Initial-irregular oblique derivative problems for nonlinear parabolic systems of second order equations with measurable coefficients

The initial-irregular oblique derivative boundary value problems for nonlinear and nondivergence parabolic systems of second order equations in multiply connected domains are dealt with where coefficients of systems of equations are meaurable. The uniqueness theorem of solutions for the above problems and somea priori estimates of solutions for the problems are given. And by using the above estimates of solutions and the Leray-Schauder theorem, the existence of solutions of the initial-boundary value problems is proved. The results are generalizations of corresponding theorems in literature. Project supported by the National Natural Science Foundation of China (Grant No. 19671006).     

Estimates of solutions of initial-irregular oblique derivative problems for linear parabolic equations of second order with measurable coefficients

The initial-irregular oblique derivative boundary value problems for linear and nondivergence parabolic complex equations of second order in multiply connected domains are dealt with, where the coefficients of equations are measurable. Firstly the uniqueness of solutions for the above problems is introduced, and then somea priori estimates of solutions for the problems are given. By using the above estimates and the Leray-Schauder theorem, the existence of solutions of the initial-boundary value problems can be proved. The results are generalizations of corresponding theorems in literature. Project supported by the National Natural Science Foundation of China (Grant No. 19671006).     

Locally extra-optimal regularizing algorithms and a posteriori estimates of the accuracy for ill-posed problems with discontinuous solutions

Local a posteriori estimates of the accuracy of approximate solutions to ill-posed inverse problems with discontinuous solutions from the classes of functions of several variables with bounded variations of the Hardy or Giusti type are studied. Unlike global estimates (in the norm), local estimates of accuracy are carried out using certain linear estimation functionals (e.g., using the mean value of the solution on a given fragment of its support). The concept of a locally extra-optimal regularizing algorithm for solving ill-posed inverse problems, which has an optimal in order local a posteriori estimate, was introduced. A method for calculating local a posteriori estimates of accuracy with the use of some distinguished classes of linear functionals for the problems with discontinuous solutions is proposed. For linear inverse problems, the method is bases on solving specialized convex optimization problems. Examples of locally extra-optimal regularizing algorithms and results of numerical experiments on a posteriori estimation of the accuracy of solutions for different linear estimation functionals are presented.     

AN IRREGULAR OBLIQUE DERIVATIVE PROBLEM FOR SOME NONLINEAR ELLIPTIC EQUATIONS OF SECOND ORDER

1FormulationofirregularObliqueDerivativeProblemLetDbean(N 1)-connecteddomainwithboundaryr=UU=,r,6Cj(0相似文献   

Positive solutions of boundary value problems for discrete and continuous beam equations

The authors study two related boundary value problems for discrete and continuous beam equations. Some upper and lower estimates for positive solutions of the problems are obtained. Sufficient conditions for the existence and nonexistence of positive solutions are established.     

Continuous invertibility of a boundary control operator for the wave equation on subcritical intervals in classes of weak generalized solutions

For the wave equation, problems with two-sided boundary controls of three basic types are considered. Constructive estimates for the bounded invertibility of the control operator with known estimated constants are obtained in classes of weak generalized solutions on intervals of subcritical length. With the use of these estimates, stable numerical solutions of the considered problems on time intervals of strictly subcritical length can be found by applying a variational method. The main difference of the present estimated constants from those considered previously in the case of strong generalized solutions is that they degenerate as the time interval length approaches its critical value.     

Comparison and regularity results for the fractional Laplacian via symmetrization methods

In this paper we establish a comparison result through symmetrization for solutions to some boundary value problems involving the fractional Laplacian. This allows to get sharp estimates for the solutions, obtained by comparing them with solutions of suitable radial problems. Furthermore, we use such result to prove a priori estimates for solutions in terms of the data, providing several regularity results which extend the well-known ones for the classical Laplacian.     

带雪崩项的半导体方程的瞬态解

研究带雪崩项的半导体方程的混合初边值问题．采用逼近过程和先验估计,在一定条件下,证明了该问题整体弱解的存在性或局部弱解的存在性．     

Differential and Difference Boundary Value Problem for Loaded Third-Order Pseudo-Parabolic Differential Equations and Difference Methods for Their Numerical Solution

Boundary value problems for loaded third-order pseudo-parabolic equations with variable coefficients are considered. A priori estimates for the solutions of the problems in the differential and difference formulations are obtained. These a priori estimates imply the uniqueness and stability of the solution with respect to the initial data and the right-hand side on a layer, as well as the convergence of the solution of each difference problem to the solution of the corresponding differential problem.     

