Global Weak Solutions to a Generalized Hyperelastic-rod Wave Equation

Abstract

Abstract. We consider a generalized hyperelastic-rod wave equation (or generalized CamassaHolm equation) describing nonlinear dispersive waves in compressible hyperelastic rods. We establish existence of a strongly continuous semigroup of global weak solutions for any initial data from H 1 (R). We also present a "weak equals strong"uniqueness result.

H1-perturbations of Smooth Solutions for a Weakly Dissipative Hyperelastic-rod Wave Equation

“…It is clear that the analysis in [3] does not cover this kind of solutions (that do not belong to L ∞ (R + ; H 1 (R))!). In this paper we extend the result of [3] to cover also (1.2). Roughly speaking the idea is to look at (1.2) as a L ∞ (R + ; H 1 (R))−perturbation of a constant state.…”

“…The Camassa-Holm equation possesses a bi-Hamiltonian structure (and thus an infinite number of conservation laws) [11,1] and is completely integrable [1]. From a mathematical point of view the Camassa-Holm equation is well studied, see [3] for a complete list of references. In particular, we recall that existence and uniqueness results for global weak solutions have been proved by Constantin and Escher [4], Constantin and Molinet [5], and Xin and Zhang [17,18], see also Danchin [9,10].…”

“…In [3] the authors consider the case δ = 0 and prove the global existence and wellposedness of solutions belonging to L ∞ (R + ; H 1 (R)). On the other hand in [8] it is showed that for δ = 0 and any constants 0 < γ < 3, c > 0 there exists a ζ ∈ R such that the following peakon like function is a traveling wave solution of (1.1)…”

An Optimal Transportation Metric for Solutions of the Camassa-Holm Equation

“…On one hand, one can add a small diffusion term in the right hand side of (1.1), and recover solutions of the original equations as a vanishing viscosity limit [CHK1,CHK2]. An alternative technique, developed in [BC2], relies on a new set of independent and dependent variables, specifically designed with the aim of "resolving" all singularities.…”

“…The vanishing viscosity approach in [CHK1,CHK2] singles out the dissipative solutions. These can also be characterized by the Oleinik type estimate…”