scholarly journals Isolated singularities for the n-Liouville equation

2021 ◽  
Vol 60 (4) ◽  
Author(s):  
Pierpaolo Esposito
Keyword(s):  
Liouville Equation ◽  
Total Mass ◽  
Entire Solutions ◽  
Isolated Singularities ◽  
Finite Point ◽  
Logarithmic Type

AbstractIn dimension n isolated singularities—at a finite point or at infinity—for solutions of finite total mass to the n-Liouville equation are of logarithmic type. As a consequence, we simplify the classification argument in Esposito (Anal Non Linéaire 35(3):781–801, 2018) and establish a quantization result for entire solutions of the singular n-Liouville equation.

scholarly journals The behaviour of solutions of the Gaussian curvature equation near an isolated boundary point

2008 ◽  
Vol 145 (3) ◽  
pp. 643-667 ◽  
Author(s):  
DANIELA KRAUS ◽  
OLIVER ROTH
Keyword(s):  
Continuous Function ◽  
Liouville Equation ◽  
Gaussian Curvature ◽  
Boundary Point ◽  
Classical Result ◽  
Riemannian Metrics ◽  
Higher Order ◽  
Isolated Singularities ◽  
Hölder Continuous ◽  
Curvature Equation

AbstractA classical result of Nitsche [22] about the behaviour of the solutions to the Liouville equation Δu= 4e2unear isolated singularities is generalized to solutions of the Gaussian curvature equation Δu= −κ(z)e2uwhere κ is a negative Hölder continuous function. As an application a higher–order version of the Yau–Ahlfors–Schwarz lemma for complete conformal Riemannian metrics is obtained.

Kinetic theory

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Distribution Function ◽  
Kinetic Theory ◽  
Liouville Equation ◽  
Vlasov Equation ◽  
Particle Distribution ◽  
Equations Of Motion ◽  
Poisson Brackets ◽  
Hamiltonian Form ◽  
First Order ◽  
Number Of Particles

This chapter covers the equations governing the evolution of particle distribution and relates the macroscopic thermodynamical quantities to the distribution function. The motion of N particles is governed by 6N equations of motion of first order in time, written in either Hamiltonian form or in terms of Poisson brackets. Thus, as this chapter shows, as the number of particles grows it becomes necessary to resort to a statistical description. The chapter first introduces the Liouville equation, which states the conservation of the probability density, before turning to the Boltzmann–Vlasov equation. Finally, it discusses the Jeans equations, which are the equations obtained by taking various averages over velocities.

scholarly journals Galaxy Formation: Some Comparisons between Theory and Observation

1983 ◽  
Vol 100 ◽  
pp. 391-399 ◽  
Author(s):  
S. Michael Fall
Keyword(s):  
Galaxy Formation ◽  
Total Mass ◽  
Virial Theorem ◽  
Velocity Dispersion ◽  
Rotation Velocity ◽  
Morphological Type ◽  
Fundamental Parameter ◽  
Mass Ratio ◽  
First Approximation ◽  
Type Form

Before theoretical ideas in this subject can be compared with observational data, it is necessary to consider the properties of galaxies that are likely to be relics of their formation. Most astronomers would agree that the list of important parameters should be headed by the total mass M, energy E and angular momentum J. Next on the list should probably be the relative contributions to these quantities from the disc and bulge components of galaxies and denoted D/B for the mass ratio. They can be estimated from the median (i.e. half-mass) radius R, velocity dispersion σ and rotation velocity v of each component, either through the virial theorem or through the luminosity L and an assumed value of M/L. As a first approximation, it is reasonable to suppose that galaxies of a given disc-to-bulge ratio or morphological type form a sequence with mass as the fundamental parameter. The comparison of theory with data is further simplified by considering the extreme cases of ellipticals, with D/B << 1, and late-type spirals, with D/B >> 1. The approach outlined below is to explore the consequences of relaxing in succession the constraints that E, J and M be conserved during the collapse of proto-galaxies. In this article I concentrate on theories that are based on some form of hierarchical clustering because the pancake and related theories are not yet refined enough for a detailed confrontation with observations.

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