Problems of Hydrodynamic Limits and Related Subjects

1. 在经验密度的第三矩均匀有界的假设下，导出了满足超稳定条件的两体势相互作用的多维布朗粒子系统的水动力极限。对于推导，我们还展示了经典吉布斯统计力学理论中的一种维里定理：在每个空间维度上，压力被表示为大数定律中维里数的归一化和的极限[K]。内山，经典统计力学中的压力与多维布朗粒子的相互作用，庞加莱年鉴（J. thetheory）。数学。phy)。1159 - 1202(2000)]。尽管如果吉布斯态的唯一性成立（如在一维或高温的情况下），这样的定理已经被证明了，但这是第一次在不假设唯一性的情况下证明。研究了一类非局域的一维演化方程，该方程是由服从经典运动定律的相互作用粒子系统在标度极限下导出的。作为基本定理，建立了解的存在唯一性、极大原理、积分解的比较定理以及对“Barenblatt”解的收敛性等。在粒子系统中相互作用的力由对数势给出的特殊情况下，得到了解的相当精细的性质。(K。一类积分-微分方程初值问题解的行为，非线性分析（2001）。这些结果如果与先前关于尺度极限的结果相结合，就可以对相互作用势具有较长范围时潜在粒子微观模型的行为有一个清晰的宏观看法。

1. The hydrodynamic limit for a system of multi-dimensional Brownian particles which interact one another through a two-body potential having a finite range and satisfying the super-stability condition is derived under the hypothesis that the third moments of the empirical densities are uniformly bounded. For the derivation we also show a kind of virial theorem in the classical theory of Gibbs statistical mechanics: the pressure is represented as a limit of normalized sums of the virials in the law of large numbers in every space dimension [K. Uchiyama, Pressure in classical statistical mechanics and interacting Brownian particles in multi-dimensions, Annales Henri Poincare (J. Theor. Math. Phys.) 1,1159-1202 (2000)]. Although such a theorem has already been proved if the uniqueness for Gibbs states holds (as in the cases of one-dimension or of high temperature ), this is for the first time of the proof without assuming the uniqueness.2. A class of one-dimensional evolution equations which are nonlocal and derived in the scaling limits from systems of interacting particles obeying the classical law of motion is studied. As fundamental theorems there are established the existence and the uniqueness of solutions, the maximum principle, a comparison theorem for integrated solutions, and the convergences to the 'Barenblatt' solutions, etc. In the particular case when the interacting force in the particle system is given by a logarithmic potential, quite fine properties of the solutions are obtained. [K.Uchiyama, Behavior of solutions to the initial value problem for a class of integro-differential equations, Nonlinear Analysis (2001).] These results if combined with former results on the scaling limits give a clear macroscopic view about the behavior of the underlying microscopic model of particles when the interaction potential has a long range.