A Lipschitz metric for the Hunter–Saxton equation

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).

Lipschitz Metric for the Periodic Camassa–holm Equation

We study stability of conservative solutions of the Cauchy problem for the periodic Camassa–Holm equation ut−uxxt +3uux−2uxuxx−uuxxx = 0 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 relationship between this metric and usual norms in H1 per and L ∞ per is clarif...

Bi-Lipschitz Decomposition of Lipschitz functions into a Metric space

We prove a quantitative version of the following statement. Given a Lipschitz function f from the k-dimensional unit cube into a general metric space, one can be decomposed f into a finite number of BiLipschitz functions f |Fi so that the k-Hausdorff content of f([0, 1] r ∪Fi) is small. We thus generalize a theorem of P. Jones [Jon88] from the setting of R to the setting of a general metric spa...

Metric Embeddings and Lipschitz Extensions

The Near-Ring of Lipschitz Functions on a Metric Space

This paper treats near-rings of zero-preserving Lipschitz functions on metric spaces that are also abelian groups, using pointwise addition of functions as addition and composition of functions as multiplication. We identify a condition on the metric ensuring that the set of all such Lipschitz functions is a near-ring, and we investigate the complications that arise from the lack of left distri...