解定解问题的新途径

<正> 对于线性偏微分方程的Cauchy问题、半无界问题、混合问题等,一般介绍积分变换法、分离变量法等求解方法,但随着方程的阶数、维数的增高,求解过程也越来越繁杂。本文先建立Cauchy问题解析解的求解公式,并由此得到一条求解上述定解问题的新途径,算子级数法。     

含两参数的二阶常微分方程Cauchy问题解的多重层性质

研究含两参数的二阶常微分方程Cauchy问题解的多重层性质,根据不同层次引用不同的伸长变量,分别构造了具有不同量级的边界层校正项,从而证得关于解的一致有效的渐近展开式和有关的余项估计.     

关于非主型偏微分方程的唯一性的一点注记

<正> 1.引言 关于主型偏微分方程的Cauchy问题的唯一性的研究,已经十分充分.事实上,经典的Cauchy-Kovalevsky定理断言,解析方程的非特征Cauchy问题具有唯一性;而Goursat定理则断言,特征Cauchy问题的解必是唯一的,如果特征支柱上只具有单特征的话(参看Hormander[1]第五章).但是,对非主型方程,情形就大不相同.1974年,Treves研     

三维不可压缩黏弹性流体系统解的逐点估计

本文考虑三维不可压缩黏弹性流体力学模型的Cauchy问题.首先引入适当的变量变换,对变换后的方程组,研究其线性化系统的Green函数.接着,根据Green函数逐点估计方法,结合方程组解的表达式,分析Riesz算子的影响,得到解关于时空的逐点估计.     

双曲平均曲率流Cauchy问题经典解的生命跨度

本文研究了双曲平均曲率流,通过凸曲线的支撑函数,导出了一个双曲型Monge-Ampère方程并将其转化成Riemann不变量满足的拟线性双曲方程组,利用拟线性双曲方程组Cauchy问题的局部解理论,讨论了双曲平均曲率流Cauchy问题经典解的生命跨度(即局部解存在的最大时间区间).     

关于混合问题的离散现象

关于始值问题的唯一性中的离散现象,Treves研究过一个有趣的例子,他证明了下列Cauchy问题具有非平凡解的充分必要条件为P=1,3,5,…。他在1978年来华访问时,曾经指出关于存在性的离散现象具有一定的重要性。为此,王光寅等研究了这个方程的Cauchy问题和Goursat问题的存在性中的离散现象,证明了问题的C~∞解存在的充要条件是p≠1,     

α范数上的半线性分数阶发展方程的Cauchy问题

主要是对α范数上半线性分数阶发展方程Cauchy问题温性解的整体存在性进行研究.这里假设线性部分算子是一个紧的解析C_0半群的生成元.     

双参数Navier-Stokes方程组Cauchy问题的奇异极限

本文引入了带权的变量,利用Navier-Stokes方程组中耗散项的特殊结构,建立了一致的先验估计,得到了关于双参数λ(Mach数的倒数),γ(粘性系数)发生奇异极限情况:当λ→+∞,γ→0时,Cauchy问题解的存在和唯一性定理。     

关于线性偏微分方程的Cauchy问题的唯一性

<正> Cauchy问题的唯一性是偏微分方程的基本问题之一.经典的Cauchy-Kowalewski定理断言,解析方程或方程组的解析解是唯一的.1901年,Holmgren证明了,线性的解析方程或方程组的光滑解的唯一性.在取消关于系数的解析性的假设这个方向上的第一个结果是由Carleman在1939年给出的,他证明了两个自变量的相应结果,其中假设方程的主部的系数是实的,以及特征根是单重的,因而特征根的虚部如果不恒为零则总不为零.     

一阶拟线性非齐次偏微分方程存在整体光滑解的条件

在文献[1],[2]中讨论了一阶拟线性齐次偏微分方程 Cauchy 问题(1)(2)关于整体光滑解的存在性问题.文献[1]得到了λ_i=λ_i(u)时 Cauchy 问题(1)、(2)存在整体光滑解的充要条件;文献[2]进而得到了λ_i=λ_i(t,x,u)时 Cauchy 问题(1)、(2)存在整体光滑解的充要条件。本文将用[1]、[2]的思想方法,讨论一阶拟线性非齐次偏微分方程 Cauchy 问题     

Existence and uniqueness of holomorphic solutions for fractional Cauchy problem

By employing majorant functions, the existence and uniqueness of holomorphic solutions to nonlinear fractional partial differential equations (the Cauchy problems) are introduced. Furthermore, the analytic continuation of solutions is studied.     

Boundary value problems for a nonstrictly hyperbolic equation of the third order

We study classical solutions of boundary value problems for a nonstrictly hyperbolic third-order equation. The equation is posed in a half-strip and a quadrant of the plane of two independent variables. The Cauchy conditions are posed on the lower boundary of the domain, and the Dirichlet conditions are posed on the lateral boundaries. By using the method of characteristics, we find the analytic form of the solution of considered problems. The uniqueness of the solutions is proved.     

