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

在本研究项目中，我们对具有自相互作用和非平衡非线性的偏微分方程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中常见的几种现象。这些问题是高能稳定湍流的平均场和规范场、里奇流、非线性抛物方程、自相互作用流体、物质-能量输运、肿瘤生长和非线性热力学。其中，我们将在细胞黏菌孢子形成的背景下导出的趋化性系统，作为热力学中受质量守恒和自由能减少影响的物质输运的基本方程，称为Smoluchowski-Poisson方程，并通过开发各种新的分析方法阐明了量化爆破机理。在此基础上，提出了膨胀包络的概念、用对偶变分法表示稳态和非稳态、定态对非稳态的分层控制等，从而对非平衡热力学中临界现象的现象学方程定解集的结构和稳定性进行了研究。规范理论和湍流理论中的平均场方程和具有临界Sobolev指数的半线性抛物方程中的子坍缩的形成和坍缩的碰撞，非局域项非线性抛物方程和归一化Ricci流中的变形量化，以及高维的质量量子化。

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.