Blow-Up Analysis for the Periodic Two-Component μ-Hunter-Saxton System

Global weak solutions for a periodic two-component μ-Hunter-Saxton system

This paper is concerned with global existence of weak solutions for a periodic twocomponent μ-Hunter-Saxton system. We first derive global existence for strong solutions to the system with smooth approximate initial data. Then, we show that the limit of approximate solutions is a global weak solution of the two-component μ-Hunter-Saxton system. 2000 Mathematics Subject Classification: 35G25, 35L05

Lipschitz Metric for the Hunter–saxton Equation

We study stability of solutions of the Cauchy problem for the Hunter–Saxton equation ut + uux = 14 ( R x −∞ u 2 x dx− R∞ x ux dx) with initial data u0. In particular, we derive a new Lipschitz metric dD with the property that for two solutions u and v of the equation we have dD(u(t), v(t)) ≤ edD(u0, v0).

The Hunter-Saxton Equation: A Geometric Approach

The Hunter–Saxton equation is the Euler equation for the geodesic flow on the quotient space of the infinite-dimensional group of orientation preserving diffeomorphisms of the unit circle modulo the subgroup of rigid rotations equipped with a right-invariant metric. We establish several properties of this quotient space: it has constant sectional curvature equal to 1, the Riemannian exponential...

On the Blow-Up of Solutions of a Weakly Dissipative Modified Two-Component Periodic Camassa-Holm System

Convergent difference schemes for the Hunter-Saxton equation

We propose and analyze several finite difference schemes for the Hunter–Saxton equation (HS) ut + uux = 1 2 ∫ x 0 (ux) 2 dx, x > 0, t > 0. This equation has been suggested as a simple model for nematic liquid crystals. We prove that the numerical approximations converge to the unique dissipative solution of (HS), as identified by Zhang and Zheng. A main aspect of the analysis, in addition to th...