Explicit expressions for a family of Bell polynomials and derivatives of some functions

In the paper, the authors first inductively establish explicit formulas for derivatives of the arc sine function, then derive from these explicit formulas explicit expressions for a family of Bell polynomials related to the square function, and finally apply these explicit expressions to find explicit formulas for derivatives of some elementary functions.Comment: 16 page

Metamaterials Mimicking Dynamic Spacetime, D-brane and Noncommutativity in String Theory

We propose an executable scheme to mimic the expanding cosmos in 1+2 dimensions in laboratory. Furthermore, we develop a general procedure to use nonlinear metamaterials to mimic D-brane and noncommutativity in string theory.Comment: 15 pages, 2 figure

Global well-posedness for axisymmetric Boussinesq system with horizontal viscosity

In this paper, we are concerned with the tridimensional anisotropic Boussinesq equations which can be described by {equation*} {{array}{ll} (\partial_{t}+u\cdot\nabla)u-\kappa\Delta_{h} u+\nabla \Pi=\rho e_{3},\quad(t,x)\in\mathbb{R}^{+}\times\mathbb{R}^{3}, (\partial_{t}+u\cdot\nabla)\rho=0, \text{div}u=0. {array}. {equation*} Under the assumption that the support of the axisymmetric initial data $\rho_{0}(r,z)$ does not intersect the axis $(Oz)$, we prove the global well-posedness for this system with axisymmetric initial data. We first show the growth of the quantity $\frac\rho r$ for large time by taking advantage of characteristic of transport equation. This growing property together with the horizontal smoothing effect enables us to establish $H^1$-estimate of the velocity via the $L^2$-energy estimate of velocity and the Maximum principle of density. Based on this, we further establish the estimate for the quantity \|\omega(t)\|_{\sqrt{\mathbb{L}}}:=\sup_{2\leq p<\infty}\frac{\norm{\omega(t)}_{L^p(\mathbb{R}^3)}}{\sqrt{p}}<\infty which implies \|\nabla u(t)\|_{\mathbb{L}^{3/2}}:=\sup_{2\leq p<\infty}\frac{\norm{\nabla u(t)}_{L^p(\mathbb{R}^3)}}{p\sqrt{p}}<\infty. However, this regularity for the flow admits forbidden singularity since $\mathbb{L}$ (see \eqref{eq-kl} for the definition) seems be the minimum space for the gradient vector field $u(x,t)$ ensuring uniqueness of flow. To bridge this gap, we exploit the space-time estimate about $\sup_{2\leq p<\infty}\int_0^t\frac{\|\nabla u(\tau)\|_{L^p(\mathbb{R}^3)}}{\sqrt{p}}\mathrm{d}\tau<\infty$ by making good use of the horizontal smoothing effect and micro-local techniques. The global well-posedness for the large initial data is achieved by establishing a new type space-time logarithmic inequality.Comment: 32pages. arXiv admin note: text overlap with arXiv:0908.0894 by other author

Dynamics of nonlinear wave equations

In this lecture, we will survey the study of dynamics of the nonlinear wave equation in recent years. We refer to some lecture notes including such as C. Kenig \cite{Kenig01,Kenig02}, C. Kenig and F. Merle \cite{KM1} J. Shatah and M. Struwe \cite{SS98}, and C. Sogge \cite{sogge:wave} etc. This lecture was written for LIASFMA School and Workshop on Harmonic Analysis and Wave Equations in Fudan universty.Comment: 80pages. This lecture was written for LIASFMA School on Harmonic Analysis and Wave Equations in Fudan universty(2017). arXiv admin note: text overlap with arXiv:1506.00788, arXiv:0710.5934, arXiv:1407.4525, arXiv:1601.01871, arXiv:1301.4835, 1509.03331, arXiv:1010.3799, arXiv:1201.3258, arXiv:0911.4534, arXiv:1411.7905, by other author

Global well-posedness for the two-dimensional Maxwell-Navier-Stokes equations

In this paper, we investigate Cauchy problem of the two-dimensional full Maxwell-Navier-Stokes system, and prove the global-in-time existence and uniqueness of solution in the borderline space which is very close to $L^2$-energy space by developing the new estimate of $\sup_{j\in\mathbb Z} 2^{2j} \int_0^t \sum_{k\in\mathbb{Z}^2} \big\| \sqrt{\phi_{i,k}} u(\tau) \big\|^2_{L^2(\mathbb{R}^2)} \text{d}\tau < \infty$. This solves the open problem in the framework of borderline space purposed by Masmoudi in \cite{Masmoudi-10}.Comment: 46page

