A boundary value problem for a class of mixed-type equations in an unbounded domain

We consider a boundary value problem for an equation of the mixed type with a singular coefficient in an unbounded domain. The uniqueness of the solution of the problem is proved with the use of the extremum principle. In the proof of the existence of a solution of the problem, we use the method of integral equations.     

Evolution of hypersurfaces in $ {\Bbb R}^N$ by Gaussian curvature

In this paper we first prove short time existence of a classical solution for the problem which describes the evolution by Gaussian curvature of a strictly convex hypersurface in . Then we give a proof of the existence of a viscosity solution for this problem in such a way as to define a generalized motion existing for each time. Received November 24, 1997     

Evolution of hypersurfaces in \Bbb RN {\Bbb R}^N by Gaussian curvature

In this paper we first prove short time existence of a classical solution for the problem which describes the evolution by Gaussian curvature of a strictly convex hypersurface in . Then we give a proof of the existence of a viscosity solution for this problem in such a way as to define a generalized motion existing for each time. Received November 24, 1997     

一类条件变分问题解的存在性

本文对一类有广泛应用价值的不规则的条件泛函极值问题的解的存在性及解的解析表示式进行了深入系统的讨论,得到了一系列在理论及应用上颇有价值的结果.对该类问题最优解的存在性的证明填补了这方面的空白.     

A Mixed Problem with Integral Condition for the Hyperbolic Equation

In this paper, we study a mixed problem for the hyperbolic equation with a boundary Neumann condition and a nonlocal integral condition. We justify the assertion that there exists a unique generalized solution of the problem under consideration. The proof of uniqueness is based on an estimate, derived a priori, in the function space introduced in the paper, while the existence of a generalized solution is proved by the Galerkin method.     

Extremal solutions of boundary value problems of the fifth order

For boundary value problems of the fifth order, we find extremal solutions that provide the best estimates in the proof of the existence of a solution of the boundary value problem by the method of a priori estimates.     

Nonlocal boundary-value problem for linear partial differential equations unsolved with respect to the higher time derivative

We study the well-posedness of the problem with general nonlocal boundary conditions in the time variable and conditions of periodicity in the space coordinates for partial differential equations unsolved with respect to the higher time derivative. We establish the conditions of existence and uniqueness of the solution of the considered problem. In the proof of existence of the solution, we use the method of divided differences. We also prove metric statements on the lower bounds of small denominators appearing in constructing the solution of the problem. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 3, pp. 370–381, March, 2007.     

非线性互补问题的组合同伦算法

针对拟P_*-映射和P(τ,α,β)-映射所对应的非线性互补问题,本文对其解的存在性及有效求解算法进行了研究.文中利用组合同伦方法给出了这两类非线性互补问题存在有界解的构造性证明,并利用预估校正方法对同伦路径进行跟踪,得到了互补问题的解.通过数值算例验证了该算法的有效性.     

一类半无穷区间奇异边值问题正解的存在性

在半无穷区间上讨论带有非齐次边界条件的奇异边值问题正解的存在性,多解性及不存在性．主要结果的证明基于上下解方法,Schauder不动点定理及拓扑度理论．     

Variational methods for the choreography solution to the three- body problem

In this paper, we give a short proof for the existence of nontrivial choreography solution to the equal-mass three-body problem, which is discovered by Chenciner and Montgomery recently.     

A mixed boundary value problem for Chaplygin's hodograph equation

In this paper we will prove existence, uniqueness and regularity of a classical solution to a mixed boundary value problem for Chaplygin's hodograph equation, which is degenerate elliptic on a part of the boundary. This problem is derived from the study of detached bow shock ahead of a straight ramp in uniform supersonic flows in the hodograph plane. The proof depends on Perron's method and some techniques from linear elliptic equations.     

Analysis of a sequential regularization method for the unsteady Navier-Stokes equations

The incompressibility constraint makes Navier-Stokes equations difficult. A reformulation to a better posed problem is needed before solving it numerically. The sequential regularization method (SRM) is a reformulation which combines the penalty method with a stabilization method in the context of constrained dynamical systems and has the benefit of both methods. In the paper, we study the existence and uniqueness for the solution of the SRM and provide a simple proof of the convergence of the solution of the SRM to the solution of the Navier-Stokes equations. We also give error estimates for the time discretized SRM formulation.

