Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6425330 | Advances in Mathematics | 2016 | 24 Pages |
Abstract
In [13] it was conjectured by Zhang and Zheng that dissipative solutions of the Hunter-Saxton equation, which are known to be unique in the class of weak solutions, dissipate the energy at the highest possible rate. The conjecture of Zhang and Zheng was proven in [4] by Dafermos for monotone increasing initial data with bounded energy. In this note we prove the conjecture in [13] in full generality. To this end we examine the evolution of the energy of any weak solution of the Hunter-Saxton equation. Our proof shows in fact that for every time t>0 the energy of the dissipative solution is not greater than the energy of any weak solution with the same initial data.
Related Topics
Physical Sciences and Engineering
Mathematics
Mathematics (General)
Authors
Tomasz CieÅlak, Grzegorz Jamróz,