Existence and Lipschitz stability for α-dissipative solutions of the two-component Hunter–Saxton system

Abstract

We establish the concept of α-dissipative solutions for the twocomponent Hunter-Saxton system under the assumption that either α(x) = 1 or 0 ≤ α(x) < 1 for all x ∈ R. Furthermore, we investigate the Lipschitz stability of solutions with respect to time by introducing a suitable parametrized family of metrics in Lagrangian coordinates. This is necessary due to the fact that the solution space is not invariant with respect to time.which has been introduced by Hunter and Saxton as a model of the director field of a nematic liquid crystal [14].Solutions of the 2HS system develop singularities in finite time, even for smooth initial data [16,18]. The appearance of singularities, known as wave breaking, means that u x tends pointwise to −∞ while u remains bounded and continuous. The phenomenon is illustrated in the following example.

Numerical conservative solutions of the Hunter–Saxton equation

“…In this section we prove that there exists a convergent subsequence of (u Δx , F Δx ), and that the limit is a conservative weak solution of (1.2), which satisfies condition (1.4). First we rigorously define, as in [2,9,19], conservative weak solutions.…”

“…1. One can check that the function u remains uniformly bounded and uniformly Hölder continuous with exponent 1 2 on [0, 2] × R. It is possible to extend weak solutions past wave breaking in various ways, see [1][2][3]9,19]. One could ignore the part of the solution that blows up.…”

“…This new system is a reformulation of the so-called two-component Hunter-Saxton (2HS) system, which not only generalizes the HS equation, but can also be studied using the same methods and ideas, see [9,19]. Moreover, every conservative solution to the HS equation can be approximated by smooth solutions of the 2HS system.…”

“…Furthermore, it turns out that the solution u of the HS equation may develop singularities in finite time in the following sense: Unless the initial data is monotone increasing, we find (1.2) inf(u x ) → −∞ as t ↑ t * = 2/ sup(−u ′ 0 ). Past wave breaking there are at least two different classes of solutions, denoted conservative (energy is conserved) and dissipative (where energy is removed locally) solutions, respectively, and this dichotomy is the source of the interesting behavior of solutions of the equation (but see also [8,10]). We will in this paper consider the so-called conservative case where the associated energy is preserved.…”

Lipschitz Stability for the Hunter-Saxton Equation

“…This is determined by how one manipulates the energy past wave breaking. In general, one has the freedom to take as much energy away as one pleases [11]. Two important cases are well studied.…”