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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.
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.
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
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
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 establish the existence of conservative solutions of the initial value problem of the two-component Hunter--Saxton system on the line. Furthermore we investigate the stability of these solutions by constructing a Lipschitz metric.