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Mathematical Analysis to Nonlinear Partial Differential Equations in Mathematical Science and Its Applications

数学科学中非线性偏微分方程的数学分析及其应用

基本信息

批准号：
13440053
负责人：
SUZUKI Takashi
金额：
$ 7.23万
依托单位：
Osaka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-13440053/
关键词：
quantized blowup mechanism self-interacting particles chemotaxis self-dual gauge field theory Chem-Simons-Higgs theory dual variation tumour growth equation parallel optimization 送化性方程式 SU(3)戸田システム変分法 chemotaxis(走化性)free energy(自

项目摘要

We have established the quantized blowup mechanism for the non-equilibrium mean field of many self interacting. First, we formulate the equilibrium state as a nonlinear eigenvalue problem with non-local tern. Then, the above mechanism is observed in this level, and thus, the problem arises as a story of nolinear quantum mechanics. We have settled down this long standing problem from the study of the weak solution arising in the backward self similar transformation, employing the method of second moment and forward self similar transformation. Furthermore, we observed that the type I blowup point has high emergence in the sense of Kauffman. To examine actual existence of such a blowup point, we evolved a numerical scheme provided with the conservation of mass, decrease of the fee energy, and the positivity of the solution. Next, we noticed that thus equilibrium state shares the same mathematical structure in the problem of self dual gauge field theory concerned with the super-conducting … More or the string theory. In use of the mathematical technique developed in the study of chemotaxis, such as the compensated compactness via the symmetiization or the theory of dual variation, first, we clarified the blowup mechanism in the Abelian-Higgs theory with vortices, in particular, quantization shift and the location of the blowup points. Next, we apply the method to SU (3)Toda system and obtained the saddle type solution. Furthermore, we constructed the periodic solution in Chern-Simons-Higgs theory, in use of the gluing method of Nolasco. From the view point of the formation of the field, we took up the system of tumour growth and obtained the solution provided with the profile expected in mathematics in medicine, globally in tune. To study other mean field theories following the story of nonlinear quantum mechanics, we provided the general theory of dual variation, and applied the study on the Euler-Poisson equation concerning the formation of nebula, and the phase separation model. of Penrose-Fife. To solve many under determined problems, we proposed the method of parallel optimization, developed the technique of clustering, and applied it to the data analysis of magnetoencephalography. We studied the profile of the interface between different media or in the. context of the free boundary problem such as the interface vanishing in the Maxwell system or the Stokes system and the boundary trial method for the problem of filtration. Less

建立了多个自相互作用非平衡平均场的量子化爆破机制。首先，我们将平衡态表示为一个非局部项的非线性特征值问题。然后，在这个水平上观察到上述机制，因此，问题作为非线性量子力学的故事出现。利用二阶矩方法和前向自相似变换，研究了后向自相似变换中的弱解，解决了这一长期存在的问题.此外，我们还观察到第一类爆破点在Kauffman意义下具有高突现性。为了检验这样一个爆破点的实际存在性，我们发展了一个数值方案，该方案提供了质量守恒、能量减少和解的正性。其次，我们注意到，这样的平衡态在与超导有关的自对偶规范场论问题中具有相同的数学结构 ...更多信息或者弦理论利用在趋化性研究中发展起来的数学方法，如通过对称化或对偶变分理论得到的补偿紧性，我们首先阐明了有涡旋的Abelian-Higgs理论中的爆破机制，特别是量子化位移和爆破点的位置。其次，我们将该方法应用于SU（3）户田系统，得到了鞍型解。在此基础上，利用Nolasco的胶合方法构造了Chern-Simons-Higgs理论的周期解.从场的形成的观点来看，我们研究了肿瘤生长的系统，并获得了具有医学数学中预期的轮廓的解决方案，全球一致。为了研究非线性量子力学之后的其他平均场理论，我们给出了对偶变分的一般理论，并将其应用于研究与星云形成有关的Euler-Poisson方程和相分离模型。彭罗斯-法夫为了解决许多不确定性问题，我们提出了并行优化的方法，发展了聚类技术，并将其应用于脑磁图数据分析。我们研究了不同介质之间的界面轮廓，或在不同的介质中。本文介绍了自由边界问题（如麦克斯韦系统或Stokes系统中的界面消失问题）的背景以及渗流问题的边界试探法。少