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Register a new userGeneralized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms
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We study an equation lying `mid-way between the periodic Hunter-Saxton and Camassa-Holm equations, and which describes evolution of rotators in liquid crystals with external magnetic field and self-interaction. We prove that it is an Euler equation on the diffeomorphism group of the circle corresponding to a natural right-invariant Sobolev metric. We show that the equation is bihamiltonian and admits both cusped, as well as smooth, traveling-wave solutions which are natural candidates for solitons. We also prove that it is locally well-posed and establish results on the lifespan of its solutions. Throughout the paper we argue that despite similarities to the KdV, CH and HS equations, the new equation manifests several distinctive features that set it apart from the other three.
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In this paper we develop an existence theory for the Cauchy problem to the stochastic Hunter-Saxton equatio, and prove several properties of the blow-up of its solutions. An important part of the paper is the continuation of solutions to the stochastic equations beyond blow-up (wave-breaking). In the linear noise case, using the method of (stochastic) characteristics, we also study random wave-breaking and stochastic effects unobserved in the deterministic problem. Notably, we derive an explicit law for the random wave-breaking time.
In the article a convergent numerical method for conservative solutions of the Hunter--Saxton equation is derived. The method is based on piecewise linear projections, followed by evolution along characteristics where the time step is chosen in order to prevent wave breaking. Convergence is obtained when the time step is proportional to the square root of the spatial step size, which is a milder restriction than the common CFL condition for conservation laws.
We study the Lipschitz stability in time for $alpha$-dissipative solutions to the Hunter-Saxton equation, where $alpha in [0,1]$ is a constant. We define metrics in both Lagrangian and Eulerian coordinates, and establish Lipschitz stability for those metrics.
We show that the Hunter-Saxton equation $u_t+uu_x=frac14big(int_{-infty}^x dmu(t,z)- int^{infty}_x dmu(t,z)big)$ and $mu_t+(umu)_x=0$ has a unique, global, weak, and conservative solution $(u,mu)$ of the Cauchy problem on the line.
We analyze stability of conservative solutions of the Cauchy problem on the line for the (integrated) Hunter-Saxton (HS) equation. Generically, the solutions of the HS equation develop singularities with steep gradients while preserving continuity of the solution itself. In order to obtain uniqueness, one is required to augment the equation itself by a measure that represents the associated energy, and the breakdown of the solution is associated with a complicated interplay where the measure becomes singular. The main result in this paper is the construction of a Lipschitz metric that compares two solutions of the HS equation with the respective initial data. The Lipschitz metric is based on the use of the Wasserstein metric.
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