一类弱耗散Camassa-Holm 方程Cauchy问题在Besov 空间解的局部适定性

利用Littlewood-Paley 理论和输运方程解的先验估计, 在Besov 空间 中证明了一类弱耗散Camassa-Holm 方程Cauchy 问题解的局部适定性, 同时给出了解的能量估计及爆破准则.     

具有强迫项Dullin-Gottwald-Holm方程整体耗散解的存在性

本文借鉴Bressan和Constantin于2007年提出的新特征线法,利用具有强迫项的Dullin-Gottwald-Holm方程的平衡律和一些新的估计,证明了该方程在H~1(R)中整体耗散解的存在性.     

带耗散的广义Camassa-Holm方程的吸引子

讨论了一类带耗散的广义Camassa-Holm方程．先将方程的解以及初始条件化为积分平均为零，然后建立与原问题相应的周期初值问题近似解的先验估计，由此得到原问题解的存在唯一性，并证明了在H^2per（Ω）中吸引子的存在性．     

一类弱耗散Camassa-Holm方程局部强解的适定性及弱解的存在性

使用Pseudoparabolic正则化方法和从弱耗散Camassa-Holm方程自身导出的估计式,在Sobolev空间Hs(R)(s3/2)中,证明了该Camassa-Holm方程解的局部适定性.同时给出了一个在空间Hs(R)(1s2\3)中确保该方程弱解存在的充分条件.     

随机耗散Camassa-Holm方程的吸引子

可以接轨道得到带白噪声的随机耗散Camassa-Holm方程的唯—解并且可以检验该解产生随机动力系统,从而证明了该随机动力系统在H₀²中存在紧的随机吸引子.     

Global well-posedness for the viscous Camassa-Holm equation

We consider the initial value problem (IVP) of the Camassa-Holm equation with viscosity. We established global solution for the IVP with u₀∈L²(R). This result improves the previous results.     

The Exact Limits and Improved Decay Estimates for All Order Derivatives of the Global Weak Solutions to a Two-Dimensional Incompressible Dissipative Quasi-Geostrophic Equation

We will accomplish the exact limits for all order derivatives of the global weak solutions to a two-dimensional incompressible dissipative quasi-geostrophic equation. We will also establish the improved decay estimates with sharp rates for all order derivatives. We will consider two cases for the initial function and the external force and prove the optimal results for both cases. We will couple together existing ideas (including the Fourier transformation and its properties, Parseval''s identity, iteration technique, Lebesgue''s dominated convergence theorem, Gagliardo-Nirenberg-Sobolev interpolation inequality, squeeze theorem, Cauchy-Schwartz''s inequality, etc) existing results (the existence of global weak solutions, the existence of local smooth solution on $(T,\infty)$ and the elementary decay estimate with a sharp rate) and a few novel ideas to obtain the main results.     

一类广义四阶非线性Camassa-Holm方程的行波解

用动力系统的分支理论研究了一类广义四阶非线性Camassa-Holm方程的动力学行为和行波解,发现方程存在一些孤立波解,周期波解和一些诸如Compacton类型的非光滑行波解.在不同的参数条件下,给出了这些解存在的条件和一些特殊条件下的精确解.     

On the Cauchy Problem of the Camassa-Holm Equation

The purpose of this paper is to investigate the Cauchy problem of the Camassa-Holm equation. By using the abstract method proposed and studied by T. Kato and priori estimates, the existence and uniqueness of the global solution to the Cauchy problem of the Camassa-Holm equation in L ^p frame under certain conditions are obtained. In addition, the continuous dependence of the solution of this equation on the linear dispersive coefficient contained in the equation is obtained.     

关于五阶非线性微分方程一致耗散解的一个结果

Criteria for the existence of uniformly dissipative solutions for a certain fifth order non-linear differential equation are given by means of the frequency domain method.     

