Related papers: A Lipschitz metric for the Hunter-Saxton equation
We analyze stability of conservative solutions of the Cauchy problem on the line for the Camassa--Holm (CH) equation. Generically, the solutions of the CH equation develop singularities with steep gradients while preserving continuity of…
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.
We study stability of solutions of the Cauchy problem for the Hunter-Saxton equation $u_t+uu_x=\frac14(\int_{-\infty}^xu_x^2 dx-\int_{x}^\infty u_x^2 dx)$ with initial data $u_0$. In particular, we derive a new Lipschitz metric $d_\D$ with…
We explore the Lipschitz stability of solutions to the Hunter-Saxton equation with respect to the initial data. In particular, we study the stability of $ \alpha $-dissipative solutions constructed using a generalised method of…
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…
We study stability of conservative solutions of the Cauchy problem for the periodic Camassa-Holm equation $u_t-u_{xxt}+\kappa u_x+3uu_x-2u_xu_{xx}-uu_{xxx}=0$ with initial data $u_0$. In particular, we derive a new Lipschitz metric $d_\D$…
In this paper, we consider the Cauchy problem for the Hunter-Saxton (HS) equation on the line. Firstly, we establish the local well-posedness for the integral form of the (HS) equation by constructing some special spaces $E^s_{p,r}$, which…
We construct a Lipschitz metric for conservative solutions of the Cauchy problem on the line for the two-component Camassa--Holm system $u_t-u_{txx}+3uu_x-2u_xu_{xx}-uu_{xxx}+\rho\rho_x=0$, and $\rho_t+(u\rho)_x=0$ with given initial data…
We study stability of solutions of the Cauchy problem on the line for the Camassa-Holm equation $u_t-u_{xxt}+3uu_x-2u_xu_{xx}-uu_{xxx}=0$ with initial data $u_0$. In particular, we derive a new Lipschitz metric $d_\D$ with the property that…
In this paper we characterize the regularity structure, as well as show the global-in-time existence and uniqueness, of (energy) conservative solutions to the Hunter-Saxton equation by using the method of characteristics. The major…
We establish the concept of $\alpha$-dissipative solutions for the two-component Hunter-Saxton system under the assumption that either $\alpha(x)=1$ or $0\leq \alpha(x)<1$ for all $x\in \mathbb{R}$. Furthermore, we investigate the Lipschitz…
The modified Hunter--Saxton equation models the propagation of short capillary-gravity waves. As it involves a mixed derivative, its initial value problem on the periodic domain is much more complicated than the standard evolutionary…
We prove Hawking's singularity theorem for spacetime metrics of local Lipschitz regularity. The proof rests on (1) new estimates for the Ricci curvature of regularising smooth metrics that are based upon a quite general Friedrichs-type…
We consider finite element approximations of unique continuation problems subject to elliptic equations in the case where the normal derivative of the exact solution is known to reside in some finite dimensional space. To give quantitative…
We show that the Hunter-Saxton equation $u_t+uu_x=\frac14\big(\int_{-\infty}^x d\mu(t,z)- \int^{\infty}_x d\mu(t,z)\big)$ and $\mu_t+(u\mu)_x=0$ has a unique, global, weak, and conservative solution $(u,\mu)$ of the Cauchy problem on the…
The Cauchy problem for a multidimensional linear transport equation with discontinuous coefficient is investigated. Provided the coefficient satisfies a one-sided Lipschitz condition, existence, uniqueness and weak stability of solutions…
This paper establishes Lipschitz stability for the simultaneous recovery of a variable density coefficient and the initial displacement in a damped biharmonic wave equation. The data consist of the boundary Cauchy data for the Laplacian of…
The goal of this paper is to prove a comparison principle for viscosity solutions of semilinear Hamilton-Jacobi equations in the space of probability measures. The method involves leveraging differentiability properties of the…
We derive a robust error estimate for a recently proposed numerical method for $\alpha$-dissipative solutions of the Hunter-Saxton equation, where $\alpha \in [0, 1]$. In particular, if the following two conditions hold: i) there exist a…
In this paper we study the large time asymptotic behavior of (energy) conservative solutions to the Hunter-Saxton equation in a generalized framework that consists of the evolutions of solution and its energy measure. We describe the large…