Well-Posed Two-Temperature Constitutive Equations for Stable Dense Fluid Shockwaves using Molecular Dynamics and Generalizations of Navier-Stokes-Fourier Continuum Mechanics Wm. G. Hoover and Carol G. Hoover Ruby Valley Research Institute Highway Contract 60, Box 598, Ruby Valley 89833, NV USA (Dated: September 12, 2018) Abstract Guided by molecular dynamics simulations, we generalize the Navier-Stokes-Fourier constitu- tive equations and the continuum motion equations to include both transverse and longitudinal temperatures. To do so we partition the contributions of the heat transfer, the work done, and the heat flux vector between the longitudinal and transverse temperatures. With shockwave boundary conditions time-dependent solutions of these equations converge to give stationary shockwave profiles. The profiles include anisotropic temperature and can be fitted to molecular dynamics results, demonstrating the utility and simplicity of a two-temperature description of far-from-equilibrium states. PACS numbers: Keywords: Temperature, Shockwaves, Molecular Dynamics, Computational Methods arXiv:1001.1015v3 [cond-mat.stat-mech] 8 Mar 2010