On the Maxwell problem

We study the large-time behavior of global smooth solutions to the Cauchy problem for hyperbolic regularization of conservation laws. An attracting manifold of special smooth global solutions is determined by the Chapman–Enskog projection onto the phase space of consolidated variables. For small initial data we construct the Chapman–Enskog projection and describe its properties in the case of the Cauchy problem for moment approximations of kinetic equations. The existence conditions for the Chapman–Enskog projection are expressed in terms of the solvability of the Riccati matrix equations with parameter. Bibliography: 21 titles. Translated from Problems in Mathematical Analysis 39 February, 2009, pp. 27–63.     

一维Navier-Stokes方程组整体经典解

本文作者考虑了一维Navier-Stokes方程组Cauchy问题,得到了其经典解的整体光滑可解性。     

Error estimates for the Galerkin method as applied to time-dependent equations

A projection method is studied as applied to the Cauchy problem for an operator-differential equation with a non-self-adjoint operator. The operator is assumed to be sufficiently smooth. The linear spans of eigenelements of a self-adjoint operator are used as projection subspaces. New asymptotic estimates for the convergence rate of approximate solutions and their derivatives are obtained. The method is applied to initial-boundary value problems for parabolic equations.     

Global Smooth Solutions to the Cauchy Problem of Nonlinear Thermoelastio Equations with Dissipation

This paper considers the Cauchy problem for the nonlinear thermoelastic equations with dissipation. By means of energy methods, under the reasonable assumptions in mechanics the global existence, uniqueness and the decay rates of smooth solutions are proved.     

三维可压等熵Euler方程光滑解的整体存在性

研究了三维可压等熵Euler方程Cauchy问题光滑解的整体存在性.如果初值是一个常状态的小扰动并且初速度的旋度等于零,证明了三维可压等熵Euler方程Cauchy问题光滑解的整体存在性.     

Existence of global smooth solution to the relativistic Euler equations

In this paper, we prove an existence theorem of global smooth solutions to the Cauchy problem for the one-dimensional relativistic Euler equations. The analysis is based on a priori estimates which are obtained by the characteristic method and maximum principle.     

Cauchy Problem for Quasilinear Hyperbolic Systems with Higher Order Dissipative Terms

Abstract In this paper, the author studies the global existence, singularities and life span of smoothsolutions of the Cauchy probleth for a class of quasilinear hypetbolic systems with higher order dissipativeterms and gives their applications to nonlinear wave equations with higher order dissipative terms.     

ON THE CLASSIFICATION OF INITIAL DATAFOR NONLINEAR WAVE EQUATIONS

InroductionFor hyperbollc equations the most fundamental problem Is the     

一类广义长短波方程的整体适定性

In the present paper,we investigate the well-posedness of the global solution for the Cauchy problem of generalized long-short wave equations.Applying Kato's method for abstract quasi-linear evolution equations and a priori estimates of solution,we get the existence of globally smooth solution.     

一类Boussinesq型方程整体解的存在性和唯一性

证明一类6阶Boussinesq型方程Cauchy问题整体广义解和整体古典解的存在性和唯一性,给出解在有限时刻发生爆破的充分条件.     

The Cauchy problem for a spin-liquid model in three space dimensions

In this paper, we study the Cauchy problem for magnetic fluid of spin-liquid type with Mermin-Ho relation in the three-dimensional space and prove global existence and uniqueness of solutions. The proof is based on the equivalence relation between smooth solutions of the spin-liquid model and the systems of Schrödinger equations with Abelian gauge field. The Sobolev spaces with fractional derivatives are also used in our estimates.     

Global existence of smooth solutions to 2D Chaplygin gases on curved space

This paper investigates the smooth solution of 2D Chaplygin gas equations on an asymptotically flat Riemannian manifold. Under the assumption that the initial data are close to a constant state and the vorticity of the initial velocity vanishes, we prove the global existence of smooth solutions to the Cauchy problem for two‐dimensional flow of Chaplygin gases on curved space. Copyright © 2016 John Wiley & Sons, Ltd.     

KdV-KSV方程的初边值问题

本文证明了KdV－KSV方程的周期边值问题和Cauchy问题广义解和古典解的整体存在性、正则性及唯一性．     

非自治退化抛物-双曲方程的柯西问题

非自治退化抛物-双曲型方程可以描述许多自然现象.主要研究这类方程的柯西问题,建立了动力学公式,在对流函数、扩散函数适当光滑性的基础上,证明了该问题动力学解的存在唯一性.     

Entropy solutions for linearly degenerate hyperbolic systems of rich type

Consider a linearly degenerate hyperbolic system of rich type. Assuming that each eigenvalue of the system has a constant multiplicity, we construct a representation formula of entropy solutions in L^∞ to the Cauchy problem. This formula depends on the solution of an autonomous system of ordinary differential equations taking x as parameter. We prove that for smooth initial data, the Cauchy problem for such an autonomous system admits a unique global solution. By using this formula together with classical compactness arguments, we give a very simple proof on the global existence of entropy solutions. Moreover, in a particular case of the system, we obtain an another explicit expression and the uniqueness of the entropy solution. Applications include the one-dimensional Born–Infeld system and linear Lagrangian systems.     

一类具阻尼非线性波动方程的Cauchy问题

本文研究一类具阻尼非线性波动方程Cauchy问题整体广义解和整体古典解的存在唯一性,并用凸性方法给出解爆破的充分条件.     

On the Cauchy problem for the noncompact Landau-Lifshitz-Gilbert equation

Local well-posedness of the Cauchy problem for the noncompact Landau-Lifshitz-Gilbert equation is investigated via the pseudo-stereographic projection. Existence of global solutions is established for small initial data. In the case of one space dimension global existence theorems are proved for large initial data.     

一般非线性扩散方程Cauchy问题广义解的渐近性质

本文研究了比较一般的非线性扩散方程的Cauchy问题广义解的渐近性质.利用试验函数与积分估计的方法,获得了广义解当t→∞时的渐近性质以及其广义整体解u(x,t),推广了该类方程Cauchy问题广义解的性质.     

Existence and uniqueness of global classical solutions to 3D isentropic compressible Navier-Stokes equations with general initial data

In this paper we study the global existence and uniqueness of classical solutions to the Cauchy problem for 3D isentropic compressible Navier-Stokes equations with general initial data which could contain vacuum.We give the relation between the viscosity coefficients and the initial energy,which implies that the Cauchy problem under consideration has a global classical solution.     

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.