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

我们建立了多自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具有非局部TERN的非线性本征值问题。然后，上面的机制在这个水平上被观察到，因此，问题作为一个非线性量子力学的故事而出现。我们通过研究后向自相似变换产生的弱解，利用二阶矩和前向自相似变换的方法解决了这一长期存在的问题。此外，我们还观察到I型爆破点在Kauffman意义下具有很高的涌现性。为了检验这种爆破点的实际存在，我们发展了一个具有质量守恒、费用能量降低和解的正性的数值格式。接下来，我们注意到平衡态与超导…的自对偶规范场论问题具有相同的数学结构莫尔或者弦理论。利用化学趋化性研究中发展起来的数学手段，如对称化补偿紧性或对偶变分理论，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。接下来，我们将该方法应用于SU(3)Toda系统，得到了鞍型解。此外，我们还利用Nolasco的粘合方法构造了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系统，并获得了与医学数学中预期的轮廓相一致的解，全局一致。为了研究其他紧随非线性量子力学故事的平均场理论，我们提供了对偶变分的一般理论，并应用于关于星云形成的欧拉-泊松方程和相分离模型的研究。彭罗斯-法夫的。为了解决许多欠确定问题，我们提出了并行优化方法，发展了聚类技术，并将其应用到脑磁图的数据分析中。我们研究了不同介质之间或中的界面轮廓。麦克斯韦系统或斯托克斯系统中的界面消失等自由边界问题的背景和过滤问题的边界试探法。较少

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