Asymptotic Analysis for a Class of Infinite Systems of First-Order PDE: Nonlinear Parabolic PDE in the Singular Limit

Abstract

We study the asymptotic behavior of solutions of the Cauchy problem for a functional partial differential equation with a small parameter as the parameter tends to zero. We establish a convergence theorem in which the limit problem is identified with the Cauchy problem for a nonlinear parabolic partial differential equation. We also present comparison and existence results for the Cauchy problem for the functional partial differential equation and the limit problem.     

Existence of a solution of the cauchy problem for an integrodifferential equation of viscoelasticity

Mechanics of Composite Materials - The Cauchy problem for a nonlinear integrodifferential equation is reduced to a nonlinear integral equation for which a theorem of existence and uniqueness is...     

Dynamics of solutions of the Cauchy problem for semilinear parabolic stochastic partial differential equations with power-law singularities

We prove a comparison theorem for bounded solutions of the Cauchy problem for stochastic partial differential equations of the parabolic type with linear leading part. The drift and diffusion coefficients have locally bounded derivatives with respect to the state variable. We use this comparison theorem to study the dynamics of solutions of an equation with an absorber and an equation with a source.     

非局部非线性发展方程解的存在性与唯一性

本文利用Ｂａｎａｃｈ不动点定理证明一类非线性发展方程的非局部Ｃａｕｃｈｙ问题解的存在性与唯一性，及其对非局部拟线性抛物方程的应用。     

一类高阶超双曲型方程的中量定理及其逆定理

Asgeirsson中量定理表明超双曲型方程的Cauchy问题一般是不适定的,对Asgeirsson中量定理的推广就有重要意义。目前关于高阶方程解的中量只有初步探讨,尚未得到具体结果,本文直接利用Asgeirsson中量定理结果和积分、微分的性质与关系,得到了高阶方程解的中量满足广义双轴对称位势方程,同时还证明了其逆定理。利用关于广义双轴对称位势方程正则解的表达式及雅可比多项式的特殊性质,得到了高阶方程解的中量公式,从而使得关于解的拓展性和适定性的讨论将有可能。     

Condition of uniform well-posedness of the Cauchy problem in the case of spaces with indefinite metric

In a Hilbert space H with indefinite J-metric, the uniform well-posedness of the Cauchy problem is studied for the equation dx/dt=Ax, where A is a maximal J-dissipative operator. In particular, in the case of a Pontryagin space an analog of Phillips's theorem on the relation between the maximal dissipative operator A and the uniform well-posedness of the Cauchy problem is obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 697–700, May, 1990.     

On Inhomogeneous GBBM Equations

In this paper we consider the Cauchy problem and the initial boundary value (IBV) problem for the inhomogeneous GBBM equations. For any bounded or unbounded smooth domain, the existence and uniqueness of global strong solution for the Cauchy problem and IBV problem for the inhomogeneous GBBM equations in W^{2,p}(Ω) are established by using Banach fixed point theorem and some a priori estimates. These results have improved the known results even in the case of GBBM equation. Meanwhile, we also discuss the regularity of the Strong solution and the system of inhomogeneous GBBM equations.     

Cauchy problem in a scale of Banach spaces

The concept of quasidifferential operator in a scale of Banach spaces is formulated. A theorem of existence and uniqueness of a solution to the Cauchy problem for the equation with a nonlinear quasidifferential operator is proved. As an example of application of the theorem, the correctness of the nonlinear nonlocal problem of plane-parallel unsteady potential motion of a liquid with free boundary is proved.     

Complete integrability of the Kadomtsev-Petviashvili equation

The total hierarchy of the Kadomtsev-Petviashvili (KP) equation is transformed to the system of linear partial differential equations with constant coefficients. The complete integrability of the KP equation is proved by using this linear system. The existence and uniqueness theorem of the Cauchy problem of the KP hierarchy is obtained.     

积分型Cauchy中值定理的推广及逆问题

给出了一个积分型Cauchy中值定理的推广,并讨论了连续函数的积分型Cauchy中值定理的逆问题.     

