2.
The Boltzmann Equation Near Equilibrium States in $\mathbb R^n$ - Duan, Renjun
In this paper, we review some recent results on the Boltzmann equation near the equilibrium states in the whole space $\mathbb R^n$. The emphasize is put on the well-posedness of the solution in some Sobolev space without time derivatives and its uniform stability and optimal decay rates, and also on the existence and asymptotical stability of the time-periodic solution. Most of results obtained here are proved by combining the energy estimates and the spectral analysis.
3.
Asymptotic Completeness for Relativistic Kinetic Equations with Short-range Interaction Forces - Ha, Seung-Yeal; Kim, Yong Duck; Lee, Ho; Noh, Se Eun
We present an $L^1$-asymptotic completeness results for relativistic kinetic equations with short range interaction forces. We show that the uniform phase space-time bound for nonlinear terms to the relativistic nonlinear kinetic equations yields the asymptotic completeness of the relativistic kinetic equations. For this space-time bound, we employ dispersive estimates and explicit construction of a Lyapunov functional.
4.
Deformation of Surfaces Induced by Motions of Curves in Higherdimensional Similarity Geometries - Qu, Changzheng; Li, Yanyan
In this paper, we study deformation of surfaces induced by adding one and two extra space variables to the motions of space curves in higher-dimensional similarity geometries. It is shown that the 2+1- and 3+1-dimensional nonlinear evolution equations including the 2+1-dimensional mKdV equation and a generalization to the mKdV-Burgers system arise from such motions.
5.
The Sharp Interface Limit of a Phase Field Model for Moving Contact Line Problem - Wang, Xiao-Ping; Wang, Ya-Guang
Using method of matched asymptotic expansions, we derive the sharp interface limit for the diffusive interface model with the generalized Navier boundary condition recently proposed by Qian, Wang and Sheng in "Molecular scale contact line hydrodynamics of immiscible flows," and "Power-law slip profile of the moving contact line in two-phase immiscible flows," for the moving contact line problem. We show that the leading order problem satisfies a boundary value problem for a coupled Hale-Shaw and Navier-Stokes equations with the interface being a free boundary, and the leading order dynamic contact angle is the same as the static one satisfying the...
7.
Pointwise convergence of the boundary layer of the Boltzmann equation for the cutoff hard potential - Deng, Shijin; Wang, Weike; Yu, Shih-Hsien
In this paper, we consider the nonlinear stability of a boundary layer of the Boltzmann
equation with the cutoff hard potential when Mach number at far-field is greater than 1. Based on
the Greens function for the Cauchy problem constructed in M.-Y. Lee, T.-P. Liu and S.-H. Yu and the weighted energy method,
we obtain the estimates for the Greens function of the initial boundary problem and use it to obtain
the nonlinear stability with an almost exponential convergent rate to the nonlinear Knudsen layer.
9.
Multidimensional Hahn polynomials, intertwining functions on the symmetric group and Clebsch-Gordon coefficients - Scarabotti, Fabio
We generalize a construction of Dunkl, obtaining a wide class intertwining functions
on the symmetric group Sn and a related family of multidimensional Hahn polynomials. Following
a suggestion of Vilenkin and Klimyk, we develop a tree-method approach for those intertwining
functions. Moreover, using our theory of $S_n$-intertwining functions and James version of the Schur-
Weyl duality, we give a proof of the relation between Hahn polynomials and $SU(2)$ Clebsch-Gordan
coefficients, previously obtained by Koornwinder and by Nikiforov, Smorodinski? and Suslov in the
$SU(2)$-setting. Such relation is also extended to the multidimensional case.
10.
Hypoelliptic convolution equations in $\cal{S}' (\Bbb R)$ for the Dunkl theory on $\Bbb R$ - Ben Farah, Slaim; Mokni, Kamel
The aim of this paper is to characterize hypoelliptic convolution-equations in $\cal{S}' (\Bbb R)$ for
the Dunkl theory on the real line. For this we determine the spaces of convolution and multiplication
operators in $\cal{S}' (\Bbb R)$ for the Dunkl convolution and we show that the Fourier-Dunkl transform is a
topological isomorphism between them.
11.
Viscosity approximation methods for equilibrium problems and fixed point problems of nonexpansive mappings and inverse-strongly monotone mappings - Wang, Shenghua; Zhou, Haiyun; Song, Jianmin
In this paper, we introduce an iterative scheme by viscosity approximation method
for obtaining a common element of the set of solutions of an equilibrium problem and the set of
fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for
an inverse-strongly monotone mapping in a Hilbert space. We obtain a strong convergence which
improves and extends S. Takahashi and W. Takahashis result [S. Takahashi, W. Takahashi, Viscosity
approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math.
Anal. Appl. 331 (2007) 506-515].
13.
An Approximation Lemma about the Cut Locus, with Applications in Optimal Transport Theory - Figalli, Alessio; Villani, Cedric
A path in a Riemannian manifold can be approximated by a path meeting only
finitely many times the cut locus of a given point. The proof of this property uses recent works
of Itoh–Tanaka and Li–Nirenberg about the differential structure of the cut locus. We present
applications in the regularity theory of optimal transport.
14.
Hamilton-Jacobi Equations in the Wasserstein Space - Gangbo, Wilfrid; Nguyen, Truyen; Tudorascu, Adrian
We introduce a concept of viscosity solutions for Hamilton-Jacobi equations (HJE)
in the Wasserstein space. We prove existence of solutions for the Cauchy problem for certain Hamiltonians
defined on the Wasserstein space over the real line. In order to illustrate the link between
HJE in the Wasserstein space and Fluid Mechanics, in the last part of the paper we focus on a
special Hamiltonian. The characteristics for these HJE are solutions of physical systems in finite
dimensional spaces.
15.
On Instant Blow-up for Semilinear Heat Equations with Growing Initial Data - Giga, Yoshikazu; Umeda, Noriaki
For a semilinear heat equation admitting blow-up solutions a sufficient condition for
nonexistence of local-in-time solutions are obtained. In particular, it is shown that if an initial data
tends to infinity at space infinity then there is no local-in-time solution. As an application if the
solution blows up at space infinity with least blow-up time, the solution cannot be extendable after
blow-up time.
17.
Partial Regularity of Weak Solutions to Maxwell's Equations in a Quasi-static Electromagnetic Field - Hong, Min-Chun; Tonegawa, Yoshihiro; Yassin, Alzubaidi
We study Maxwell’s equations in a quasi-static electromagnetic field, where the
electrical conductivity of the material depends on the temperature. By establishing the reverse
Hölder inequality, we prove partial regularity of weak solutions to the non-linear elliptic system and
the non-linear parabolic system in a quasi-static electromagnetic field.
18.
The Large-time Behavior of Solutions of Hamilton-Jacobi Equations on the Real Line - Ichihara, Naoyuki; Ishii, Hitoshi
We investigate the large-time behavior of solutions of the Cauchy problem for
Hamilton-Jacobi equations on the real line $R$. We establish a result on convergence of the solutions
to asymptotic solutions as time $t$ goes to infinity.
20.
A Class of Sobolev Type Inequalities - Tian, Gu-Ji; Wang, Xu-Jia
In this paper, we prove that for any strictly convex polynomial, or more generally
any strictly convex function satisfying appropriate conditions, there is an associated Sobolev type
inequality.
1.
