Fokker-Planck models for Bose-Einstein particles

Duration: 57 mins 29 secs

Share this media item:

Embed this media item:

<iframe width="_width_" height="_height_" src="https://sms.cam.ac.uk/media/1065915/embed" frameborder="0" scrolling="no" allowfullscreen></iframe>

Choose size:

About this item

Available Formats

About this item

Description:	Toscani, G (Pavia) Thursday 09 September 2010, 16:30-17:30

Created:

2010-09-14 16:14

Collection:

Partial Differential Equations in Kinetic Theories
Rothschild Seminars

Publisher:

Isaac Newton Institute

Toscani, G

Language:

eng (English)

Credits:

Author:	Toscani, G
Producer:	Steve Greenham


Abstract:	We study nonnegative, measure-valued solutions of the initial value problem for one-dimensional drift-diffusion equations where the linear drift has a driving potential with a quadratic growth at infinity, and the nonlinear diffusion is governed by an increasing continuous and bounded function. The initial value problem is studied in correspondence to initial densities that belong to the space of nonnegative Borel measures with finite mass and finite quadratic momentum and it is the gradient flow of a suitable entropy functional with respect to the Wasserstein distance. Due to the degeneracy of diffusion for large densities, concentration of masses can occur, whose support is transported by the drift. We shall show that the large-time behavior of solutions depends on a critical mass which can be explicitly characterized in terms of the diffusion function and of the drift term. If the initial mass is less than the critical mass, the entropy has a unique minimizer which is absolutely continuous with respect to the Lebesgue measure. Conversely, when the total mass of the solutions is greater than the critical one, the steady state has a singular part in which the exceeding mass is accumulated.

Abstract:

We study nonnegative, measure-valued solutions of the initial value problem for one-dimensional drift-diffusion equations where the linear drift has a driving potential with a quadratic growth at infinity, and the nonlinear diffusion is governed by an increasing continuous and bounded function. The initial value problem is studied in correspondence to initial densities that belong to the space of nonnegative Borel measures with finite mass and finite quadratic momentum and it is the gradient flow of a suitable entropy functional with respect to the Wasserstein distance. Due to the degeneracy of diffusion for large densities, concentration of masses can occur, whose support is transported by the drift. We shall show that the large-time behavior of solutions depends on a critical mass which can be explicitly characterized in terms of the diffusion function and of the drift term. If the initial mass is less than the critical mass, the entropy has a unique minimizer which is absolutely continuous with respect to the Lebesgue measure. Conversely, when the total mass of the solutions is greater than the critical one, the steady state has a singular part in which the exceeding mass is accumulated.

Available Formats

Format	Quality	Bitrate	Size
MPEG-4 Video	640x360	1.84 Mbits/sec	797.01 MB	View	Download
WebM	480x360	742.38 kbits/sec	312.56 MB	View	Download
Flash Video	484x272	568.75 kbits/sec	239.74 MB	View	Download
iPod Video	480x270	506.27 kbits/sec	213.40 MB	View	Download
MP3	44100 Hz	125.02 kbits/sec	52.50 MB	Listen	Download
Auto *	(Allows browser to choose a format it supports)