3,121 research outputs found
Thermodynamics in the universe described by the emergence of the space and the energy balance relation
It has previously been shown that it is more general to describe the
evolution of the universe based on the emergence of the space and the energy
balance relation. Here we investigate the thermodynamic properties of the
universe described by such a model. We show that the first law of
thermodynamics and the generalized second law of thermodynamics (GSLT) are both
satisfied and the weak energy condition are also fulfilled for two typical
examples. Finally we examine the physical consistency for the present model.Comment: 9 pages, 2 figure
Steady Euler Flows with Large Vorticity and Characteristic Discontinuities in Arbitrary Infinitely Long Nozzles
We establish the existence and uniqueness of smooth solutions with large
vorticity and weak solutions with vortex sheets/entropy waves for the steady
Euler equations for both compressible and incompressible fluids in arbitrary
infinitely long nozzles. We first develop a new approach to establish the
existence of smooth solutions without assumptions on the sign of the second
derivatives of the horizontal velocity, or the Bernoulli and entropy functions,
at the inlet for the smooth case. Then the existence for the smooth case can be
applied to construct approximate solutions to establish the existence of weak
solutions with vortex sheets/entropy waves by nonlinear arguments. This is the
first result on the global existence of solutions of the multidimensional
steady compressible full Euler equations with free boundaries, which are not
necessarily small perturbations of piecewise constant background solutions. The
subsonic-sonic limit of the solutions is also shown. Finally, through the
incompressible limit, we establish the existence and uniqueness of
incompressible Euler flows in arbitrary infinitely long nozzles for both the
smooth solutions with large vorticity and the weak solutions with vortex
sheets. The methods and techniques developed here will be useful for solving
other problems involving similar difficulties.Comment: 43 pages; 2 figures; To be published in Advances in Mathematics
(2019
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