Large-time Behavior of Solutions for Parabolic Conservation Laws with Large Initial Data

In this paper,we study the large-time behavior of periodic solutions for parabolic conservation laws.There is no smallness assumption on the initial data.We firstly get the local existence of the solution by the iterative scheme,then we get the exponential decay estimates for the solution by energy method and maximum principle,and obtain the global solution in the same time.

FIRST ORDER QUASILINEAR EQUATIONS IN SEVERAL INDEPENDENT VARIABLES WITH SINGULAR INITIAL DATA L^p（P＜∞）

We study the existence problem for the equations of first order quasilinearequations in several inpendent variables with singular initial data Lp(P<∞). We the convergence of the Lp(P<∞) bounded approximating sequences generatedby the method of vanishing viscosity. The uniqueness of the generalized solutions whichcan be obtained by the method of vanishing viscosity is also obtained.

Global axisymmetric classical solutions of full compressible magnetohydrodynamic equations with vacuum free boundary and large initial data

In this paper,the global existence of the classical solution to the vacuum free boundary problem of full compressible magnetohydrodynamic equations with large initial data and axial symmetry is studied.The solutions to the system(1.6)–(1.8) are in the class of radius-dependent solutions,i.e.,independent of the axial variable and the angular variable.In particular,the expanding rate of the moving boundary is obtained.The main difficulty of this problem lies in the strong coupling of the magnetic field,velocity,temperature and the degenerate density near the free boundary.We overcome the obstacle by establishing the lower bound of the temperature by using different Lagrangian coordinates,and deriving the uniform-in-time upper and lower bounds of the Lagrangian deformation variable r;by weighted estimates,and also the uniform-in-time weighted estimates of the higher-order derivatives of solutions by delicate analysis.

Large-Time Behavior of Periodic Solutions to Fractal Burgers Equation with Large Initial Data

The asymptotic behavior of periodic solutions to fractal nonlinear Burgers equation is considered and the initial data are allowed to be arbitrarily large.The exponential decay estimates of the solutions are obtained for the power of Laplacian α∈[1/2,1).

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.

Large Solutions to the Generalized Debye-Hückel System in Fourier-Besov Spaces

The special structure in some coupled equations makes it possible to drop partial smallness assumption of the initial data to gain the global well-posedness.In this paper,we study the Cauchy problem for generalized Debye-Hückel system in Fourier-Besov spaces.Under more generalized index range,we obtain the global solution with small initial data and local solution with arbitrary initial.Besides,by constructing some weighted function,we prove that the global well-posedness still holds under the small assumption of the charge of initial data.Thus we show that although the initial densities and the hole in electrolytes are large,the equation is still global well-posedness.

GLOBAL WELL-POSEDNESS FOR THE DENSITY-DEPENDENT INCOMPRESSIBLE MAGNETOHYDRODYNAMIC FLOWS IN BOUNDED DOMAINS

In this paper, we study the three-dimensional incompressible magnetohydrody-namic equations in a smooth bounded domains, in which the viscosity of the fluid and themagnetic diffusivity are concerned with density. The existence of global strong solutions isestablished in vacuum cases, provided the assumption that （｜ ｜μ（ρ0）｜｜Lp＋｜｜ v（P0）｜｜Lq＋｜｜b0｜｜L^3 ＋｜｜ρO｜｜L^∞） （p,q〉3） is small enough, there is not any smallness condition on thevelocity.

Global Smooth Solution to the Incompressible Navier-Stokes-Landau-Lifshitz Equations

In this paper, we will investigate the incompressible Navier-Stokes-Landau-Lifshitz equations, which is a system of the incompressible Navier-Stokes equations coupled with the Landau-Lifshitz-Gilbert equations. We will prove global existence of the smooth solution to the incompressible Navier-Stokes-Landau-Lifshitz equation with small initial data in T2or R2and R3.

输运系数依赖温度的可压缩辐射流体解的整体存在性

该文主要研究了当粘性系数λ和热传导系数κ依赖于温度θ,即λ(θ)=θ^(α),κ(θ)=1+θ^(β),其中α∈[0,+∞),β∈(2,+∞)时,带有辐射项的可压缩Navier-Stokes方程解的全局存在性和非线性稳定性.在关于参数α和初值的某些假设下,该文得到了强解的全局存在唯一性.此外,在基于时间的一直估计下,本文还证明了解的非线性指数稳定性.需要指出的是,如果α较小,并且增长指数β任意大,则初值可以任意大.

