Global existence for coupled systems of nonlinear wave and Klein–Gordon equations in three space dimensions

We consider the Cauchy problem for coupled systems of wave and Klein–Gordon equations with quadratic nonlinearity in three space dimensions. We show global existence of small amplitude solutions under certain condition including the null condition on self-interactions between wave equations. Our condition is much weaker than the strong null condition introduced by Georgiev for this kind of coupled system. Consequently our result is applicable to certain physical systems, such as the Dirac–Klein–Gordon equations, the Dirac–Proca equations, and the Klein–Gordon–Zakharov equations.     

A Liouville Theorem for the Planer Navier-Stokes Equations with the No-Slip Boundary Condition and Its Application to a Geometric Regularity Criterion

We establish a Liouville type result for a backward global solution to the Navier-Stokes equations in the half plane with the no-slip boundary condition. No assumptions on spatial decay for the vorticity nor the velocity field are imposed. We study the vorticity equations instead of the original Navier-Stokes equations. As an application, we extend the geometric regularity criterion for the Navier-Stokes equations in the three-dimensional half space under the no-slip boundary condition.     

On the global periodicity of discrete dynamical systems and application to rational difference equations

A new necessary condition for global periodicity of discrete dynamical systems and of difference equations is obtained here. This condition will be applied to contribute to solving the problem of global periodicity for second order rational difference equations.     

中立型随机泛函微分方程的Khasminskii型定理

本文对中立型随机泛函微分方程建立了Khasminskii型定理,这个定理显示在局部Lipschitz条件但是不要求线性增长的条件下,中立型随机泛函微分方程存在一个全局解.本文的这个解存在性条件可以包含更广的一类非线性中立型随机泛函微分方程.最后,本文给出一个例子来阐述我们的思想.     

Approximation properties for solutions to non‐Lipschitz stochastic differential equations with Lévy noise

In this paper, we consider the non‐Lipschitz stochastic differential equations and stochastic functional differential equations with delays driven by Lévy noise, and the approximation theorems for the solutions to these two kinds of equations will be proposed respectively. Non‐Lipschitz condition is much weaker condition than the Lipschitz one. The simplified equations will be defined to make its solutions converge to that of the corresponding original equations both in the sense of mean square and probability, which constitute the approximation theorems. Copyright © 2014 John Wiley & Sons, Ltd.     

An experimental adaptation of Higdon-type non-reflecting boundary conditions to linear first-order systems

Experiments in adapting the Higdon non-reflecting boundary condition (NRBC) method to linear 2-D first-order systems are presented. Finite difference implementations are developed for the free-space Maxwell equations, the linearized shallow-water equations with Coriolis, and the linearized Euler equations with uniform advection. This NRBC technique removes up to 99% of the reflection error generated by the Sommerfeld radiation condition with only a modest increase in computational overhead.     

Asymptotic Solutions of Hamilton-Jacobi Equations with State Constraints

We study Hamilton-Jacobi equations in a bounded domain with the state constraint boundary condition. We establish a general convergence result for viscosity solutions of the Cauchy problem for Hamilton-Jacobi equations with the state constraint boundary condition to asymptotic solutions as time goes to infinity.     

Solving the reaction–diffusion equations with nonlocal boundary conditions based on reproducing kernel space

The reaction–diffusion equations with initial condition and nonlocal boundary conditions are discussed in this article. A reproducing kernel space is constructed, in which an arbitrary function satisfies the initial condition and nonlocal boundary conditions of the reaction‐diffusion equations. Based on the reproducing kernel space, a new algorithm for solving the reaction–diffusion equations with initial condition and nonlocal boundary conditions is presented. Some examples are displayed to demonstrate the validity and applicability of the proposed method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Estimates for Solutions to Model Problems Connected with Quasistationary Approximation for the Stefan Problem

Model problems for elliptic equations are considered. The time-derivative of a solution can be contained in the boundary condition or in the conjugation condition. Suich problems appear, for example, in the study of free boundary problems for elliptic equations that can be considered as quasistationary approximations to free boundary problems for parabolic equations. Estimates for solutions to the model problems are obtained. Bibliography: 11 titles.     

On the removability of isolated singular points for degenerating nonlinear elliptic equations

The well known results on the removable singularity of elliptic equations are generalized to the class of degenerating nonlinear elliptic equations. A sufficient condition for the isolated singular point to be removable has been found. In the absence of degeneration, this condition coincides with already known results.     

