A Lipschitz metric for conservative solutions of the two-component Hunter–Saxton system

Abstract

We establish the existence of conservative solutions of the initial value problem of the two-component Hunter-Saxton system on the line. Furthermore we investigate the stability of these solutions by constructing a Lipschitz metric.

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

“…Now, we state the existence results of conservative solutions to the two-component HunterSaxton system (3.1)-(3.3), c.f. [15]. Theorem 3.1.…”

“…15) where f ϑ 1 , f ϑ 1 , f ϑ 3 are the perturbations of the right hand side of (2.7) 1 , (2.7) 2 and (2.7) 3 . In light of [10] …”

“…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].…”

“…These results give us a thorough understand on the properties of generic singularities, which are among the most physically relevant singularities, for systems (1.1) and (1.2). As stated before, some geodesic Lipschitz metrics for conservative solutions, using variables under new coordinates instead of (t, x) coordinate, were established in [5,15]. One reason why we still wish to construct the Lipschitz metric through our method, under the help of the generic regularity results, is because our method which works mainly on the original (t, x)-coordinates can provide readers a new intuitive way to understand the construction of metrics through other ways.…”

“…Especially, we expect that the construction of the metrics in this paper, especially the metrics for the smooth cases (in Subsections 2.3.1 and 3.3.1), can be easily understood even by readers in a broader field. In this paper, we study the initial boundary value problems, instead of the initial value problems considered by [5,15].…”