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Reseach for the singularities and regularity of solutions to crtical nonlinear partial differential equations

临界非线性偏微分方程解的奇异性和正则性研究

基本信息

批准号：
15340056
负责人：
OGAWA Takayoshi
金额：
$ 8.26万
依托单位：
Tohoku University (2004-2006)Kyushu University (2003)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2006
项目状态：
已结题

项目摘要

The main researcher, Prof.Ogawa obtained the following results. He researched for the Sobolev type inequality of the critical type, especially for the real interpolation spaces such as Besov and Triebel-Lirzorkin spaces and generalized it for the abstract Besov and Lorentz space. Those inqualities involving the logarithmic interpolation order can be applied for the regularity and uniqueness criterion of the seimilinear partial differential equation. In a series of collaboration with the research colabolators, he shows that the reguarlity and uniquness criterion for the weak solution of the 3 dimensional Navier-Stokes equations and break down condition for the Euer equation. In a similar method, he also showed the regularity criterion for the smooth solution of the 2 dimensional harmonic heat flow into a sphere. In particular, for the weak solution of the harmonic heat flow, the similar regularity criterion is also holds. The result is obtained by establishing the "monotonicity formula" … More for the mean oscillation of the energy density of the solutions.He also consider the asymptotic behavior of the solution for the semi-lineear parabolic equation of the non-local type. Those system appeared in a various Physical scaling such as semi-conductor simulation model, Chemotaxis model and the birth of star in Astronomy. The system is involving Poisson equation as the field generated by the dencity of the charge or mucous ameba and the non-local effect is essential for the analysis of the solution. He particulariy investigated to the critical situation, 2-dimensional case, and showed that there exists a time local solution in the critical Hardy space, time global solution upto the threshold initial density and finite time blow-up for the system of forcusing drift-diffusion case. Besides, the asymootitic behavior of the solution for small data is characterized by the heat kernel. Moreover if the field equation is purterbed in a certain nonlinear way, then there exist two solutions for the same initial data in a radially symmetric case.He also studied for the asymptotic behavior of the solution for the semi-linear damped wave equation in whole and half spaces and exterior domains and show the small solution is going to be decomposed into the solutions of the linear heat equation, some combination of linear wave equation with nonlinear effect. This was shown for 1 and 3 dimensional cases before, however the mothod there could not be applicable for the 2dimensional case. Less

主要研究者小川教授得到了以下结果。他研究了临界型的Sobolev型不等式，特别是对于Besov和triiebel - lirzorkin等实际插值空间，并将其推广到抽象的Besov和Lorentz空间。这些涉及对数插值阶数的判定可应用于半线性偏微分方程的正则性和唯一性判据。在与研究合作者的一系列合作中，他证明了三维Navier-Stokes方程弱解的正则性和唯一性判据以及Euer方程的分解条件。在一个类似的方法中，他还给出了二维调和热流进入球体的光滑解的正则性准则。特别地，对于调和热流的弱解，也具有类似的正则性准则。通过建立解的能量密度平均振荡的“单调性公式”得到了结果。他还考虑了非局部型半线性抛物方程解的渐近性态。这些系统以各种物理尺度出现，如半导体模拟模型、趋化性模型和天文学中的恒星诞生模型。该系统涉及泊松方程作为电荷或黏液阿米巴密度产生的场，非局部效应对溶液的分析至关重要。特别对临界情况、二维情况进行了研究，证明了强迫漂移扩散情况系统在临界Hardy空间存在时间局部解，在达到阈值初始密度的时间全局解和有限时间爆破。此外，对于小数据解的非对称性质用热核来表征。此外，如果场方程以一定的非线性方式进行假设，则在径向对称情况下，对于相同的初始数据存在两个解。他还研究了半线性阻尼波动方程解在全空间、半空间和外域的渐近行为，并表明小解将分解成线性热方程的解，即线性波动方程与非线性效应的某种组合。这在1维和3维的情况下已经证明了，但是这种方法不适用于2维的情况。少