International Conference on Partial Differential Equations
Universal Inequality and Upper Bounds of Eigenvalues for Fractional
Laplacian
Hua Chen (•z)
Wuhan University, China
e-mail: chenhua@whu.edu.cn
Abstract
In this talk, we shall mention some recent results on the estimates of Dirichlet eigenvalues for a class of elliptic operatos and degnerate elliptic operators. In particular we
shall talk some new results for the eigenvalues estimate of fractional Laplacian.
Fourth-order nonlinear Schrödinger equation in the critical case
Nakao Hayashi
Department of Mathematics, Graduate School of Science, Osaka University, Osaka,
Tokyonaka 560-0043, Japan
e-mail: nhayashi@math.sci.osaka-u.ac.jp
Abstract
We consider the Cauchy problem for the fourth-order nonlinear Schrödinger equation
1
i∂t u − ∂x4 u = λ|u|4 u,
4
u(0, x) = u0 (x).
Our purpose in this talk is to prove the large time asymptotic behavior of solutions with
a logarithmic correction under the non total mass condition and defocusing case λ > 0.
Denote the kernel
Z
i 4
1
G(x) = √
eixξ− 4 ξ dξ
2π
and
Z
1
|G(x)|4 G(x)dx.
b= √
2π
Note that Imb < 0. We prove the following result.
0,1
Theorem 0.1 Let λ > 0, the initial data u0 ∈ H
ε > 0 is sufficiently small. We also assume that
and kub0 kL∞ + kxu0 kL2 ≤ ε, where
5
inf |ub0 (ξ)| ≥ δ = ε 4 .
|ξ|≤1
it
4
Then there exists a unique solution e− 4 ∂x u ∈ C [0, ∞); H0,1
Cauchy problem satisfying the time decay estimate
1
T
0,1
C [0, ∞); H∞
of the
1
t 4 ku(t)kL∞ + t 2 k∂x u(t)kL∞ ≤ Cε.
Jinhua, March 24–27th, 2016
1
International Conference on Partial Differential Equations
Moreover the asymptotic
1
u(t, x) =
iReb
log(1−4λImb|b
u(1,0)|4 log t))
u
b(1,0)G(xt− 4 ) exp(− 4Imb
1
1
u(1,0)|4 log t) 4
t 4 (1−4λImb|b
1
5
+O t− 4 (log t)− 4
is valid for t → ∞ uniformly with respect to x ∈ R.
(This is a joint work with J.A. Mendez-Navarro and P.I. Naumkin.)
On stabilizing effect of the magnetic field in the magnetic Rayleigh-Taylor
problem
Song Jiang(ôt)
Institute of Applied Physics and Computational Mathematics,Beijing,China
e-mail: jiang@iapcm.ac.cn
Abstract
We investigate the stabilizing effect of the vertical equilibrium magnetic field in the
Rayleigh-Taylor (RT) problem for a nonhomogeneous incompressible viscous magnetohydrodynamic (MHD) fluid of zero resistivity in the presence of a uniform gravitational
field in a horizontally periodic domain, in which the velocity of the fluid is non-slip on
both upper and lower flat boundaries. When an initial perturbation around a magnetic
RT equilibrium state satisfies some relations, and the strength of the vertical magnetic
field of the equilibrium state is bigger than the critical number, we can use the Bogovskii
function in the standing-wave form and adapt a two-tier energy method in Lagrangian
coordinates to show the existence of a unique global-in-time (perturbed) stability solution
to the magnetic RT problem.
For the case that the strength of the vertical magnetic field is smaller than the
critical number, by developing new analysis technique based on the method of bootstrap
instability, we show that the nonlinear RT instability will occur. The current result
reveals from the mathematical point of view that the sufficiently large vertical equilibrium
magnetic field has a stabilizing effect and can prevent the RT instability in MHD flows
from occurring. Similar conclusions can be also verified for the horizontal magnetic field
when the domain is vertically periodic, which shows the horizontal magnetic field has the
same stabilizing effect as the vertical one.
Jinhua, March 24–27th, 2016
2
International Conference on Partial Differential Equations
Wave packet transform and singularities of solutions to Schrödinger
equations
Keiichi Kato
Tokyo University of Science, Japan
e-mail: kato@ma.kagu.tus.ac.jp
Abstract
In this talk, we introduce the wave packet transform and its application to Schrödinger
equations. As an application, we study the singularities of solutions to Schrödinger equations. We determine the wave front set of solutions by its initial function.
Hyperbolic balance laws with relaxation
Shuichi Kawashima
Faculty of Mathematics, Kyushu University,
Fukuoka 819-0395, Japan
e-mail: kawashim@math.kyushu-u.ac.jp
Abstract
In this talk, I report the general theory for hyperbolic balance laws which was developed in [1]. The theory assumes two structural conditions for the system: One is
the existence of a mathematical entropy and the other is the stability condition (called
Shizuta-Kawashima Condition). Under these structural conditions we can show the global
existence and optimal decay of solutions for small initial data. The above general theory is valid for systems with symmetric relaxation. Recently, however, we found several
interesting examples which have nonsymmetric relaxation and hence the general theory
in [1] is not applicable: Among such examples, we mention the Timoshenko system and
the Euler-Maxwell system. I also would like to explain the difficulty in developing a new
general theory which covers those examples.
Reference.
[1] S. Kawashima and W.-A. Yong, Dissipative structure and entropy for hyperbolic systems of balance laws, Arch. Rat. Mech. Anal., 174 (2004), 345-364.
