|
|||||
 |
|
|
|
|
|
|
|
Convergence rate in
Wasserstein distance and semiclassical limit for the
defocusing logarithmic Schrödinger equation
Guillaume Ferriere |
|
Vol. 14 (2021), No. 2, 617–666
|
Abstract |
|
We consider the dispersive logarithmic Schrödinger equation in a semiclassical scaling. We extend the results of Carles and Gallagher (Duke Math. J. 167:9 (2018), 1761–1801) about the large-time behavior of the solution (dispersion faster than usual with an additional logarithmic factor and convergence of the rescaled modulus of the solution to a universal Gaussian profile) to the case with semiclassical constant. We also provide a sharp convergence rate to the Gaussian profile in the Kantorovich–Rubinstein metric through a detailed analysis of the Fokker–Planck equation satisfied by this modulus. Moreover, we perform the semiclassical limit of this equation thanks to the Wigner transform in order to get a (Wigner) measure. We show that those two features are compatible and the density of a Wigner measure has the same large-time behavior as the modulus of the solution of the logarithmic Schrödinger equation. Lastly, we discuss about the related kinetic equation (which is the kinetic isothermal Euler system) and its formal properties, enlightened by the previous results and a new class of explicit solutions. |
PDF Access Denied
We have not been able to recognize your IP address
3.14.141.228
as that of a subscriber to this journal.
Online access to the content of recent issues is by
subscription, or purchase of single articles.
Please contact your institution's librarian suggesting a subscription, for example by using our journal-recommendation form. Or, visit our subscription page for instructions on purchasing a subscription.
You may also contact us at
contact@msp.org
or by using our
contact form.
Or, you may purchase this single article for USD 40.00:
Keywords
logarithmic Schrödinger equation, large-time behavior,
semiclassical limit, Wigner measures, Wasserstein
distances, harmonic Fokker–Planck operator, kinetic
isothermal Euler system
|
Mathematical Subject Classification 2010
Primary: 35Q55, 35Q83, 81Q20
Secondary: 35B25, 35B35, 35B40, 35Q35, 35Q84
|
Milestones
Received: 21 March 2019
Accepted: 25 October 2019
Published: 20 March 2021
|
Authors |
| Guillaume Ferriere | |
| Institut Montpelliérain Alexander
Grothendieck Université Montpellier CNRS Montpellier France |
|
