Problems of Hydrodynamic Limits and Related Subjects

1.在经验密度的三阶矩一致有界的假设下，导出了通过有限值域的二体势相互作用且满足超稳定性条件的多维布朗粒子系统的流体动力学极限.在推导过程中，我们还证明了Gibbs统计力学经典理论中的一种维里定理：压力在每个空间维[K]上都表示为大数定律中维里正规化和的极限。内山，经典统计力学中的压力和多维中的相互作用布朗粒子，Annales Henri Poincare（J. Theor.（数学物理）1，1159 -1202（2000）]。虽然这样一个定理已经被证明，如果吉布斯状态的唯一性成立（如在一维或高温的情况下），这是第一次证明没有假设的唯一性。2.本文研究了一类由服从经典运动定律的相互作用粒子系统在标度极限内导出的非局部一维演化方程。作为基本定理，建立了解的存在唯一性、最大值原理、积分解的比较定理以及收敛于Barenblatt解的定理等。在粒子系统中的相互作用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.Uchiyama，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，如果结合以前的结果的标度限制给出了一个清晰的宏观的行为时，相互作用势有一个长的范围内的粒子的微观模型。

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.