     

Constructive proof for existence of nonlinear two-point boundary value problems

Owing to the importance of differential equations in physics, the existence of solutions for differential equations has been paid much attention. In this paper, the existence of solution are obtained for the nonlinear second order two-point boundary value problem in the reproducing kernel space. Under certain assumptions on right-hand side, we propose constructive proof for the existence result, and a method is presented to obtain the exact solution expressed by the form of series. This paper is a extension of previous paper [Wei Jiang, Minggen Cui, The exact solution and stability analysis for integral equation of third or first kind with singular kernel, Appl. Math. Comput. 202 (2) (2008) 666-674], which extends a method of solving linear problems to present method for solving nonlinear problems.     

Singular nonhomogeneous quasilinear elliptic equations with a convection term

In this work we establish existence results for a class of nonhomogeneous and singular quasilinear elliptic equations involving a convection term. The gradient term makes the problem non variational, and in addition to this difficulty we have to handle the singular term with a sign changing nonlinearity. The proof of the results are made combining the sub‐super solution method, fixed point theorem, Leray–Schauder degree theory and comparison theorems.     

Degenerate parabolic stochastic partial differential equations

We study the Cauchy problem for a scalar semilinear degenerate parabolic partial differential equation with stochastic forcing. In particular, we are concerned with the well-posedness in any space dimension. We adapt the notion of kinetic solution which is well suited for degenerate parabolic problems and supplies a good technical framework to prove the comparison principle. The proof of existence is based on the vanishing viscosity method: the solution is obtained by a compactness argument as the limit of solutions of nondegenerate approximations.     

A strong solution of a high-order mixed type partial differential equation with integral conditions

In this article, we study a mixed problem with integral boundary conditions for a high-order partial differential equation of mixed type. We prove the existence and uniqueness of a strong solution. The proof is based on energy inequality and on the density of the range of the operator generated by the considered problem.     

A Primal-Dual Active Set Method for Bilaterally Control Constrained Optimal Control of the Navier–Stokes Equations

Abstract

This paper is devoted to the analysis and numerical solution of distributed optimal control of the Navier–Stokes equations in presence of bilateral pointwise control constraints. The analysis of the problem involves the proof of existence of an optimal solution, as well as the presentation of necessary and sufficient conditions for optimality. For the numerical solution of the problem we apply a primal-dual active set strategy and show global and local convergence properties of the method. Finally, some numerical experiments, which illustrate the performance of the method, are discussed.     

具Gilbert耗散项的多变量铁磁链旋方程组解的Blow Up

本文考虑具Gilbert耗散项的多变量铁磁链旋方程组 Z_t=εLZ-αZ×(Z×LZ) βZ×LZ f(x,Z)Z初边值问题 Z|_=0, t≥0 Z(x,0)=Z_0(x), x∈ΩR~m强解的Blow Up.这里Z(x,t)=(u(x,t),v(x,t),w(x,t)),L=sum from i,j=1 to nε>0,α>0.采用Galerkin方法和紧致性原理证明局部强解的存在性,利用凸性方法证明强解的Blow Up性质。     

具非均匀介电常数的半导体方程组的初边值问题

考虑具非均匀介电常数的半导体方程组的初边值问题．采用逼近解过程,利用不动点原理证明逼近解的存在性,经过对逼近解建立先验估计,根据紧致性原理证明了该问题整体弱解的存在性．     

Existence of a Fixed Point of a Nonsmooth Function Arising in Numerical Mechanics

A recent work (Acary et al. 2010) introduces a formulation as a nonsmooth fixed-point problem of a basic problem in numerical mechanics (namely the dynamical Coulomb friction problem in finite dimension with discretized time). Using this new formulation, the existence of a solution to the problem and its numerical resolution are then guaranteed under a strong assumption on the data of this problem. In this paper, we show that the fixed point problem admits solution under a natural, weaker assumption. This existence proof uses a perturbation argument combined with continuity properties of a set-valued mapping associated with the constraints of the problem.