Can spectral gap form in soft potentials?

Duration: 1 hour 6 mins

Share this media item:

Embed this media item:

<iframe width="_width_" height="_height_" src="https://sms.cam.ac.uk/media/3763992/embed" frameborder="0" scrolling="no" allowfullscreen></iframe>

Choose size:

About this item

Available Formats

About this item

Description:	Renjun Duan, Chinese University of Hong Kong 13 January 2022 – 14:30 to 15:30


Created:	2022-01-17 09:25
Collection:	Frontiers in analysis of kinetic equations
Publisher:	Isaac Newton Institute
Copyright:	Renjun Duan
Language:	eng (English)


Abstract:	In the talk I will consider the spectral gap for the linearized Boltzmann or Landau equation with soft potentials. It is known that the corresponding collision operators admit only the degenerated spectral gap. We rather prove the formation of spectral gap in the spatially inhomogeneous setting where the space domain is bounded with an inflow boundary condition. The key strategy is to introduce a new Hilbert space with an exponential weight function that involves the inner product of space and velocity variables and also has the strictly positive upper and lower bounds. The action of the transport operator on such space-velocity dependent weight function induces an extra non-degenerate relaxation dissipation in large velocity that can be employed to compensate the degenerate spectral gap and hence give the exponential decay for solutions in contrast with the sub-exponential decay in either the spatially homogeneous case or the case of torus domain. The result reveals a new insight of hypocoercivity for kinetic equations with soft potentials in the specified situation.

Abstract:

In the talk I will consider the spectral gap for the linearized Boltzmann or Landau equation with soft potentials. It is known that the corresponding collision operators admit only the degenerated spectral gap. We rather prove the formation of spectral gap in the spatially inhomogeneous setting where the space domain is bounded with an inflow boundary condition. The key strategy is to introduce a new Hilbert space with an exponential weight function that involves the inner product of space and velocity variables and also has the strictly positive upper and lower bounds. The action of the transport operator on such space-velocity dependent weight function induces an extra non-degenerate relaxation dissipation in large velocity that can be employed to compensate the degenerate spectral gap and hence give the exponential decay for solutions in contrast with the sub-exponential decay in either the spatially homogeneous case or the case of torus domain. The result reveals a new insight of hypocoercivity for kinetic equations with soft potentials in the specified situation.

Available Formats

Format	Quality	Bitrate	Size
MPEG-4 Video	1408x720	1.35 Mbits/sec	670.98 MB	View	Download
MPEG-4 Video	704x360	371.39 kbits/sec	179.53 MB	View	Download
WebM	1408x720	674.14 kbits/sec	325.88 MB	View	Download
WebM	704x360	227.35 kbits/sec	109.90 MB	View	Download
iPod Video	480x270	480.98 kbits/sec	232.51 MB	View	Download
MP3	44100 Hz	251.08 kbits/sec	121.37 MB	Listen	Download
Auto *	(Allows browser to choose a format it supports)