Exact Boundary Controllability of Weak Solutions for a Kind of First Order Hyperbolic System — the HUM Method*

The exact boundary controllability and the exact boundary observability for the 1-D first order linear hyperbolic system were studied by the constructive method in the framework of weak solutions in the work [Lu, X. and Li, T. T., Exact boundary controllability of weak solutions for a kind of first order hyperbolic system — the constructive method, Chin. Ann. Math. Ser. B, 42(5), 2021, 643–676]. In this paper, in order to study these problems from the viewpoint of duality, the authors establish ...     

Strong （Weak） Exact Controllability and Strong （Weak） Exact Observability for Quasilinear Hyperbolic Systems

Numerical approximations of Cahn-Hilliard phase-field model for the two-phase incompressible flows are considered in this paper.Several efficient and energy stable time discretization schemes for the coupled nonlinear Cahn-Hilliard phase-field system for both the matched density case and the variable density case are constructed,and are shown to satisfy discrete energy laws which are analogous to the continuous energy laws.     

Exact Boundary Observability of Unsteady Supercritical Flows in a Tree-Like Network of Open Canals

The author establishes the exact boundary observability of unsteady supercritical flows in a tree-like network of open canals with general topology. An implicit duality between the exact boundary controllability and the exact boundary observability is also given for unsteady supercritical flows.     

Weak and Strong Solutions for the Stokes Approximation of Non-homogeneous Incompressible Navier-Stokes Equations

In this paper,the Dirichlet problem of Stokes approximate of non-homogeneous incompressibleNavier-Stokes equations is studied.It is shown that there exist global weak solutions as well as global andunique strong solution for this problem,under the assumption that initial density ρ_0(x)is bounded away from0 and other appropriate assumptions(see Theorem 1 and Theorem 2).The semi-Galerkin method is applied toconstruct the approximate solutions and a prior estimates are made to elaborate upon the compactness of theapproximate solutions.     

Existence and Nonuniqueness of Weak Solutions of the Initial-Boundary Value Problem for ut=u^σ div（｜△↓u｜^（p-2）△↓u）

In this paper, we study the existence and nonuniqueness of weak solutions of the initial and boundary value problem for ut =u^σ div(|△↓u|^(p-2)△↓u) with σ≥1. Localization property of weak solutions is also discussed.     

The Existence and Long-Time Behavior of Weak Solution to Bipolar Quantum Drift-Diffusion Model

The authors study the existence and long-time behavior of weak solutions to the bipolar transient quantum drift-diffusion model,a fourth order parabolic system.Using semi-discretization in time and entropy estimate,the authors get the global existence of nonnegative weak solutions to the one-dimensional model with nonnegative initial and homogenous Neumann(or periodic)boundary conditions.Furthermore,by a logarithmic Sobolev inequality,it is proved that the periodic weak solution exponentially approaches its mean value as time increases to infinity.     

Global Weak Solutions of Initial Boundary Value Problem for Boltzmann－Poisson System with Absorbing Boundary

This paper deals with the initial boundary value problem for the Boltzmann-Poisson system, which arises in semiconductor physics, with absorbing boundary. The global existence of weak solutions is proved by using the stability of velocity averages and the compactness results on L1-theory under weaker conditons on initial boundary values.     

Exact Boundary Controllability and Exact Boundary Observability for a Coupled System of Quasilinear Wave Equations

Based on the theory of semi-global classical solutions to quasilinear hyperbolic systems, the authors apply a unified constructive method to establish the local exact boundary(null) controllability and the local boundary(weak) observability for a coupled system of 1-D quasilinear wave equations with various types of boundary conditions.     

Energy Equality and Uniqueness of Weak Solutions to MHD Equations in L∞（0, T; L^n（Ω））

In this paper, we study the energy equality and the uniqueness of weak solutions to the MHD equations in the critical space L∞（0, T; L^n（Ω）. We prove that if the velocity u belongs to the critical space L∞（0, T; L^n（Ω）, the energy equality holds. On the basis of the energy equality, we further prove that the weak solution to the MHD equations is unique.     

Weak Hopf Algebras Corresponding to Borcherds-Cartan Matrices

Let y be a generalized Kac-Moody algebra with an integral Borcherds-Cartan matrix. In this paper, we define a d-type weak quantum generalized Kac-Moody algebra wUq^d（y）, which is a weak Hopf algebra. We also study the highest weight module over the weak quantum algebra wUdq^d（y） and weak A-forms of wUq^d（y）.     

Weak and Strong Limit Values

The classical results about the boundary values of holomorphic or harmonic functions on a domain D state that under additional integrability assumptions these functions have limits along specific sets approaching boundary. The proofs of these results are based on properties of smooth boundaries used to define the approach regions and on estimates of representing kernels along these regions. This paper attempts to look at the situation when no assumptions about the boundary smoothness are made and, consequently, no natural definitions of approach regions could be given.     

Weak and Strong Density of Compositions

The convergence in various topologies of sequences of inner superposition (composition) operators acting between Lebesgue spaces and of their linear combinations is studied. In particular, the sequential density results for the linear span of such operators is proved for the weak, weak continuous and strong operator topologies.

     

When Are Weak Permutation Polynomials Strong?

For a commutative finite ring with identity R, the two definitions of "permutaIion polynomial in several indeterminates over R" coincide if and only if R is a direct sum of finite fields.     

Weak and Strong Subgradients of Set-Valued Maps

We consider weak subgradients of a set-valued map and present a new notion of strong subgradient. We study their properties and compare our constructions and results with other developments. We give existence conditions of both types and establish several optimality conditions in terms of set optimization.     

Weak Equality and Exact Sequences in H v -modules

The largest class of multivalued systems satisfying the module-like axioms is the H_v-module. H_v-modules first were introduced by Vougiouklis. In this paper we define weak equality between two subsets of an H_v-module and introduced the notion of exact sequences of H_v-modules. Also some results on the weak equality and exact sequences are given.     

Weak and Strong Interactions between Internal Solitary Waves

The interaction of weakly nonlinear long internal gravity waves is studied. Weak interactions occur when the wave phase speeds are unequal; this case includes that of a head-on collision. It is shown that each wave satisfies a Korteweg-de Vries equation, and the main effect of the interaction is described by a phase shift. Strong interactions occur when the wave phase speeds are nearly equal although the waves belong to different modes. This case is described by a pair of coupled Korteweg-de Vries equations, for which some preliminary numerical results are presented.     

一类带强色散项DGH方程的精确解

利用改进的F-展开法,求出了一类带强色散项DGH方程的一系列类孤子解,三角函数周期解和有理数解,方程结合了KdV方程的线性色散项和C-H方程的非线性色散项.而且改进的F-展开法在借助于计算机符号系统Mathematica(Maple)下,操作方便,适用于大量的非线性偏微分方程(组),并有助于发现新解.     

L-拓扑空间的弱局部强F紧性及单点强F紧化

以R.Lowen的强F紧性为基础,定义了L-拓扑空间的弱局部强F紧性及单点强F紧化,推广了有关弱局部紧拓扑空间和拓扑空间的单点紧化的若干结果,证明了L-拓扑空间的弱局部强F紧性是拓扑空间的弱局部紧性的L-推广。     

强收敛与弱收敛的微小区别

对于独立同分布随机变量序列{Xn,n≥1},证明了其满足弱大数律的一个必要条件是:对于任何r∈(0,1),E|X|r<∞都成立.另外我们还举例说明了这样的条件不是充分的.