10,319 research outputs found
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 does not intersect the axis ,
we prove the global well-posedness for this system with axisymmetric initial
data. We first show the growth of the quantity for large time by
taking advantage of characteristic of transport equation. This growing property
together with the horizontal smoothing effect enables us to establish
-estimate of the velocity via the -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 (see \eqref{eq-kl} for the definition) seems be the minimum space
for the gradient vector field ensuring uniqueness of flow. To bridge
this gap, we exploit the space-time estimate about 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
-energy space by developing the new estimate of . 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)
in dimension . We prove that if the solution
is apriorily bounded in the critical Sobolev space, that is, with all if
is an even integer or otherwise, then 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 in
dimension . We show that if the solution is apriorily bounded in
the critical Sobolev space, that is, with , then
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 First, we construct a
global-time forward self-similar solutions to the fractional Navier-Stokes
equations with 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 . In particular, when
, 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
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