentials

We construct multisoliton solutions to the defocusing energy critical wave equation with potentials in $\mathbb{R}^{3}$ based on regular and reversed Strichartz estimates developed in \cite{GC3} for wave equations with charge transfer Hamiltonians. We also show the asymptotic stability of multisoliton solutions. The multisoliton structures with both stable and unstable solitons are covered. Since each soliton decays slowly with rate $\frac{1}{\left\langle x\right\rangle }$, the interactions among the solitons are strong. Some reversed Strichartz estimates and local decay estimates for the charge transfer model are established to handle strong interactions.Comment: 33 page

Non-divergence Parabolic Equations of Second Order with Critical Drift in Morrey Spaces

We consider uniformly parabolic equations and inequalities of second order in the non-divergence form with drift $$-u_{t}+Lu=-u_{t}+\sum_{ij}a_{ij}D_{ij}u+\sum b_{i}D_{i}u=0\,(\geq0,\,\leq0)$$ in some domain $\Omega\subset \mathbb{R}^{n+1}$. We prove a variant of Aleksandrov-Bakelman-Pucci-Krylov-Tso estimate with $L^{p}$ norm of the inhomogeneous term for some number $p<n+1$. Based on it, we derive the growth theorems and the interior Harnack inequality. In this paper, we will only assume the drift $b$ is in certain Morrey spaces defined below which are critical under the parabolic scaling but not necessarily to be bounded. This is a continuation of the work in \cite{GC}.Comment: 23 pages. arXiv admin note: substantial text overlap with arXiv:1511.0121

Non-divergence Parabolic Equations of Second Order with Critical Drift in Lebesgue Spaces

We consider uniformly parabolic equations and inequalities of second order in the non-divergence form with drift $$-u_{t}+Lu=-u_{t}+\sum_{ij}a_{ij}D_{ij}u+\sum b_{i}D_{i}u=0\,(\geq0,\,\leq0)$$ in some domain $Q\subset \mathbb{R}^{n+1}$. We prove growth theorems and the interior Harnack inequality as the main results. In this paper, we will only assume the drift $b$ is in certain Lebesgue spaces which are critical under the parabolic scaling but not necessarily to be bounded. In the last section, some applications of the interior Harnack inequality are presented. In particular, we show there is a "universal" spectral gap for the associated elliptic operator. The counterpart for uniformly elliptic equations of second order in non-divergence form is shown in \cite{S10}.Comment: 30 pages, the introduction is revise

Cosmological constraints on ultra-light axion fields

Ultra-light axions (ULAs) with mass less than 10^-20 eV have interesting behaviors that may contribute to either dark energy or dark matter at different epochs of the Universe. Its properties can be explored by cosmological observations, such as expansion history of the Universe, cosmic large-scale structure, cosmic microwave background, etc. In this work, we study the ULAs with a mass around 10^-33 eV, which means the ULA field still rolls slowly at present with the equation of state w=-1 as dark energy. In order to investigate the mass and other properties of this kind of ULA field, we adopt the measurements of Type Ia supernova (SN Ia), baryon acoustic oscillation (BAO), and Hubble parameter H(z). The Markov Chain Monte Carlo (MCMC) technique is employed to perform the constraints on the parameters. Finally, by exploring four cases of the model, we find that the mass of this ULA field is about 3x10^-33 eV if assuming the initial axion field phi_i=M_pl. We also investigate a general case by assuming phi_i< M_pl and find that the fitting results of phi_i/M_pl are consistent with or close to 1 for the datasets we use.Comment: 10 pages, 4 figures, 5 tables. Accepted by Research in Astronomy and Astrophysic

