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...

The notion of a scalar equation describing pseudo-spherical surfaces is reviewed. It is shown that if an equation admits this structure, the existence of conservation laws, symmetries, and quadratic pseudo-potentials, can be studied by geometrical means. As an application, it is pointed out that the important Camassa–Holm and Hunter–Saxton equations possess features considered to be characteris...

We show that two families of equations, the generalized inviscid Proudman–Johnson equation and r-Hunter–Saxton (recently introduced by Cotter et al.), coincide for a certain range parameters. This gives new geometric interpretation these equations as geodesic right invariant homogeneous $$W^{1,r}$$ -Finsler metrics on diffeomorphism group. Generalizing construction Lenells Hunter–Saxton equatio...

In this paper, we study the periodic Hunter–Saxton equation with weak dissipation. We first establish local existence of strong solutions, blow-up scenario and criteria equation. Then, investigate rate for blowing-up solutions to Finally, prove that has global solutions.

In this paper we characterize the regularity structure, as well show global-in-time existence and uniqueness, of (energy) conservative solutions to Hunter-Saxton equation by using method characteristics. The major difference between current work previous results is that are able singularities energy measure their nature in a very precise manner. particular, singularities, whose temporal spatial...