On the Hunter--Saxton system

Generic regularity and Lipschitz metric for the Hunter–Saxton type equations

“…It is also a special case of the Gurevich-Zybin system modelling the dynamics of nondissipative dark matter [16]. Its local well-posedness, global existence and blow-up phenomena were discussed recently in [18]. Moreover, Munsch [19] proved that there exist global dissipative solutions to the two-component Hunter-Saxton system on R. In [15], Nordli established the existence of conservative solutions and the Lipschitz continuous dependence of the solutions with respect to the initial data using similar method as in [6].…”

Analytic Solutions of the Cauchy Problem for the Generalized Two-Component Hunter-Saxton System

“…Furthermore, mathematical properties of this system have been also studied further in many works, for example [10,16,17,27,28,29,30].…”

“…Its local well-posedness, global existence and blow up phenomena were discussed recently in [28]. Lenells-Lechtenfeld [16] showed that it can be interpreted as the Euler equation on the superconformal algebra of contact vector fields on the 1|2-dimensional supercircle.…”

Global Existence for the Generalized Two-Component Hunter–Saxton System

“…L ∞ to get the uniform estimate (5.6). Besides, in contrast with [46,Proposition 6.1], we have shown that in order to have global existence in H 2 × H 1 , one does not need to impose certain smallness assumptions on the initial data, and it only requires that ρ 0 is strictly nonzero (cf. (5.1)).…”

“…This system was first studied in full generality by Wunsch [47], where it was coined the generalized Hunter-Saxton system, since for (α, κ) = (−1, ±1) it becomes the Hunter-Saxton system [46]. The latter is a particular case of the Gurevich-Zybin system pertaining to nonlinear one-dimensional dynamics of dark matter as well as nonlinear ion-acoustic waves (cf.…”