带松弛项的单个守恒律方程解的时态渐近性质

本文研究一维空间中带松弛项的单个守恒律方程解的大时间状态估计.在松弛项满足耗散条件下通过对线性化方程Green函数的逐点估计得到方程解在时间充分大时的衰减估计,并由此反映出“弱”惠更斯原理.     

NONLINEAR EVOLUTION SYSTEMS AND GREEN’S FUNCTION

In this paper, we will introduce how to apply Green’s function method to get the pointwise estimates for the solutions of Cauchy problem of nonlinear evolution equations with dissipative structure. First of all, we introduce the pointwise estimates of the time-asymptotic shape of the solutions of the isentropic Navier-Stokes equations and show to exhibit the generalized Huygen’s principle. Then, for other nonlinear dissipative evolution equations, we will only introduce the result and give some brief explanations. Our approach is based on the detailed analysis of the Green’s function of the linearized system and micro-local analysis, such as frequency decomposition and so on.     

带阻尼项一维非等熵流方程组初边值问题解的收敛速度

研究了带阻尼项一维非等熵流方程组的初边值问题，利用能量估计的办法证明了整体解的存在性和得到在L2－模及L∞－模意义下解的大时间状态稳定性估计。     

半线性偏微分方程的弱奇性行波的干扰

本文考虑在多维空间的情形下半线性偏微分方程解的不同弱奇性行波的干扰.证明了若两个载有不同弱奇性的特征曲面横截相交,且交口处还通过另外几个特征曲面,那么方程解的奇性在相交后除继续沿原特征面传播外,另外的几个特征面上也将产生新的奇性,其奇异程度满足所谓的“和规律”.     

带非局部耗散项的守恒律方程大扰动解的整体存在性

本文考虑带非局部耗散项的单个守恒律方程大扰动解的整体存在性.首先,针对方程次临界和临界两种不同的情形,利用Green函数方法和环形分解的技巧,构造开放式高频估计方法,得到了一个新的解的正则性准则.然后,利用极大值原理得到方程解的极大模的有界性,验证了次临界情形下解满足相应的正则性准则.对于临界这一更困难的情形,本文应用非线性极大值原理方法得到了更好一点的有界性估计,验证了临界情形下解也满足相应的正则性准则,从而得到了Cauchy问题大扰动经典解的整体存在性.     

具超音速边界可压Navier-Stokes方程解的指数衰减

考虑了二维空间上具超音速物理边界的可压Navier-Stokes方程的初边值问题.给定常数平衡态(ρ~*,0),得到了所考虑问题解的整体存在性.在平衡态附近的小扰动下,利用加权能量估计方法得到解的指数衰减性.     

POINTWISE ESTIMATE OF SOLUTIONS OF ISENTROPIC NMIER-STOKES EQUATIONS IN EVEN SPACE-DIMENSIONS

1 IntroductionIn this paPer, we are interested in tlie time-asymptotic behavior of solutions to isentropicNavier-stokes equations in even space-dimensions. Recently many people began to study it'sasymptotic behavier when time is large enough. This can be manifested in tlie after refer-ences. For the pointwise estimate of solution to generaJ hyperbolic-parabolic systems in onespace dimension see Liu and Zeng[']. In several space variables, Kawashima[31 has studiedgeneral hyperbolic-parabolic…     

测不准原理与Cauchy问题的唯一性

本文讨论具非光滑特征的二阶椭圆偏微分算子Cauchy问题的唯一性.借用测不准原理的思想,通过将方程的解的精细微局部分解,把唯一性证明中最关键的Carleman估计在微局部的层次展开,从而可以在相差一个低阶项的意义下“凝固”某些奇异点的系数,克服因非光滑特征带来的困难,在较以往文章更一般的条件下证明了二阶椭圆微分算子Cauchy 问题的唯一性.     

半线性方程强弱余法奇性波的相互干扰

首先定义并考查了一类新的函数空间,即所谓二重余法分布．在得到了一些建立在此类余法分布空间上的能量估计之后,证明了半线性严格双曲的一阶偏微分方程组的弱奇性与强奇性相互作用后将会产生透射与反射等新奇性,而且其强度不强于原弱奇性的强度．     

POINTWISE ESTIMATES OF SOLUTION FOR NON-ISENTROPIC NAVIER-STOKES-POISSON EQUATIONS IN MULTI-DIMENSIONS

The Cauchy problem of the non-isentropic Navier-Stokes-Poisson equations in multi-dimensions is considered.The global existence and pointwise estimates of the classical solution are given,which extend the optimal decay rate in L 2-norm in [27] to the L p(R n)(p > n/n-1)-norm.