A necessary condition of solvability for the capillarity boundary of Monge-Ampere equations in two dimensions

In this paper we consider a class of Monge-Ampere equations with a prescribed contact angle boundary value problem on a bounded strictly convex domain in two dimensions. The purpose is to give a sharp necessary condition of solvability for the above mentioned equations. This is achieved by using the maximum principle and introducing a curvilinear coordinate system for Monge-Ampere equations in two dimensions. An interesting feature of our necessary condition is the need for a certain strong restriction between the curvature of the boundary of domain and the boundary condition, which does not appear in the Dirichlet and Neumann boundary values.

     

Derivation of Macroscale Equations for a Class of Physical Applications

A macroscale formulation is constructed from a system of partialdifferential equations which govern the microscale dependent variables. Theconstruction is based upon transformations of equations on second ordercontact manifolds under conditions of integrability. Necessary conditions onthe structure of the macroscale equations are obtained under the requirementthat the solutions of the macroscale equations satisfy, in some approximatesense, the equations associated with the microscale. This approach offers amethod whereby one can construct only those macroscale equations that can bevalidated by a condition of consistency based on the model error. Themethodology is employed to construct a turbulence closure model forincompressible flow. It is shown that the large eddy viscosity, whichsatifies contemporary tests based on Galilean invariance, fails theconsistency condition defined here.     

Generalized BSDEs and nonlinear Neumann boundary value problems

Summary. We study a new class of backward stochastic differential equations, which involves the integral with respect to a continuous increasing process. This allows us to give a probabilistic formula for solutions of semilinear partial differential equations with Neumann boundary condition, where the boundary condition itself is nonlinear. We consider both parabolic and elliptic equations. Received: 27 September 1996 / In revised form: 1 December 1997     

On the removability of isolated singular points for degenerating nonlinear elliptic equations

The well known results on the removable singularity of elliptic equations are generalized to the class of degenerating nonlinear elliptic equations. A sufficient condition for the isolated singular point to be removable has been found. In the absence of degeneration, this condition coincides with already known results.     

MHD Boundary Layers Theory in Sobolev Spaces Without Monotonicity I: Well-Posedness Theory

We study the well-posedness theory for the MHD boundary layer. The boundary layer equations are governed by the Prandtl-type equations that are derived from the incompressible MHD system with non-slip boundary condition on the velocity and perfectly conducting condition on the magnetic field. Under the assumption that the initial tangential magnetic field is not zero, we establish the local-i-time existence, uniqueness of solutions for the nonlinear MHD boundary layer equations. Compared with the well-posedness theory of the classical Prandtl equations for which the monotonicity condition of the tangential velocity plays a crucial role, this monotonicity condition is not needed for the MHD boundary layer. This justifies the physical understanding that the magnetic field has a stabilizing effect on MHD boundary layer in rigorous mathematics. © 2018 Wiley Periodicals, Inc.     

A blow-up criterion for 3D Boussinesq equations in Besov spaces

In this paper, we consider the 3D Boussinesq equations with the incompressibility condition. We obtain a regularity condition for the three-dimensional Boussinesq equations by means of the Littlewood-Paley theory and Bony’s paradifferential calculus.     

Linear Problems Associated with Evolution Equations of the Type utt = F(u,ux, uxx,ut)

Evolution equations which describe pseudospherical surfaces are considered. A systematic procedure to determine a linear problem for which a given equation is the integrability condition is obtained. The theory is applied to families of equations. In particular, the Liouville equation is obtained as the integrability condition of a linear problem.     

一类大气演化方程组的Cauchy问题

在分层理论的框架下讨论一类大气演化方程组的Cauchy问题,证明了:1) 惯性力对一类大气演化方程组Cauchy问题的适定性判别标准没有影响;2) 可压缩性对粘性大气方程组Cauchy问题的适定性判别标准没有影响,但对无粘大气方程组,可压缩性改变Cauchy问题适定性判别标准;3) 所论方程组在t=0超平面上的Cauchy问题均是不适定的,并不受粘性和可压缩性的影响;4) 可压无粘大气方程与运动静止初始条件构成的Cauchy问题是不适定的．     

带有多变滞量的高阶非线性中立型差分方程的振动性判据

研究了较为广泛的一类带有多个变滞量的高阶非线性中立型差分方程的振动性,建 立了这类方程振动的一个充要条件和一个充分条件．     

On the existence of solution to one–dimensional forward–backward sdes

In this paper, we study the existence of the solution to one-dimensional forward–backward stochastic differential equations with neither the smooth condition nor the monotonicity condition for the coefficients. Under the nondegeneracy condition for the forward equation, we prove the existence of the solution to one-dimensional forward–backward stochastic differential equations. And we apply this result to establish the existence of the viscosity solution to a certain one-dimensional quasilinear parabolic partial differential equation