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Mathematical Study of the Nonlinear Partial Differential Equations Arising in the Statistical Mechanics

统计力学中非线性偏微分方程的数学研究

基本信息

批准号：
16340047
负责人：
SUZUKI Takashi
金额：
$ 6.67万
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：
2004
资助国家：
日本
起止时间：
2004 至 2007
项目状态：
已结题

项目摘要

In this research project, we provided a unified analysis to the critical phenomena, arising in the solution to the partial differential equation provided with the nonlinearity due to the self-interaction and the non-equilibrium. We formulate the mathematical principle across the hierarchy to control several phenomena common to many nonlinear problems. These problems are the mean field of stationary turbulence in high energy and of gauge field, Ricci flow, nonlinear parabolic equations, self-interacting fluids, material-energy transport, tumor growth, and nonlinear thermodynamics. Among them, we formulated the system of chemotaxis, derived in the context of th the formation of spores of the cellular slime molds, as the fundamental equation of the material transport subject to the mass conservation and the decrease of the free energy in the thermodynamics called Smoluchowski-Poisson equation and clarified the quantized blowup mechanism by developing various new methods of analysis. Then, we obtained the notions of the blowup envelope, formulation of the stationary and non-stationary states by the dual variation, hierarchical control of the stationary states upon the non-stationary states, which motivates the study on the structure and the stability of the set of stationary solutions of phenomenological equations concerning the critical phenomena arising in the non-equilibrium thermodynamics, formation of sub-collapses and the collision of collapses in the mean field equation arising in the gauge theory and turbulent theory and the semilinear parabolic equation with the critical Sobolev exponent, deformed quantization in the nonlinear parabolic equation with non-local term and the normalized Ricci flow, and mass quantization in higher dimensions.

在本研究项目中，我们对偏微分方程解由于自相互作用和非平衡而产生的非线性提供了一个统一的分析。我们制定了跨层次的数学原理，以控制许多非线性问题中常见的几个现象。这些问题包括高能定常湍流和规范场的平均场、Ricci流、非线性抛物方程、自相互作用流体、物质-能量传输、肿瘤生长和非线性热力学。其中，我们根据细胞黏菌孢子的形成推导出趋化性系统，作为物质输运受质量守恒和自由能降低的热力学基本方程，称为Smoluchowski-Poisson方程，并通过发展各种新的分析方法阐明了量子化的膨胀机理。然后，我们得到了爆破包络的概念，定态和非定态的对偶变分公式，定态对非定态的分级控制，从而推动了关于非平衡热力学中的临界现象的唯象方程的定常解的结构和稳定性的研究，规范理论和湍流理论中平均场方程的次坍缩的形成和崩塌的碰撞，具有临界索勃列夫指数的半线性抛物型方程，含非局域项的非线性抛物型方程的形变量子化和归一化Ricci流，以及高维的质量量子化。