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

Abstract

Abstract. The Hunter-Saxton equation determines a flow of conservative solutions taking values in the space H 1 (R + ). However, the solution typically includes finite time gradient blowups, which make the solution flow not continuous w.r.t. the natural H 1 distance. The aim of this paper is to first study the generic properties of conservative solutions of some initial boundary value problems to the Hunter-Saxton type equations. Then using these properties, we give a new way to construct a Finsler type metric which renders the flow uniformly Lipschitz continuous on bounded subsets of H 1 (R + ).

Lipschitz metric for conservative solutions of the two-component Camassa–Holm system

“…Thus, by virtue of Proposition 3.1, we report directly the proof of Theorem 1.2. Similar as in [4], we have Proposition 4.1. For any (u, ρ), (û,ρ) ∈ H 1 (R) × L 2 (R), there exists some positive constant C depends only on E 1 , such that,…”

“…The method in this paper is very robust. Similar metrics were established for other models whose solutions include cusp singularities, such as the variational wave equation and Novikov equation, [2,3,4]. …”

“…By the smooth coefficients of (3.2)-(3.4), we have that the first order perturbations satisfy a linearized system and are well defined for (t, ξ) ∈ R + × R. Similar to the argument of [3,4], we express the quantities w, v, σ appearing in (2.7) in terms of (X, U, L, A, B, Q). More specifically, we see that (1) The shift in x is computed by…”