Global well-posedness and large-time behavior of 1D compressible Navier-Stokes equations with density-depending viscosity and vacuum in unbounded domains

We consider the Cauchy problem for one-dimensional(1D)barotropic compressible Navier-Stokes equations with density-depending viscosity and large external forces.Under a general assumption on the densitydepending viscosity,we prove that the Cauchy problem admits a unique global strong(classical)solution for the large initial data with vacuum.Moreover,the density is proved to be bounded from above time-independently.As a consequence,we obtain the large time behavior of the solution without external forces.

带粘性含不活泼项Cahn-Hilliard方程解的存在性

本文研究了带粘性的含不活泼项Cahn-Hilliard方程柯西问题.利用格林函数不同频域的详细分析及不动点原理,得到了方程大初值情形经典解的整体存在性,推广了以往工作集中在弱解或拟强解或小初值情形下的经典解.

非线性波动方程的长时间解

主要讨论非线性波动方程u_(?)-△u=f(t,x,u) (*)的小初值问题的几乎整体解(almost global solution)的存在问题,证明了,若|f(t,x,u)|≤C|u|~p,x∈ R^(?),1<p≤1+(2/(n-4))(n>4);1<p<+∞(n≤4),则对任意给定的正数 T,有依赖于 T 的δ>0存在,当初值 u(0,x),u_(?)(0,x)的范数‖u(0)‖(?)(R^(?))+‖u_(?)‖(?)(R^(?))≤δ时,(*)式的 Cauchy 问题的小振幅解在 C^1(0,T;H^1)∩ C(0,T;H^2)中唯一存在.

一维可压缩Navier-Stokes-Korteweg方程组的大初值整体光滑解（英文）

本文研究了当粘性系数和毛细系数是密度函数的一般光滑函数时,一维等温的可压缩Navier-Stokes-Korteweg方程的Cauchy问题.利用基本能量方法和Kanel的技巧,得到了大初值、非真空光滑解的整体存在性与时间渐近行为.本文结果推广了已有文献中的结论.

四阶非线性Schrodinger方程在L^(p)初值空间中解的整体存在性研究

本文考虑当初值u0∈L^(p)时,四阶非线性Schrodinger方程的柯西问题的整体适定性.在p≠2的情形下,关于该柯西问题的解的存在性结论较少.受已有二阶Schrodinger方程启发,借助一个关于L^(p)初值的分解引理和Strichartz估计,在p>2且非线性项指数满足一定条件下,本文得到了该四阶非线性Schrodinger方程的柯西问题的解的整体存在性.

具大始值一阶拟线性双曲方程组Cauchy问题的整体光滑解

本文证明在大初值条件一类拟线性双曲方程组Ｃａｕｃｈｙ问题整体光滑解的存在性

非严格双曲型Kirchhoff方程

作者讨论了一个非齐次Kirchhoff方程在非严格双曲型条件下的Cauchy问题。在适当假设下，得到了此问题整体解的存在唯一性。其结果适用于一类依赖小参数的kirchhoff方程。

初值间断的可压缩Navier-Stokes方程弱解的存在性

本文研究了粘性系数依赖密度的一维可压缩Navier-Stokes方程的初值间断问题.当初始密度间断任意大时,证明了一维可压缩Navier-Stokes方程固定边界问题整体弱解的存在唯一性,分段正则性,并给出了弱解的大时间行为等.

黏性依赖于密度的一维可压缩黏性辐射反应气体方程组的Cauchy问题

本文考虑一维可压缩黏性辐射反应气体方程组,对一类依赖于密度的、退化的黏性系数,得到了其Cauchy问题大初值非真空整体解的存在性.问题的关键在于如何得到密度和温度的正的上下界估计.

初始值具有大震荡的椭圆BGK模型整体解的存在性

椭圆BGK模型的提出是为了正确匹配经典BGK模型的Navier-Stokes方程逼近中的Prandtl数.本文证明了在初始值有较大震荡情形下该模型Cauchy问题温和解的整体存在性,证明方法来源于最近关于Boltzmann方程类似问题的研究.

ＯＮ ＴＨＥ ＣＬＡＳＳＩＦＩＣＡＴＩＯＮ ＯＦ ＩＮＩＴＩＡＬ ＤＡＴＡＦＯＲ ＮＯＮＬＩＮＥＡＲ ＷＡＶＥ ＥＱＵＡＴＩＯＮＳ

The purpose of the present paper is to call for attention to the following question: Which of the initial data (nonsmall) admit global smooth solutions to the Cauchy problem for nonlinear wave equations. A few cases and examples are sketched, showing that the general answer of this question may be quite complicated.