Maximal dissipation in Hunter-Saxton equation for bounded energy initial data

TL;DR

This paper proves that dissipative solutions of the Hunter-Saxton equation dissipate energy at the maximal rate for all bounded energy initial data, confirming a longstanding conjecture and establishing energy comparison results.

Contribution

It generalizes previous results by proving the maximal dissipation conjecture for all bounded energy initial data in the Hunter-Saxton equation.

Findings

Dissipative solutions dissipate energy at the maximal rate.

Energy of dissipative solutions is not greater than that of any weak solution with same initial data.

The proof applies to all bounded energy initial data, not just monotone cases.

Abstract

In [Zhang and Zheng, 2005] 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 [Dafermos, 2012] by Dafermos for monotone increasing initial data with bounded energy. In this note we prove the conjecture in [Zhang and Zheng, 2005] 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.