Partial regularity for harmonic maps of evolution into spheres

We consider the global Cauchy problem for generalized Kirchhoff equations with small non-linear terms or small data. We solve this problem in the space of functions which are twice differentiable with respect to time coordinate and uniformly analytic with respect to other coordinates. We determine, in two different situations, estimates of lifespan of solutions for some problems with perturbations and we give stability result of the solution for small perturbations.     

The problem of obtaining prescribed distributions of gas parameters

Using one-dimensional cylindrically and spherically symmetric flows as examples, the following problem is investigated in which a prescribed density distribution in a gas is obtained: given a known gas flow (background flow), it is required to continuously attach another, unknown gas flow whose density distribution at a fixed instant of time is described by some previously given function. It is shown that this problem is a characteristic Cauchy problem of standard type for which there is a valid analogue of the Kovalevskaya theorem, provided that the input data are analytic. Other problems of the same type are considered: to ensure that the unknown flow will have a prescribed gas velocity distribution and the prescribed density of the gas in the unknown flow is strictly greater than that of the gas in the background flow (flow with discontinuity). On the assumption that the input data to these problems are analytic, the existence and uniqueness of solutions are proved, in fact—of piecewise analytic solutions. The theorems proved are extended to the case of non-one-dimensional flows.     

A solution for antiplane‐strain local problems using elliptic integrals of Cauchy type

Quasi‐periodic piecewise analytic solutions, without poles, are found for the local antiplane‐strain problems. Such problems arise from applying the asymptotic homogenization method to an elastic problem in a parallel fiber‐reinforced periodic composite that presents an imperfect contact of spring type between the fiber and the matrix. Our methodology consists of rewriting the contact conditions in a complex appropriate form that allow us to use the elliptic integrals of Cauchy type. Several general conditions are assumed including that the fibers are disposed of arbitrary manner in the unit cell, that all fibers present imperfect contact with different constants of imperfection, and that their cross section is smooth closed arbitrary curves. Finally, we obtain a family of piecewise analytic solutions for the local antiplane‐strain problems that depend of a real parameter. When we vary this parameter, it is possible to improve classic bounds for the effective coefficients. Copyright © 2017 John Wiley & Sons, Ltd.     

The Schwarz problem for analytic functions in torus-related domains

The Cauchy kernel is one of the two significant tools for solving the Riemann boundary value problem for analytic functions. For poly-domains, the Cauchy kernel is modified in such a way that it corresponds to a certain symmetry of the boundary values of holomorphic functions in poly-domains. This symmetry is lost if the classical counterpart of the one-dimensional form of the Cauchy kernel is applied. It is also decisive for the establishment of connection between the Riemann–Hilbert problem and the Riemann problem. Thus, not only the Schwarz problem for holomorphic functions in poly-domains is solved, but also the basis is established for solving some other problems. The boundary values of functions, holomorphic in poly-domains, are classified in the Wiener algebra. The general integral representation formulas for these functions, the solvability conditions and the solutions of the corresponding Schwarz problems are given explicitly. A necessary and sufficient condition for the boundary values of a holomorphic function for arbitrary poly-domains is given. At the end, well-posed formulations of the torus-related problems are considered.     

Functions averages over the roots of unity,cauchy problems and addition formulas

An averaging operator over the roots of unity is defined on a class of analytic functions and its algebraic and analytic properties are investigated. A Cauchy like integral formula for this is obtained. This operator and its properties are then employed to solve higher order Cauchy problems, to derive addition formulas for hypergeometric functions and to obtain integral representations for special classes of hypergeometric functions.     

On the multisummability of divergent solutions of linear partial differential equations with constant coefficients

We consider the Cauchy problem for general linear partial differential equations in two complex variables with constant coefficients. We obtain necessary and sufficient conditions for the multisummability of formal solutions in terms of analytic continuation properties and growth estimates of the Cauchy data.     

On solvability of singular integral-differential equations with convolution

In this paper, we study a class of singular integral-different equations of convolution type with Cauchy kernel. By means of the classical boundary value theory, of the theory of Fourier analysis, and of the principle of analytic continuation, we transform the equations into the Riemann-Hilbert problems with discontinuous coefficients and obtain the general solutions and conditions of solvability in class $\{0\}$. Thus, the result in this paper generalizes the classical theory of integral equations and boundary value problems.     

On the Cauchy problem for the fractional drift-diffusion system in critical Besov spaces

In this paper, we establish global well-posedness and asymptotic stability of mild solutions for the Cauchy problem of the fractional drift-diffusion system with small initial data in critical Besov spaces. The regularizing-decay rate estimates of mild solutions are also proved, which imply that mild solutions are analytic in space variables.