二维Stokes近似系统弱解的全局存在性

本文主要考虑了二维可压流体的Stokes近似系统,我们运用类似于Eduard Feireisl的方法证明了Stokes近似系统在,γ=1与γ>1时弱解的全局存在性.     

The regularity of local solutions for a generalized Camassa-Holm type equation

The regularity of the Cauchy problem for a generalized Camassa-Holm type equation is investigated. The pseudoparabolic regularization approach is employed to obtain some prior estimates under certain assumptions on the initial value of the equation. The local existence of its solution in Sobolev space Hs （R） with 1 〈 s ≤ 3/2 is derived.     

Gibbs Measure for the Higher Order Modified Camassa-Holm Equation

This paper is devoted to constructing a globally rough solution for the higher order modified Camassa-Holm equation with randomization on initial data and periodic boundary condition.Motivated by the works of Thomann and Tzvetkov(Nonlinearity,23(2010),2771–2791),Tzvetkov(Probab.Theory Relat.Fields,146(2010),4679–4714),Burq,Thomann and Tzvetkov(Ann.Fac.Sci.Toulouse Math.,27(2018),527–597),the authors first construct the Borel measure of Gibbs type in the Sobolev spaces with lower regularity,and then establish the existence of global solution to the equation with the helps of Prokhorov compactness theorem,Skorokhod convergence theorem and Gibbs measure.     

Existence Uniqueness and Decay of the Global Weak Solutions for a Class of Parabolic Systems with Nonlinear Boundary Conditions

In this paper, a class of parabolic systems with nonlinear boundary conditions is discussed. By introducing a complete metric space with decay of W^s_p-norm, we obiain the existence uniqueness of global weak solutions, our method is simpler than before. A decay estimate of the global weak solutions is obtained simultaneously.     

Investigation of Multi-soliton, Multi-cuspon Solutions to the Camassa-Holm Equation and Their Interaction

The authors study the multi-soliton,multi-cuspon solutions to the CamassaHolm equation and their interaction.According to the solution formula due to Li in 2004and 2005,the authors give the proper choi...     

二阶Camassa-Holm方程行波解

对二阶Camassa-Holm方程行波解的情况进行了讨论.利用解的唯一性,得到了如下结论:二阶CH方程的行波解唯一存在,但不具有u(x,t)=kem(x-ct)形式.还为二阶CH方程行波解的研究提供了一种新途径和方法.     

一类非线性发展方程整体解的存在性、渐近性与解的爆破

研究一类非线性发展方程初边值问题整体弱解的存在性，渐近性和解的爆破问题，证明在关于非线性项的不同条件下，上述初边值问题分别在大初值和小初始能量的情况下存在整体弱解，并且讨论了弱解的渐近性。还证明：在相反的条件下，上述弱解在有限时刻爆破，并且给出了一个实例。     

抽象空间常微分方程弱解的存在和唯一性

本文研究了抽象空间中初值问题ｘ′＝ ｆ（ｔ,ｘ）,ｘ（ｔ０）＝ ｘ０ 的弱解存在和唯一性,并推广了文［１］、［２］中的有关结果     

具任意次非线性项的Camassa-Holm方程的双孤子新解

给出辅助方程、函数变换与变量分离解相结合的方法,构造了具任意次非线性项的Camassa-Holm方程的双孤子和双周期新解.首先,通过两个辅助方程、函数变换与变量分离解,将具任意次非线性项的Camassa-Holm方程的求解问题转化为非线性代数方程的求解问题.然后,借助符号计算系统Mathematica求出该方程组的解,并用辅助方程的相关结论,构造了双周期解和双孤子新解.     

带耗散项的浅水波方程的整体边界稳定性

在[0,1]区间上研究带耗散项的浅水波方程由边界反馈引起的整体指数稳定性.在控制边界条件u(0)=u_x(1)=u_(xx)(0)=δu_(xxx)(1)-εu(1)u_(xx)(1)-(?)(u(1))=0之下,证明了解的H1整体指数稳定性和H4整体渐进稳定性.