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

主要的研究员Ogawa教授获得了以下结果。他研究了关键类型的Sobolev类型不平等，尤其是对于真正的插值空间，例如Besov和Triebel-Lirzorkin空间，并将其概括为抽象的Besov和Lorentz空间。那些涉及对数插值顺序的查询可以用于标准层偏微分方程的规律性和独特标准。在与研究共蛋白公司的一系列合作中，他表明，navier-stokes方程的弱解决方案的易害性和UNICES标准，并破坏了EUER方程的条件。在类似的方法中，他还展示了两个维度的平滑溶液谐波热流入球体的规律性标准。特别是，对于谐波热流的较弱溶液，还持有相似的规律性标准。通过建立“单调性”公式来获得结果……更多用于溶液的能量密度的平均振荡。由电荷或粘液ameba的齿轮和非本地效应产生的田地对于对解决方案的分析至关重要。此外，如果磁场方程以某种非线性方式启动，则在径向对称的情况下存在两个解决方案，用于相同的初始数据。与非线性效应的等效性。在1和3维情况下显示了这一点，但是该基序不适用于2维情况。较少的