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，我们对由于自相互作用和非平衡而具有非线性的偏微分方程的解中出现的临界现象进行了统一的分析。我们制定了跨层次的数学原理来控制许多非线性问题常见的几种现象。这些问题是高能稳态湍流和规范场的平均场、里奇流、非线性抛物线方程、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，我们制定了在细胞粘菌孢子形成的背景下推导的趋化系统，作为热力学中受质量守恒和自由能减少影响的物质传输的基本方程，称为Smoluchowski-Poisson 方程并通过开发各种新的分析方法阐明了量子化爆炸机制。然后，我们得到了爆炸包络线的概念，通过对偶变分表示稳态和非稳态状态，稳态对非稳态的层次控制，这激发了对结构和稳定性的研究。唯象方程的一组平稳解，涉及非平衡热力学中出现的临界现象、规范理论和湍流理论中出现的平均场方程中的子塌陷的形成和塌陷的碰撞以及具有临界 Sobolev 指数的半线性抛物型方程，具有非局部项和归一化 Ricci 流的非线性抛物型方程中的变形量化，以及更高维度的质量量化。