A Low-rank Tensor Dictionary Learning Method for Multi-spectral Images Denoising

As a 3-order tensor, a multi-spectral image (MSI) has dozens of spectral bands, which can deliver more information for real scenes. However, real MSIs are often corrupted by noises in the sensing process, which will further deteriorate the performance of higher-level classification and recognition tasks. In this paper, we propose a Low-rank Tensor Dictionary Learning (LTDL) method for MSI denoising. Firstly, we extract blocks from the MSI and cluster them into groups. Then instead of using the exactly low-rank model, we consider a nearly low-rank approximation, which is closer to the latent low-rank structure of the clean groups of real MSIs. In addition, we propose to learn an spatial dictionary and an spectral dictionary, which contain the spatial features and spectral features respectively of the whole MSI and are shared among different groups. Hence the LTDL method utilizes both the latent low-rank prior of each group and the correlation of different groups via the shared dictionaries. Experiments on synthetic data validate the effectiveness of dictionary learning by the LTDL. Experiments on real MSIs demonstrate the superior denoising performance of the proposed method in comparison to state-of-the-art methods

On Finite Block-Length Quantization Distortion

We investigate the upper and lower bounds on the quantization distortions for independent and identically distributed sources in the finite block-length regime. Based on the convex optimization framework of the rate-distortion theory, we derive a lower bound on the quantization distortion under finite block-length, which is shown to be greater than the asymptotic distortion given by the rate-distortion theory. We also derive two upper bounds on the quantization distortion based on random quantization codebooks, which can achieve any distortion above the asymptotic one. Moreover, we apply the new upper and lower bounds to two types of sources, the discrete binary symmetric source and the continuous Gaussian source. For the binary symmetric source, we obtain the closed-form expressions of the upper and lower bounds. For the Gaussian source, we propose a computational tractable method to numerically compute the upper and lower bounds, for both bounded and unbounded quantization codebooks.Numerical results show that the gap between the upper and lower bounds is small for reasonable block length and hence the bounds are tight

On Second Order Elliptic and Parabolic Equations of Mixed Type

It is known that solutions to second order uniformly elliptic and parabolic equations, either in divergence or nondivergence (general) form, are H\"{o}lder continuous and satisfy the interior Harnack inequality. We show that even in the one-dimensional case ($x\in R^1$), these properties are not preserved for equations of mixed divergence-nondivergence structure: for elliptic equations. \begin{equation*} D_i(a^1_{ij}D_ju)+a^2_{ij}D_{ij}u=0, \end{equation*} and parabolic equations \begin{equation*} p\partial_t u=D_i(a_{ij}D_ju), \end{equation*} where $p=p(t,x)$ is a bounded strictly positive function. The H\"{o}lder continuity and Harnack inequality are known if $p$ does not depend either on $t$ or on $x$. We essentially use homogenization techniques in our construction. Bibliography: 23 titles.Comment: 16 page

Long-time asymptotics of the modified KdV equation in weighted Sobolev spaces

The long time behavior of solutions to the defocusing modified Korteweg-de vries (MKdV) equation is established for initial conditions in some weighted Sobolev spaces. Our approach is based on the nonlinear steepest descent method of Deift and Zhou and its reformulation by Dieng and McLaughlin through $\overline{\partial}$-derivatives. To extend the asymptotics to solutions with initial data in lower regularity spaces, we apply a global approximation via PDE techniques.Comment: 51 page

The limit on w′w' for tachyon dark energy

We get the same degeneracy relation between $w_0$ and $w_a$ for the tachyon fields as for quintessence and phantom fields. Our results show that the dynamics of scalar fields with different origins becomes indistinguishable when the equation of state parameter $w$ does not deviate too far away from -1, and the time variation $w'$ satisfies the same bound for the same class of models. For the tachyon fields, a limit on $w'$ exists due to the Hubble damping and we derived the generic bounds on $w'$ for different classes of models. We may distinguish different models in the phase plane of $w-w'$. The current constraints on $w$ and $w'$ are consistent with all classes of models. We need to improve the constraint on $w'$ by 50% to distinguish different models.Comment: use revtex, 2 figure