Scattering theory for the defocusing fourth-order Schr\"odinger equation

In this paper, we study the global well-posedness and scattering theory for the defocusing fourth-order nonlinear Schr\"odinger equation (FNLS) $iu_t+\Delta^2 u+|u|^pu=0$ in dimension $d\geq9$. We prove that if the solution $u$ is apriorily bounded in the critical Sobolev space, that is, $u\in L_t^\infty(I;\dot H^{s_c}_x(\R^d))$ with all $s_c:=\frac{d}2-\frac4p\geq1$ if $p$ is an even integer or $s_c\in[1,2+p)$ otherwise, then $u$ is global and scatters. The impetus to consider this problem stems from a series of recent works for the energy-supercritical and energy-subcritical nonlinear Schr\"odinger equation (NLS) and nonlinear wave equation (NLW). We will give a uniform way to treat the energy-subcritical, energy-critical and energy-supercritical FNLS, where we utilize the strategy derived from concentration compactness ideas to show that the proof of the global well-posedness and scattering is reduced to exclude the existence of three scenarios: finite time blowup; soliton-like solution and low to high frequency cascade. Making use of the No-waste Duhamel formula, we deduce that the energy or mass of the finite time blow-up solution is zero and so get a contradiction. Finally, we adopt the double Duhamel trick, the interaction Morawetz estimate and interpolation to kill the last two scenarios.Comment: 40pages. arXiv admin note: text overlap with arXiv:0812.2084 by other author

Scattering theory for energy-supercritical Klein-Gordon equation

In this paper, we consider the question of the global well-posedness and scattering for the cubic Klein-Gordon equation $u_{tt}-\Delta u+u+|u|^2u=0$ in dimension $d\geq5$. We show that if the solution $u$ is apriorily bounded in the critical Sobolev space, that is, $(u, u_t)\in L_t^\infty(I; H^{s_c}_x(\R^d)\times H_x^{s_c-1}(\R^d))$ with $s_c:=\frac{d}2-1>1$, then $u$ is global and scatters. The impetus to consider this problem stems from a series of recent works for the energy-supercritical nonlinear wave equation and nonlinear Schr\"odinger equation. However, the scaling invariance is broken in the Klein-Gordon equation. We will utilize the concentration compactness ideas to show that the proof of the global well-posedness and scattering is reduced to disprove the existence of the scenario: soliton-like solutions. And such solutions are precluded by making use of the Morawetz inequality, finite speed of propagation and concentration of potential energy.Comment: 24page

Thermodynamic approach to field equations in Lovelock gravity and f(R) gravity revisited

The first law of thermodynamics at black hole horizons is known to be obtainable from the gravitational field equations. A recent study claims that the contributions at inner horizons should be considered in order to give the conventional first law of black hole thermodynamics. Following this method, we revisit the thermodynamic aspects of field equations in the Lovelock gravity and f(R) gravity by focusing on two typical classes of charged black holes in the two theories.Comment: 10 pages, no figures; v2: clarifications and references added, to appear in Int. J. Mod. Phys.

Evaluating the Effectiveness of Health Awareness Events by Google Search Frequency

Over two hundreds health awareness events take place in the United States in order to raise attention and educate the public about diseases. It would be informative and instructive for the organization to know the impact of these events, although such information could be difficult to measure. Here 46 events are selected and their data from 2004 to 2017 are downloaded from Google Trend(GT). We investigate whether the events effectively attract the public attention by increasing the search frequencies of certain keywords which we call queries. Three statistical methods including Transfer Function Noise modeling, Wilcoxon Rank Sum test, and Binomial inference are conducted on 46 GT data sets. Our study show that 10 health awareness events are effective with evidence of a significant increase in search frequencies in the event months, and 28 events are ineffective, with the rest being classified as unclear

Forward self-similar solutions of the fractional Navier-Stokes Equations

We study forward self-similar solutions to the 3-D Navier-Stokes equations with the fractional diffusion $(-\Delta)^{\alpha}.$ First, we construct a global-time forward self-similar solutions to the fractional Navier-Stokes equations with $5/6<\alpha\leq1$ for arbitrarily large self-similar initial data by making use of the so called blow-up argument. Moreover, we prove that this solution is smooth in $\mathbb R^3\times (0,+\infty)$. In particular, when $\alpha=1$, we prove that the solution constructed by Korobkov-Tsai [Anal. PDE 9 (2016), 1811-1827] satisfies the decay estimate by establishing regularity of solution for the corresponding elliptic system, which implies this solution has the same properties as a solution which was constructed in [Jia and \v{S}ver\'{a}k, Invent. Math. 196 (2014), 233-265].Comment: 46page