Vanishing Viscosity Limit of the compressible Isentropic Navier-Stokes
Equations with degenerate viscosities
Yachun Li (oæX)
Shanghai Jiao Tong University, China
e-mail: ycli@sjtu.edu.cn
Abstract
Jinhua, March 24–27th, 2016
3
International Conference on Partial Differential Equations
In this talk we first establish the local-in-time well-posedness of the unique regular
solution to the compressible isentropic Navier-Stokes equations with density-dependent
viscosities in a power law and with vacuum appearing in some open set or at the far
field, then after establishing uniform energy-type estimates with respect to the viscosity
coefficients for the regular solutions we prove the convergence of the regular solution of
the Navier-Stokes equations to that of the Euler equations with arbitrarily large data
containing vacuum.
Evolution of diffusion in a mutation-selection model
Yuan Lou (¢ )
Renmin University of China, The Ohio State University
e-mail: yuanlou2011@gmail.com
Abstract
We consider a mutation-selection model of a population structured by the spatial
variables and a trait variable which is the diffusion rate. Competition for resource is local
in spatial variables, but nonlocal in trait. We show that in the limit of small mutation
rate, steady state solutions remain regular in the spatial variables and yet concentrates
in the trait variable and forms a Dirac mass supported at the lowest diffusion rate. This
is a joint work with King-Yueng Lam (Ohio State University).
äk‡²•
³
Schrödinger•
•§
Ñ
nØ
Changxing Miao (¢•,)
Institute of Applied Physics and Computational Mathematics,Beijing,China
e-mail: miao changxing@iapcm.ac.cn
Abstract
¯¤±•§Bourgain Uþ8BE↔ƒC” ng•ïÄ .ÚÑ•§ Ñ
nØm8
´. Tao-ìèÏLmuƒpŠ^ Morawetz. O9ÙÛÜz O)
ûšà
Uþ .š‚5ÚÑÅ•§Ñ nØù˜úm¯K"Kenig-MerleÄuž
˜Profiles ©)§uÐ ¤¢ 8¥;—†f5•{§¤J/)û à š‚5 .
ÚÑÅ•§ Ñ nØ"XÛ)ûäk‡²• ³ Schrödinger•§ Ñ ßŽ´
T+•ÿ™)û úm¯K"T ẇ® †ÜŠöÏLuÐ †ä‡²• ³
LaplaceŽfƒéA NÚ©Û•{§(Ü8¥;—†f5•{§)û äk‡²•
³ Schrödinger•§ Ñ ßŽ"ƒ'©ÙŒ„µ
1. Multipliers and Riesz transforms for the Schrödinger operator with inverse-square
potential£arXiv:1503.02716 ¤
R. Killip, C. Miao, M. Visan, J. Zhang, J. Zheng
2. The energy-critical NLS with inverse-square potential £arXiv:1509.05822¤
R. Killip, C. Miao, M. Visan, J. Zhang, J. Zheng
Jinhua, March 24–27th, 2016
4
International Conference on Partial Differential Equations
Global behavior of solution to the drift-diffusion system in the higher space
dimensions
Takayoshi Ogawa
Tohoku University, Japan
e-mail: ogawa@math.tohoku.ac.jp
Abstract
The drift-diffusion system arises from a various field of physics and chemotaxis. For
the two dimensional case, the global behavior of the solution is classified by the threshold
value of the critical L1 -norm by 8π. For the higher dimensional case, it is not clear for the
threshold of the global behavior. We show the possible candidate of the threshold for the
critical norm of the solution and under some sufficient condition, the solution blows up in
a finite time. The proof relies on the usage of the sharp form of the Shannon inequality for
entropy functional and some concentration phenomena is observed with the best possible
constant of the Hardy-Littlewood inequality.
Life span of solutions to nonlinear Schrödinger equations on torus
Tohru Ozawa
Waseda University, Japan
e-mail: txozawa@waseda.jp
Abstract
We give an explicit upper bound of life span of solutions to non gauge invariant
nonlinear Schrödinger equations on n-dimensional tours. This talk is based on a recent
joint work with Kazumasa Fujiwara.
On elliptic systems with Sobolev critical exponent
Shuangjie Peng ($V )
Central China Normal University
e-mail: sjpeng@mail.ccnu.edu.cn
Abstract
We will talk about some basic results on elliptic systems with Sobolev critical growth.
Firstly, we provide a uniqueness result on the least energy solutions and show that a
manifold of a type of positive solutions is non-degenerate for the system for some ranges
of the parameters. Moreover, we establish a global compactness result and by means of
this global compactness result, we extend a classical result of Coron on the existence of
positive solutions of scalar equations with critical exponent on domains with nontrivial
topology to the elliptic system.
Jinhua, March 24–27th, 2016
5
International Conference on Partial Differential Equations
Global Large Solutions to a Viscous Heat-Conducting One-Dimensional Gas
with Temperature-Dependent Viscosity
Huijiang Zhao (ë¬ô)
Wuhan University, China
e-mail: hhjjzhao@hotmail.com
Abstract
This talk is concerned with our recent results on the construction of global large
solutions to one-dimensional compressible Navier-Stokes equations with temperature and
density dependent transport coefficients.
Almost optimal local existence for radial symmetric minimal surface
equation in 1+3 dimensional Minkowski space
Yi Zhou (±Á)
Fudan University, China
e-mail: yizhou@fudan.edu.cn
Abstract
We study the Cauchy problem for the radially symmetrical minimal surface equation in 1+3 dimensional Minkowski space, which falls into the class of 1+2 dimensional
quasilinear wave equation. We show that under a classical characteristic coordinate transformation, the equation can be converted to a semi-linear one. A bootstrap argument
combined with delicate analysis for the transformation coefficients and the nonlinearities
generates a series of crucial a priori estimates, which finally gives us a local result at the
almost optimal regularity for the original minimal surface equation.
Jinhua, March 24–27th, 2016
6