Existence, uniqueness and long-time behavior of the $\lambda$-dissipative solutions to the two-component Hunter-Saxton system

Abstract

In this paper, we construct the explicit characteristics for the $\lambda$-dissipative solutions ($\lambda \in [0,1]$) to the two-component Hunter--Saxton (2HS) system. Using these characteristics, we provide a comprehensive study of the existence, uniqueness, and asymptotic behavior of these solutions. For the fully dissipative case ($\lambda =1$), uniqueness follows from the absence of outgoing cusps. In the partially dissipative regime ($0<\lambda <1$), outgoing cusps are present. We formulate an exact Eulerian dissipation rule which identifies the energy that has already passed through wave breaking by the intrinsic condition $\rho =0$ and $u_x\ge 2/t$. This rule determines the dissipated part of the energy measure and yields uniqueness. This uniqueness applies to the classical Hunter-Saxton equation and seems to be the first uniqueness result for the general $\lambda$-dissipative solutions. Concerning the large-time dynamics, we show that the density $\rho$ and the singular part of the energy measure decay to zero as $t\to \infty$, indicating that all energy is eventually concentrated in the $u$-component. Moreover, we derive the leading-order asymptotic term, which takes the form of a kink-wave determined by the system's remaining energy. This kink-wave can be explicitly computed from the initial data and the dissipation parameter $\lambda$.