Lipschitz metric for the Hunter–Saxton equation
Alberto Bressan, Helge Holden and Xavier Raynaud
Abstract: We study stability of solutions of the Cauchy problem for the Hunter–Saxton equation
ut+uux=¼(∫(−∞,x) ux2dx−∫(x,∞) ux2dx)
with initial data u0. In particular, we derive a new Lipschitz metric dD with the property that for two solutions u and v of the equation we have dD(u(t),v(t))≤eCtdD(u0,v0).