On Solvability of Some Initial-Boundary Problems for Mathematical Models of the Motion of Nonlinearly Viscous and Viscoelastic Fluids

In the paper, the settings of initial-boundary and initial value problems arising in a number of models of movement of nonlinearly viscous or viscoelastic incompressible fluid are considered, and existence theorems for these problems are presented. In particular, the settings of initial-boundary value problems appearing in the regularized model of the movement of viscoelastic fluid with Jeffris constitutive relation are described. The theorems for the existence of weak and strong solutions for these problems in bounded domains are given. The initial value problem for a nonlinearly viscous fluid on the whole space is considered. The estimates on the right-hand side and initial conditions under which there exist local and global solutions of this problem are presented. The modification of Litvinov's model for laminar and turbulent flows with a memory is described. The existence theorem for weak solutions of initial-boundary value problem appearing in this model is given.     

二阶非线性无穷边值问题的奇摄动(Ⅱ)

本文讨论了含小参数ε>0的二阶非线性奇摄动无穷边值问题(其中a_i,β为常数,i=0,1)之解的存在性,并且给出了解的渐近估计式.     

Oblique derivative problems for linear mixed equations of second order

Oblique derivative boundary value problems for the linear mixed (elliptic-hyperbolic) equation of the second order, i.e. the generalized Lavrent’ev-Bitsadze equation with weak conditions are discussed. The representation of solutions for the above boundary value problem is given, the uniqueness and existence of solutions of the above problem are proved, and a priori estimates of the solutions of the above problem are obtained. Project supported by the National Natural Science Foundation of China.     

Upper and lower estimates of the positive solutions of a higher order boundary value problem

The authors consider a higher order boundary value problem that originates from the study of boundary value problems for the beam equation. Upper and lower estimates for positive solutions of the problem are obtained.     

Kuramoto-Sivashinsky方程解的定性分析

研究Kuramoto-Sivashinsky方程的两种初边值问题,运用Galerkin方法给出一系列先验估计结果,得到广义解和古典解的存在唯一性、正则性及某些条件下的渐近性质。     

Optimal summability of solutions to nonlinear elliptic problems

The strongest possible norms in a priori estimates for solutions to a class of second-order nonlinear elliptic boundary value problems are exhibited.     

Long-time uniform stability of solution to magnetohydrodynamics equation

The long-time uniform stability of solutions to the initial value problems for 2-dimensional Magnetohydrodynamics equations is studied. The decay estimates are given.     

Artificial boundary conditions for Petrovsky systems of second order in exterior domains and in other domains of conical type

The approximation of solutions to boundary value problems on unbounded domains by those on bounded domains is one of the main applications for artificial boundary conditions. Based on asymptotic analysis, here a new method is presented to construct local artificial boundary conditions for a very general class of elliptic problems where the main asymptotic term is not known explicitly. Existence and uniqueness of approximating solutions are proved together with asymptotically precise error estimates. One class of important examples includes boundary value problems for anisotropic elasticity and piezoelectricity. Copyright © 2004 John Wiley & Sons, Ltd.     

Superconvergent recovery of gradients on subdomains from piecewise linear finite-element approximations

Engineers have been aware for some time of the phenomenon of superconvergence, whereby there exist (stress) points at which the accuracy of a finite-element solution is superior to that of the approximation generally. This phenomenon has been treated in recent years by mathematicians who have proved, for certain two-dimensional second-order elliptic problems, superconvergent error estimates for retrieved finite-element derivatives. These results have demanded high global regularity of the solutions of the boundary value problems. In this present article cut-off functions are used to prove similar superconvergence results over interior subdomains. This allows superconvergence estimates to be derived for problems with solutions of low global regularity, particularly those involving singularities.     

Superconvergent recovery of gradients on subdomains from piecewise linear finite-element approximations

Engineers have been aware for some time of the phenomenon of superconvergence, whereby there exist (stress) points at which the accuracy of a finite-element solution is superior to that of the approximation generally. This phenomenon has been treated in recent years by mathematicians who have proved, for certain two-dimensional secondorder elliptic problems, superconvergent error estimates for retrieved finite-element derivatives. These results have demanded high global regularity of the solutions of the bondary value problems. In this present article cut-off functions are used to prove similar superconvergence results over interior subdomains. This allows superconvergence estimates to be derived for problems with solutions of low global regularity, particularly those involving singularities.     

Some boundary value problems for pseudodifferential equations with degeneration

Boundary value problems for a new class of degenerate pseudodifferential equations containing a variable-symbol degenerate pseudodifferential operator based on a special integral transform and the first derivative with respect to one of the variables are studied. Existence theorems for these problems are proved. A priori estimates for their solutions are obtained in special weighted spaces similar to Sobolev ones.