Equations containing locally Henstock-Kurzweil integrable functions

A fixed point theorem in ordered spaces and a recently proved monotone convergence theorem are applied to derive existence and comparison results for solutions of a functional integral equation of Volterra type and a functional impulsive Cauchy problem in an ordered Banach space. A novel feature is that equations contain locally Henstock-Kurzweil integrable functions.     

The Cauchy Problem for Certain Inhomogeneous Difference-Differential Parabolic Equations

We prove an existence and uniqueness theorem for classical solutions of the Cauchy problem for an inhomogeneous equation of parabolic type with shifted space variables.     

A New Class of Uniformly Second Order Accurate Difference Schemes for 2D Scalar Conservation Laws

In this paper, concerned with the Cauchy problem for 2D nonlinear hyperbolic conservation laws, we construct a class of uniformly second order accurate finite difference schemes, which are based on the E-schemes. By applying the convergence theorem of Coquel-Le Floch [1], the family of approximate solutions defined by the scheme is proven to converge to the unique entropy weak $L^{\infty}$-solution. Furthermore, some numerical experiments on the Cauchy problem for the advection equation and the Riemann problem for the 2D Burgers equation are given and the relatively satisfied result is obtained.     

非线性发展方程非局部初值问题解的存在性

本文讨论非线性方程的非局部初值问题，利用半群理论、单调算子理论以及Banach不动点定理，分别得到两个问题的温和解和弱解的存在唯一性定理．     

弱耗散Camassa-Holm方程解的Blowup及衰减性质

本文研究弱耗散Camassa-Holm方程的Cauchy问题,由Kato理论得到了局部适定性的结果,证明了解的blowup及整体存在性,并证明了当耗散系数满足适当条件时,整体解具有衰减性质.     

Formal analysis of the Cauchy problem for a system associated with the (2+1)-dimensional Krichever-Novikov equation

The singularity manifold equation of the Kadomtsev-Petviashvili equation, the so-called Krichever-Novikov equation, has an exact linearization to an overdetermined system of partial differential equations in three independent variables. We study in detail the Cauchy problem for this system as an example for the use of the formal theory of differential equations. A general existence and uniqueness theorem is established. Formal theory is then contrasted with Janet-Riquier theory in the formulation of Reid. Finally, the implications of the results for the Krichever-Novikov equation are outlined.     

Time-independence of the solution of the problem of determining the hydraulic conductivity field

We consider the determination of the hydraulic conductivity field of a nonhomogeneous layer with linear seepage of an incompressible fluid in a nonelastic layer. In mathematical terms, the problem is formulated as a Cauchy problem for a partial differential equation of a special form. A theorem is proved which establishes that the solutions obtained for different times are identical.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 57, pp. 67–70, 1985.     

Homogenization Theorem for Problems in Domains of the “Infusorian” Type with Uncoordinated Structure

The model averaging problem for the Poisson equation in a perforated domain with rapidly oscillating exterior boundary is considered. The perforation and the boundary oscillation are assumed to be locally periodic, their structures are concordant, and have different orders of smallness. The averaged problem is constructed and the theorem of the convergence of solutions of the initial problem to the solution of the averaged problem is proved.     

The Tartar Equation for Homogenization of a Model of the Dynamics of Fine-Dispersion Mixtures

We consider a mathematical model describing a nonstationary Stokes flow in a fine-dispersion mixture of viscous incompressible fluids with rapidly oscillating initial data. We perform homogenization of the model as the frequency of oscillations tends to infinity; this leads to the problem of finding effective coefficients of the averaged equations. To solve this problem, we propose and implement a method which bases on supplementing the averaged system with the Cauchy problem for the kinetic Tartar equation whose unique solution is the Tartar H-measure. Thereby we construct a correct closed model for describing the motion of a homogeneous mixture, because the effective coefficients of the averaged equations are uniquely expressed in terms of the H-measure.     

Estimates for the Rate of Convergence in Ordinary Differential Equations under the Action of Random Processes with Fast Time

We study the procedure of averaging in the Cauchy problem for an ordinary differential equation perturbed by a certain Markov ergodic process. We establish several estimates for the rate of convergence of solutions of the original problem to solutions of the averaged one. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 4, pp. 435–457, April, 2005.