scholarly journals Uniform Controllability of a Stokes Problem with a Transport Term in the Zero-Diffusion Limit

2020 ◽  
Vol 58 (3) ◽  
pp. 1597-1625
Author(s):  
Jon Asier Bárcena-Petisco
Keyword(s):  
Stokes Problem ◽  
Diffusion Limit ◽  
Zero Diffusion Limit ◽  
Uniform Controllability ◽  
Transport Term

Gravito-inertial waves in a rotating stratified sphere or spherical shell

1999 ◽  
Vol 398 ◽  
pp. 271-297 ◽  
Author(s):  
B. DINTRANS ◽  
M. RIEUTORD ◽  
L. VALDETTARO
Keyword(s):  
Periodic Orbit ◽  
Spherical Shell ◽  
Periodic Orbits ◽  
Point Spectrum ◽  
Boussinesq Approximation ◽  
Diffusion Limit ◽  
Order Partial Differential Equation ◽  
Inertial Waves ◽  
Zero Diffusion Limit ◽  
Hyperbolic Domain

The properties of gravito-inertial waves propagating in a stably stratified rotating spherical shell or sphere are investigated using the Boussinesq approximation. In the perfect fluid limit, these modes obey a second-order partial differential equation of mixed type. Characteristics propagating in the hyperbolic domain are shown to follow three kinds of orbits: quasi-periodic orbits which cover the whole hyperbolic domain; periodic orbits which are strongly attractive; and finally, orbits ending in a wedge formed by one of the boundaries and a turning surface. To these three types of orbits, our calculations show that there correspond three kinds of modes and give support to the following conclusions. First, with quasi-periodic orbits are associated regular modes which exist at the zero-diffusion limit as smooth square-integrable velocity fields associated with a discrete set of eigenvalues, probably dense in some subintervals of [0, N], N being the Brunt–Väisälä frequency. Second, with periodic orbits are associated singular modes which feature a shear layer following the periodic orbit; as the zero-diffusion limit is taken, the eigenfunction becomes singular on a line tracing the periodic orbit and is no longer square-integrable; as a consequence the point spectrum is empty in some subintervals of [0, N]. It is also shown that these internal shear layers contain the two scales E1/3 and E1/4 as pure inertial modes (E is the Ekman number). Finally, modes associated with characteristics trapped by a wedge also disappear at the zero-diffusion limit; eigenfunctions are not square-integrable and the corresponding point spectrum is also empty.

THE VLASOV–POISSON EQUATIONS AS THE SEMICLASSICAL LIMIT OF THE SCHRÖDINGER–POISSON EQUATIONS: A NUMERICAL STUDY

2008 ◽  
Vol 05 (03) ◽  
pp. 569-587 ◽  
Author(s):  
SHI JIN ◽  
XIAOMEI LIAO ◽  
XU YANG
Keyword(s):  
Weak Solution ◽  
Space Dimension ◽  
Numerical Study ◽  
Semiclassical Limit ◽  
Planck Equation ◽  
Diffusion Limit ◽  
Poisson Equations ◽  
Multivalued Solution ◽  
Moment System ◽  
Zero Diffusion Limit

In this paper, we numerically study the semiclassical limit of the Schrödinger–Poisson equations as a selection principle for the weak solution of the Vlasov–Poisson in one space dimension. Our numerical results show that this limit gives the weak solution that agrees with the zero diffusion limit of the Fokker–Planck equation. We also numerically justify the multivalued solution given by a moment system of the Vlasov–Poisson equations as the semiclassical limit of the Schrödinger–Poisson equations.

THE ZERO DIFFUSION LIMIT FOR NONLINEAR HYPERBOLIC SYSTEM WITH DAMPING AND DIFFUSION

2008 ◽  
Vol 05 (04) ◽  
pp. 767-783 ◽  
Author(s):  
KAIMAO CHEN ◽  
CHANGJIANG ZHU
Keyword(s):  
Hyperbolic System ◽  
Existence Of Solutions ◽  
Diffusion Limit ◽  
Linear Diffusion ◽  
Diffusion Waves ◽  
Global Existence Of Solutions ◽  
Nonlinear Hyperbolic System ◽  
The Cauchy Problem ◽  
Zero Diffusion Limit ◽  
And Diffusion

We consider the Cauchy problem for a nonlinear hyperbolic system with damping and diffusion. Thanks to a suitably constructed corrector function, we can eliminate the layer at infinity and by using the energy method we establish the global existence of solutions if the initial data is a small perturbation around the corresponding linear diffusion waves. Furthermore, we study the zero diffusion limit and, precisely, we show that the sequence of solutions converges to the corresponding hyperbolic system as the diffusion parameter tends to zero.

scholarly journals Steady-State Analysis of the Join-the-Shortest-Queue Model in the Halfin–Whitt Regime

2020 ◽  
Vol 45 (3) ◽  
pp. 1069-1103
Author(s):  
Anton Braverman
Keyword(s):  
Steady State ◽  
Diffusion Limit ◽  
Fluid Limit ◽  
Time Intervals ◽  
Dimensional Diffusion ◽  
Queue Model ◽  
Join The Shortest Queue ◽  
Shortest Queue ◽  
Process Level ◽  
General Tool

This paper studies the steady-state properties of the join-the-shortest-queue model in the Halfin–Whitt regime. We focus on the process tracking the number of idle servers and the number of servers with nonempty buffers. Recently, Eschenfeldt and Gamarnik proved that a scaled version of this process converges, over finite time intervals, to a two-dimensional diffusion limit as the number of servers goes to infinity. In this paper, we prove that the diffusion limit is exponentially ergodic and that the diffusion scaled sequence of the steady-state number of idle servers and nonempty buffers is tight. Combined with the process-level convergence proved by Eschenfeldt and Gamarnik, our results imply convergence of steady-state distributions. The methodology used is the generator expansion framework based on Stein’s method, also referred to as the drift-based fluid limit Lyapunov function approach in Stolyar. One technical contribution to the framework is to show how it can be used as a general tool to establish exponential ergodicity.

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