On hydrodynamic equations and their relation to kinetic theory and statistical mechanics

Abstract

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] We establish conditions on the Hamiltonian evolution of interacting molecules that imply hydrodynamic equations at the limit of infinitely many molecules and show that these conditions are satisfied whenever the solutions of the classical equations for N interacting molecules obey uniform in N bounds. We show that this holds when the initial conditions are bounded and the molecule interaction is weak enough at the initial time. We then obtain hydrodynamic equations that coincide with Maxwell's. We then construct explicit examples of spontaneous energy generation and nonuniqueness for the standard compressible Euler system, with and without pressure, agagin by taking limits of Hamiltonian dynamics as the number of molecules increases to infinity. The examples come from rescaling of well-posed, deterministic systems of molecules that either collide elastically or interact via singular pair potentials. We also obtain Percus macroscopic equation as the limit of a sequence of single systems of N hard rods with the number of hard rods going to infinity. Finally, we establish the strict convexity of the pressure as a thermodynamic limit for continuous systems. As a result we show the existence of a local bijection between macroscopic density, velocity, and energy on one hand and thermodynamic parameters on the other, for continuous systems.

Geotechnical Engineering Control on the Scenic Area — Research on the Method for Geological Disasters-Rockfall, Landslide and Debris Flow in Laoshan District of Qingdao

“…The stronger slope scour, the more loose materials is contained, the steeper the slope, the more prone to landslides. Because there is no river basin of measured data, the surface water flow according to the Chinese science which highway, the proposed method of experience formula [16].…”

“…(2)Slope surface displacement Because there is no river basin of measured data, the Surface water exchange flow is Calculated by the China highway science institute's Empirical formula(2) [16].…”