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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

主要研究员小川教授取得了以下成果。他研究了临界型的Sobolev型不等式，特别是Besov和Triebel-Lirzorkin空间等实插值空间，并将其推广到抽象的Besov和Lorentz空间。那些涉及对数插值阶的不等式可以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3维Navier-Stokes方程弱解的正则性和唯一性准则以及Euer方程的分解条件。在类似的方法中，他还展示了二维谐波热流平滑求解球体的正则性判据。特别是，对于简谐热流的弱解，类似的规律性判据也成立。其结果是通过建立解的能量密度的平均振荡的“单调性公式”得到的。他还考虑了非局部型半线性抛物型方程解的渐近行为。这些系统出现在各种物理尺度中，如半导体模拟模型、趋化性模型和天文学中恒星的诞生。该系统涉及泊松方程，因为电荷或粘液阿米巴的密度产生的场和非局部效应对于解决方案的分析至关重要。他专门研究了临界情况，即二维情况，证明了强迫漂移扩散情况下的系统在临界Hardy空间中存在时间局部解、达到阈值初始密度的时间全局解和有限时间爆炸。此外，小数据解的渐近行为由热核来表征。此外，如果场方程以某种非线性方式进行扰动，则在径向对称的情况下，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、半空间和外域中的渐近行为，并表明小解将被分解为线性热方程的解，即线性波动方程与非线性效应的某种组合。之前已经针对 1 维和 3 维情况展示了这一点，但是那里的方法不适用于 2 维情况。较少的