函数的可测性问题

讨论了函数的可测性问题 ,特别是函数的 Borel可测和 Lebesgue可测的关系 .对相应结论给出了明确和严格的证明 .利用本文的结果 ,我们严格证明了 Lebesgue积分的变量替换公式 .此外还明确指出了分布函数密度的等价唯一性问题 .  

Optimal Control for the Convective Cahn–Hilliard Equation in 2D Case

In this paper, for 2D convective Cahn–Hilliard equation, the optimal control problem is considered, the existence of optimal solution is proved and the optimality system is established.  

Time-periodic solution of a 2D fourth-order nonlinear parabolic equation

By using the Galerkin method, we study the the existence and uniqueness of time-periodic generalized solutions and time-periodic classical solutions to a fourth-order nonlinear parabolic equation in 2D case.  

Optimal control problem for viscous Cahn-Hilliard equation

This paper is concerned with the viscous Cahn-Hilliard equation, which arises in the dynamics of viscous first order phase transitions in cooling binary solutions. The optimal control under boundary condition is given and the existence of optimal solution to the equation is proved.  

Asymptotic Behavior of Solutions to a New Hall-MHD System

In this paper, by using Fourier splitting method and the properties of decay character \(r^{*}\), we consider the algebraic time decay rate of solutions to a new Hall-MHD system in Sobolev space \(H^{m}(\mathbb{R} ^{3})\times H^{m+1}(\mathbb{R}^{3})\) for \(m\geq 0\).  

Decay of solutions to a new Hall–MHD system in R3

This paper discusses the large-time behavior of solutions for a new Hall–MHD system in ${\mathbb{R}}^3$. Using the Fourier splitting method, we establish the upper bound of the time-decay rate in $L^2({\mathbb{R}}^3)$ for weak solutions.  

The existence of global attractor for a fourth-order parabolic equation

This article is concerned with a fourth-order parabolic equation. Based on the regularity estimates for the semigroups and the classical existence theorem of global attractors, we prove that the fourth-order parabolic equation possesses a global attractor in H ^k (0 ≤ k < 5) space, which attracts any bounded subset of H ^k (Ω) in the H ^k -norm.  

Optimal control of the convective Cahn–Hilliard equation

This article studies the problem for optimal control of the convective Cahn–Hilliard equation in one-space dimension. The optimal control under boundary condition is given, the existence of optimal solution to the equation is proved and the optimality system is established.  

Finite difference discretization of a fourth-order parabolic equation describing crystal surface growth

In this paper, by using finite difference method, we consider the approximate solutions for a fourth-order parabolic equation describing crystal surface growth.