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

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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