Two problems in mathematical physics: Villani's conjecture and trace inequality for the fractional Laplacian.

Fractional laplacian;Villani's conjecture;Entropy production;Kac's model;Trace inequality

Einav, Amit ; Mathematics

University:Georgia Institute of Technology

Department:Mathematics

关键词: Fractional laplacian;    Villani's conjecture;    Entropy production;    Kac's model;    Trace inequality;   

Others  :  https://smartech.gatech.edu/bitstream/1853/42788/1/einav_amit_201112_phd.pdf

美国|英语

来源: SMARTech Repository

【 摘 要 】

The presented work deals with two distinct problems in the field of Mathematical Physics.The first part is dedicated to an 'almost' solution of Villani's conjecture, a knownconjecture related to a Statistical Mechanics model invented by Kac in 1956, giving a rigorous explanation of some simple cases of the Boltzmann equation. In 2003 Villani conjectured that the time it will take the system of particles in Kac's model to equilibrate is proportional to the number of particles in the system. Our main result in this part is a proof, up to an epsilon, of that conjecture, showing that for all practical purposes we can consider it to be true.The second part of the presentation is based on a joint work with Prof. Michael Loss and is dedicated to a newly developed trace inequality for the fractional Laplacian, connecting between the fractional Laplacian of a function and its restriction to intersection of hyperplanes. The newly found inequality is sharp and the functions that attain equality in it are completely classified.